Search result: Catalogue data in Spring Semester 2023

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Abstract

This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).

Learning objective

Learning Algebraic Geometry.

Literature

Primary References:
* John Ottem, Geir Ellingsrud: Introduction to algebraic varieties,  https://www.uio.no/studier/emner/matnat/math/MAT4210/data/mastermat4210.pdf
* James Milne: Algebraic Geometry,
http://www.jmilne.org/math/CourseNotes/AG.pdf 

Secondary References:
* Miles Reid: Undergraduate Algebraic Geometry, Cambridge University Press.
* Ravi Vakil: Foundations of Algebraic Geometry,
http://math.stanford.edu/~vakil/216blog/
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer. 

Other textbooks:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* Igor Shafarevich: Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.

Prerequisites / Notice

Requirement: Basic knowledge of Commutative Algebra.

401-8142-21L

Algebraic Geometry II (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT517

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

9 credits

University lecturers

Abstract

We continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc.

Learning objective

Prerequisites / Notice

MAT507 Algebraic Geometry = 401-8141-00L Algebraic Geometry (University of Zurich) offered in the Autumn Semester 2022.
This is also a good follow-up course unit for ETH students who visited Maria Yakerson's lecture Algebraic Geometry in Spring 2022.

401-3226-00L

Symmetric Spaces

8 credits

A. Iozzi

Abstract

* Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples
* Symmetric spaces of non-compact type: flats and rank, roots and root spaces
* Iwasawa decomposition, Weyl group, Cartan decomposition
* Geometry at infinity

Learning objective

Learn the basics of symmetric spaces

401-3532-08L

Differential Geometry II

10 credits

J. Serra

Abstract

This is a continuation course of Differential Geometry I. Topics covered include:
Introduction to Riemannian geometry: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, and isoperimetric inequalities.

Learning objective

Providing an introductory invitation to Riemannian geometry.

Literature

- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,

Prerequisites / Notice

Differential Geometry I (or basics of differentiable manifolds)

401-3462-00L

Functional Analysis II

10 credits

Core Courses: Applied Mathematics and Further Appl.-Oriented Fields

P. Hintz

Abstract

Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles. Basic results for hyperbolic PDE.

Learning objective

Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods to the study of elliptic boundary value problems, and to initial value problems for hyperbolic PDE.

Literature

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001.

Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.

Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.

Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

Prerequisites / Notice

Functional Analysis I and fluency in multivariable calculus.

Number

Title

Type

ECTS

Hours

Lecturers

401-3052-10L

Graph Theory

10 credits

B. Sudakov

Abstract

Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem

Learning objective

The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.

Lecture notes

Lecture will be only at the blackboard.

Literature

West, D.: "Introduction to Graph Theory"
Diestel, R.: "Graph Theory"

Further literature links will be provided in the lecture.

Prerequisites / Notice

Students are expected to have a mathematical background and should be able to write rigorous proofs.

401-3642-00L

Brownian Motion and Stochastic Calculus

10 credits

D. Possamaï

Abstract

This course gives an introduction to Brownian motion and stochastic calculus. It includes the construction and properties of Brownian motion, basics of Markov processes in continuous time and of Levy processes, and stochastic calculus for continuous semimartingales.

Learning objective

This course gives an introduction to Brownian motion and stochastic calculus. The following topics are planned:
- Definition and construction of Brownian motion
- Some important properties of Brownian motion
- Basics of Markov processes in continuous time
- Stochastic calculus, including stochastic integration for continuous semimartingales, Ito's formula, Girsanov's theorem, stochastic differential equations and connections with partial differential equations
- Basics of Levy processes

Lecture notes

Lecture notes will be made available in class.

Literature

- R.F. Bass, Stochastic Processes, Cambidge University Press (2001).
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).

Prerequisites / Notice

Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).

401-3632-00L

Computational Statistics

8 credits

M. Mächler

Abstract

We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.

Learning objective

The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.

Content

See the class website

Prerequisites / Notice

At least one semester of (basic) probability and statistics.

Programming experience is helpful but not required.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Media and Digital Technologies	assessed
	Problem-solving	assessed
Personal Competencies	Creative Thinking	assessed
	Critical Thinking	assessed

401-3602-00L

Applied Stochastic Processes

8 credits

V. Tassion

Abstract

Poisson processes; renewal processes; Markov chains in discrete and in continuous time; some applications.

Learning objective

Stochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We present several classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable".

Literature

R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997
R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1
M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1
S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)

Prerequisites / Notice

Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).

401-3652-00L

Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT827

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

10 credits

Electives: Pure Mathematics

University lecturers

Abstract

This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB.

Learning objective

The goal of this course is familiarity with the fundamental ideas and mathematical
consideration underlying modern numerical methods for conservation laws and wave equations.

Content

* Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering.

* Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes.

* Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes.

* Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods.

* Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes.

* Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.

Lecture notes

Lecture slides will be made available to participants. However, additional material might be covered in the course.

Literature

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online.

R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online.

E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.

Prerequisites / Notice

Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite.

Programming exercises in MATLAB

Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"

Electives

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic

Number

Title

Type

ECTS

Hours

Lecturers

401-3058-00L

Combinatorics I

4 credits

N. Hungerbühler

Abstract

The course Combinatorics I and II is an introduction into the field of enumerative combinatorics.

Learning objective

Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.

Content

Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers.

Prerequisites / Notice

Recognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008).

401-3035-00L

Forcing: An Introduction to Independence Proofs

8 credits

Selection: Further Realms and Some UZH Courses

L. Halbeisen

Abstract

Mit Hilfe der Forcing-Technik werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese von den Axiomen der Mengenlehre unabhaengig ist.

Learning objective

Die Forcing-Technik kennenlernen und verschiedene Unabhaengigkeitsbeweise fuehren koennen.

Content

Mit Hilfe der sogenannten Forcing-Technik, welche anfangs der 1960er Jahre von Paul Cohen entwickelt wurde, werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese CH von den Axiomen der Mengenlehre ZFC unabhaengig ist. Weiter wird in Modellen von ZFC, in denen CH nicht gilt, die Groesse verschiedener Kardinalzahlcharakteristiken untersucht. Zum Schluss der Vorlesung wird ein Modell von ZFC konstruiert, in dem es (bis auf Isomorphie) genau n Ramsey-Ultrafilter gibt, wobei n fuer irgend eine nicht-negative ganze Zahl steht.

Lecture notes

Ich werde mich weitgehend an mein Buch "Combinatorial Set Theory" halten, aus dem einige Kapitel aus Part III & IV behandelt werden.

Literature

"Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2017)

http://www.springer.com/de/book/9783319602301

Prerequisites / Notice

Voraussetzung ist die Vorlesung "Axiomatische Mengenlehre" (Herbstsemester 2017) bzw. die entsprechenden Kapitel aus meinem Buch.

Selection: Geometry

Number

Title

Type

ECTS

Hours

Lecturers

401-3056-00L

Finite Geometries I

Does not take place this semester.

4 credits

N. Hungerbühler

Abstract

Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.

Learning objective

Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.

Content

Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design

Literature

- Max Jeger, Endliche Geometrien, ETH Skript 1988

- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983

- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press

- Dembowski: Finite Geometries.

401-4146-22L

Derived Algebraic Geometry

4 credits

A. Bojko

Abstract

The main goal is to introduce this subject to a wider audience in a more intuitive way. The course should ideally end with applications of derived algebraic geometry to constructing virtual fundamental classes in enumerative geometry using perverse sheaves.

Learning objective

A keen listener should understand by the end of the course why derived algebraic geometry is useful and have an idea of where to begin in applying it to problems in enumerative geometry.

Literature

B. Toën, Derived Algebraic Geometry, arXiv:1401.1044, 2014.
J. Lurie. Higher topos theory, Annals of Mathematics Studies. Princeton
University Press, Princeton, NJ, 2009.
J. Lurie, On Infinity Topoi, arXiv:math/0306109, 2003.
J. Lurie, Derived Algebraic Geometry, Ph.D. thesis, Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
B. Toën and G. Vezzosi. Homotopical algebraic geometry I: Topos theory”, Advances in mathematics, 2005.
B. Toën and G. Vezzosi, From HAG to DAG: Derived Moduli Stacks:Ax-iomatic, Enriched and Motivic Homotopy Theory, 2004.
B. Toën and M. Vaquié, Moduli of objects in dg-categories, Annales scien-tifiques de l’Ecole normale supérieure, 2007.
C. Brav, V. Bussi, and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, Journal of the American Mathematical Society, 2019.
D. Joyce , P. Safronov, A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes, In Annales de la Faculté des sciences de Toulouse: Mathématiques, 2019.
D. Borisov, and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geometry & Topology, 2017.
K. Tasuki, Virtual classes via vanishing cycles, arXiv:2109.06468, 2021.

Prerequisites / Notice

One should have some understanding of algebraic geometry (in particular intersection theory), algebraic topology and category theory.

Familiarity with some enumerative geometry using virtual fundamental classes would be helpful for understanding the goal of the course.

Competencies

Subject-specific Competencies

Concepts and Theories

assessed

401-3574-61L

Introduction to Knot Theory

Does not take place this semester.

6 credits

Abstract

Introduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.

Learning objective

The aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.

Content

Definition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)

Literature

An extensive bibliography will be handed out in the course.

Prerequisites / Notice

Prerequisites are some elementary knowledge of algebra and topology.

Selection: Analysis

Number

Title

Type

ECTS

Hours

Lecturers

401-4494-23L

Variational Problems and PDEs

4 credits

A. Figalli

Abstract

In this class, we will study some classical variational problems to motivate the introduction and study of a series of important tools in analysis, such as:
1) the direct method of the calculus of variations
2) the use of geometric measure theory to deal with non-smooth objects
3) the regularity theory for elliptic PDEs with measurable and smooth coefficients

Learning objective

Learn important tools in the calculus of variations, elliptic PDEs, and geometric measure theory.

Prerequisites / Notice

Functional Analysis I

401-4834-23L

Nonlinear Wave Equations with Applications to General Relativity

4 credits

C. Kehle

Abstract

Introduction to linear and nonlinear wave equations, aspects of Lorentzian geometry and the Einstein equations in general relativity.

Learning objective

In the first part of the course, the basics of linear wave equations on Minkowski space are covered. We then go beyond to nonlinear equations as well as to curved backgrounds. To this end, the relevant background from Lorentzian geometry is introduced. The ultimate goal is to rigorously study dynamics of the Einstein equations of general relativity.

Prerequisites / Notice

It may be helpful if the participants have some familiarity with the basics of differential manifolds and basic functional analysis.

Competencies

Subject-specific Competencies

Concepts and Theories

assessed

401-3374-23L

Dynamical Systems and Ergodic Theory (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT733

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

9 credits

4V + 2U

University lecturers

Abstract

Learning objective

By the end of the unit the student: will have developed a good background in the area of dynamical systems; will be familiar with the basic concepts, results, and techniques relevant to the area; will have detailed knowledge of a number of fundamental examples that help clarify and motivate the main concepts in the theory; will understand the proofs of the fundamental theorems in the area; will have mastered the application of dynamical systems techniques for solving a range of standard problems; will have a firm foundation for further study in the area.

Content

Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it in average. We will give a strong emphasis on presenting many fundamental examples of dynamical systems. Driven by the examples, we will first introduce some of the phenomena and main concepts which one is interested in studying.
We will then formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications for example to number theory, information theory and Internet search engines. Topics which will be covered include:

-Basic examples of dynamical systems (e.g. rotations and doubling map; baker’s map, CAT map and hyperbolic toral automorphisms; the Gauss map and continued fractions);
-Elements of topological dynamics (minimality; topological conjugacy; topological mixing; topological entropy);
-Elements of symbolic dynamics (shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding);
-Introduction to ergodic theory: invariant measures; Poincare' recurrence; ergodicity; mixing; the Birkhoff Ergodic Theorem and applications; Markov measures; the ergodic theorem for Markov chains and applications to Internet Search; measure theoretic entropy;
-Selected topics (time permitting): Shannon-McMillan-Breiman theorem; Lyapunov exponents and multiplicative ergodic theorem; continuous time dynamical systems and some mathematical billiards.

Lecture notes

Lecture notes for several of the topics covered will be provided.

Literature

Textbooks which can be used as additional reference for some of the topics include:

-B. Hasselblatt and A. Katok, Dynamics: A first course. (Cambdirge University Press, 2003) – Chapters 7,8,10 and 15
-M. Brin and G. Stuck, Introduction to Dynamical Systems. (Cambridge University Press, 2002) – Chapters 1-4
-Omri Sarig, Lectures Notes on Ergodic Theory (Available Online), Topics from Chapter 1 and 2

Prerequisites / Notice

Prior Knowledge
Basic knowledge of measure theory and integration.

Number

Title

Type

ECTS

Hours

Lecturers

401-3502-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

2 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3503-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

3 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-02L

Reading Course (No. 2)

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-03L

Reading Course (No. 3)

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-8144-23L

Singular Foliations (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT555

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.htm

9 credits

4V + 2U

University lecturers

Abstract

Understanding the basic concepts of singular foliation, holonomy, foliation C*-algebra, and some aspects of the associated representation theory.

Learning objective

Understanding the basic concepts of singular foliation, holonomy, foliation C*-algebra, and some aspects of the associated representation theory.

Content

1. Regular foliations: The Frobenius theorem and foliation charts.
2. The holonomy groupoid of a regular foliation. Construction and examples. Explanation why the holonomy groupoid replaces the leaf space.
3. The C*-algebra of a regular foliation. Construction and examples.
4. Singular foliations: Definition and examples. Correspondence of projective modules of vector fields with almost regular foliations.
5. Bisubmersions and bisections: Definition and examples.
6. The holonomy groupoid of a singular foliation: Construction and examples. Proof that the holonomy groupoid of an almost regular foliation is always a Lie groupoid.
7. Applications in Poisson geometry: The almost regular case and log- symplectic Poisson structures. (Computation of the holonomy groupoid in this case and proof that it is a Poisson groupoid.)
8. The C*-algebra of a singular foliation: Construction and
computations (exact sequence of foliation C*-algebra)
9. Some representation theory: The desintegration theorem.
If me allows it, a choice of the following topics can be presented, depending on the interest of the audience:
- The hierarchy of singularities.
- Deformation to normal cone and the analytic index map. Also strict quantisation.
- Laplacians of singular foliations as unbounded multipliers of the
foliation C*-algebra, and their spectrum.
- More generally, longitudinal pseudodifferential operators.

Literature

- Lecture Notes by the Instructor
Further material:
- Alberto Candel and Lawrence Conlon. Foliations I. AMS Graduate Studies in Mathema cs, vol. 23.
- Kirill C. H. Mackenzie. General theory of Lie groupoids and Lie algebroids. LMS Lecture Notes Series 2013.
https://doi.org/10.1017/CBO9781107325883
- Jean Renault. A groupoid approach to C*-algebras. Lecture Notes in Mathematics vol. 798, Springer https://doi.org/10.1007/BFb0091072
- Alan L. T. Paterson. Groupoids, Inverse Semigroups and their Operator Algebras. Progress in Mathematics vol. 170, Springer https://doi.org/
10.1007/978-1-4612-1774-9
- Ieke Moerdijk. Introduction to Foliations and Lie groupoids. Cambridge University Press.

https://doi.org/10.1017/CBO9780511615450

Electives: Applied Mathematics and Further Application-Oriented Fields

Selection: Numerical Analysis

Number

Title

Type

ECTS

Hours

Lecturers

401-4658-00L

Numerical Methods for Finance

6 credits

C. Schwab, A. Stein

Abstract

Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB and Python programming
and knowledge of numerical mathematics at ETH BSc level.

Learning objective

Introduce the main methods for efficient numerical valuation of derivative contracts in a
Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility
models. Develop implementation of pricing methods in MATLAB and Python.
Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.

Content

1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic
volatility models.
2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees.
European contracts.
3. Finite Difference methods for Asian, American and Barrier type contracts.
4. Finite element methods for European and American style contracts.
5. Pricing under local and stochastic volatility in Black-Scholes Markets.
6. Finite Element Methods for option pricing under Levy processes. Treatment of
integrodifferential operators.
7. Stochastic volatility models for Levy processes.
8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and
stochastic volatility models in Black Scholes and Levy markets.
9. Introduction to sparse grid option pricing techniques.

Lecture notes

There will be english lecture notes as well as MATLAB or Python software for registered participants in the course.

Literature

Main reference (course text):
N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013.

Supplementary texts:
R. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004.

Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005.

D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008.

J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.

Prerequisites / Notice

Knowledge of Numerical Analysis/ Scientific Computing Techniques
corresponding roughly to BSc MATH or BSc RW/CSE at ETH is expected.
Basic programming skills in MATLAB or Python are required for the exercises,
and are _not_ taught in this course.

401-4656-21L

Deep Learning in Scientific Computing

Aimed at students in a Master's Programme in Mathematics, Engineering and Physics.

6 credits

Selection: Probability Theory, Statistics

S. Mishra, B. Moseley

Abstract

Machine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs.

Learning objective

The objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods.

Content

A selection of the following topics will be presented in the lectures.

1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces.

2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets.

3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors.

4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error.

5. Uncertainty Quantification for PDEs with supervised learning algorithms.

6. Deep Neural Networks as Reduced order models and prediction of solution fields.

7. Active Learning algorithms for PDE constrained optimization.

8. Recurrent Neural Networks and prediction of time series for dynamical systems.

9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs.

10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation.

All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others.

Lecture notes

Lecture notes will be provided at the end of the course.

Literature

All the material in the course is based on research articles written in last 1-2 years. The relevant references will be provided.

Prerequisites / Notice

The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications.

Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful.

401-4652-23L

Inverse Problems

4 credits

R. Alaifari

Abstract

Inverse problems arise in many applications in science & engineering. Typically, a physical model describes a forward problem and the task is to reconstruct from measurements, i.e. to perform inversion. In ill-posed problems, these inversions are troublesome as the inverse lacks e.g. stability. Regularization theory studies the controlled extraction of information from such systems.

Learning objective

The goal of this course is to give an understanding of ill-posedness and how it arises and to introduce the theory of regularization, which gives a mathematical framework to handle these delicate systems.

Content

Linear inverse problems, compact operators and singular value decompositions, regularization of linear inverse problems, regularization penalties, regularization parameters and parameter choice rules, iterative regularization schemes and stopping criteria, non-linear inverse problems.

Lecture notes

The lecture notes will be made available during the semester.

Literature

Engl, H. W., Hanke, M., & Neubauer, A. (1996). Regularization of inverse problems (Vol. 375). Springer Science & Business Media.

Prerequisites / Notice

Analysis, linear algebra, numerical analysis, ideal but not necessary: functional analysis

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	assessed
Personal Competencies	Creative Thinking	fostered
	Critical Thinking	assessed

Number

Title

Type

ECTS

Hours

Lecturers

401-4604-23L

First Passage Percolation and Large Deviations

4 credits

B. Dembin

Abstract

Keywords : First passage percolation; large deviations; concentration inequalities; noise sensitivity; fluctuations

Learning objective

The model of first passage percolation (FPP) was introduced in 1965 by Hammersley and Welsh to study the spread of a fluid through a random medium.
The model is defined on the lattice (Z^d,E^d) by assigning independently to each edge a positive random variable representing the time to cross the edge. This induces a random metric on the lattice, where the time between two vertices corresponds to the time of the shortest path. In this course, our goal is to study the asymptotic properties of this random metric, as well as the time-minimizing paths (geodesics). In particular, we will study time and spatial fluctuations of geodesics.

The objectives of this class are two-fold. First, discover an active field of research, have an overview of recent results and major open questions. Second, get familiar with various tools and concepts of probability that are not specific to this model. In particular, this class will contain an introduction to large deviations theory.

Literature

50 Years of First-Passage Percolation, Auffinger, Damron, Hanson
Aspects of first passage percolation, Kesten

Prerequisites / Notice

Basic knowledge in probability

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	assessed

401-4626-00L

Advanced Statistical Modelling: Mixed Models

Does not take place this semester.

4 credits

M. Mächler

Abstract

Mixed Models = (*| generalized| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms.

In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.

Learning objective

- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models.

- Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis:
"fit -> interpret & diagnose -> modify the fit -> interpret & ...."

- Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.

Content

The lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online.

Inference (significance of factors, confidence intervals) will focus on the more realistic *un*balanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.

Lecture notes

We will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples.

These lecture notes and all R scripts are made available from
https://github.com/mmaechler/MEMo

Literature

(see web page and lecture notes)

Prerequisites / Notice

- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..).

Typically this means at least two classes of (math based) statistics, say
1. Intro to probability and statistics
2. (Applied) regression including Matrix-Vector notation Y = X b + E

- Basic (1 semester) "Matrix calculus" / linear algebra is also assumed.

- If familiarity with [R](https://www.r-project.org/) is not given, it should be acquired during the course (by the student on own initiative).

Selection: Financial and Insurance Mathematics

Number

Title

Type

ECTS

Hours

Lecturers

401-3629-00L

Quantitative Risk Management

4 credits

P. Cheridito

Abstract

This course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, risk measures, extreme value theory, multivariate models, copulas, dependence structures and operational risk.

Learning objective

The goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.

Content

1. Introduction
2. Basic Concepts in Risk Management
3. Empirical Properties of Financial Data
4. Financial Time Series
5. Extreme Value Theory
6. Multivariate Models
7. Copulas and Dependence
8. Operational Risk

Lecture notes

Course material is available on https://people.math.ethz.ch/~patrickc/qrm

Literature

Quantitative Risk Management: Concepts, Techniques and Tools
AJ McNeil, R Frey and P Embrechts
Princeton University Press, Princeton, 2015 (Revised Edition)
http://press.princeton.edu/titles/10496.html

Prerequisites / Notice

The course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.

401-3923-00L

Selected Topics in Life Insurance Mathematics

4 credits

M. Koller

Abstract

Stochastic Models for Life insurance
1) Markov chains
2) Stochastic Processes for demography and interest rates
3) Cash flow streams and reserves
4) Mathematical Reserves and Thiele's differential equation
5) Theorem of Hattendorff
6) Unit linked policies

Learning objective

401-3917-00L

Stochastic Loss Reserving Methods

4 credits

R. Dahms

Abstract

Loss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to estimate those loss reserves.

Learning objective

Our goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.

Content

We will present the following stochastic claims reserving methods/models:
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, inlcusive one practice lesson
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios

Prerequisites / Notice

The exams ONLY take place during the official ETH examination periods.

This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Basic knowledge in probability theory is assumed, in particular conditional expectations.

401-3956-00L

Economic Theory of Financial Markets

4 credits

M. V. Wüthrich

Abstract

This lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.

Learning objective

This lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.

Content

We treat the following topics:
- Fundamental concepts in economics
- Portfolio theory
- Mean variance analysis, capital asset pricing model
- Arbitrage pricing theory
- Cash flow theory
- Valuation principles
- Stochastic discounting, deflator techniques
- Interest rate modeling
- Utility theory

Prerequisites / Notice

The exams ONLY take place during the official ETH examination period (no semester end exams), and they will only be taken in person (no remote exams).

This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.

401-3936-00L

Data Analytics for Non-Life Insurance Pricing

4 credits

M. V. Wüthrich, C. M. Buser

Abstract

We study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.

Learning objective

The student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.

Content

We present the following chapters:
- generalized linear models (GLMs)
- generalized additive models (GAMs)
- neural networks
- credibility theory
- classification and regression trees (CARTs)
- bagging, random forests and boosting

Lecture notes

The lecture notes are available from:
M.V. Wüthrich, C. Buser. Data Analytics for Non-Life Insurance Pricing
http://ssrn.com/abstract=2870308

Literature

M.V. Wüthrich, M. Merz. Statistical Foundations of Actuarial Learning and its Applications
http://ssrn.com/abstract=3822407

Prerequisites / Notice

The exams ONLY take place during the official ETH examination period (no semester end exams), and they will only be taken in person (no remote exams).

This course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch

Good knowledge in probability theory, stochastic processes and statistics is assumed.

401-4920-00L

Market-Consistent Actuarial Valuation

Does not take place this semester.

4 credits

M. V. Wüthrich

Abstract

Introduction to market-consistent actuarial valuation.
Topics: Stochastic discounting, full balance sheet approach, valuation portfolio in life and non-life insurance, technical and financial risks, risk management for insurance companies.

Learning objective

Goal is to give the basic mathematical tools for describing insurance products within a financial market and economic environment and provide the basics of solvency considerations.

Content

In this lecture we give a full balance sheet approach to the task of actuarial valuation of an insurance company. Therefore we introduce a multidimensional valuation portfolio (VaPo) on the liability side of the balance sheet. The basis of this multidimensional VaPo is a set of financial instruments. This approach makes the liability side of the balance sheet directly comparable to its asset side.

The lecture is based on four sections:
1) Stochastic discounting
2) Construction of a multidimensional Valuation Portfolio for life insurance products (with guarantees)
3) Construction of a multidimensional Valuation Portfolio for a run-off portfolio of a non-life insurance company
4) Measuring financial risks in a full balance sheet approach (ALM risks)

Literature

Market-Consistent Actuarial Valuation, 3rd edition.
Wüthrich, M.V.
EAA Series, Springer 2016.
ISBN: 978-3-319-46635-4

Wüthrich, M.V., Merz, M.
Claims run-off uncertainty: the full picture.
SSRN Manuscript ID 2524352 (2015).

England, P.D, Verrall, R.J., Wüthrich, M.V.
On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins.
Insurance: Mathematics and Economics 85 (2019), 74-88.

Wüthrich, M.V., Embrechts, P., Tsanakas, A.
Risk margin for a non-life insurance run-off.
Statistics & Risk Modeling 28 (2011), no. 4, 299--317.

Financial Modeling, Actuarial Valuation and Solvency in Insurance.
Wüthrich, M.V., Merz, M.
Springer Finance 2013.
ISBN: 978-3-642-31391-2

Cheridito, P., Ery, J., Wüthrich, M.V.
Assessing asset-liability risk with neural networks.
Risks 8/1 (2020), article 16.

Prerequisites / Notice

The exams ONLY take place during the official ETH examination period (no semester end exams), and they will only be taken in person (no remote exams).

This course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.

401-3888-00L

Introduction to Mathematical Finance

A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.

10 credits

M. Schweizer

Abstract

Introductory course on mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. Topics: arbitrage, completeness, risk-neutral pricing, utility maximisation, and maybe others. Fundamental theorem of asset pricing, hedging duality theorems in discrete time, convex duality in utility maximisation.

Learning objective

This is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and we also study convex duality in utility maximization.

Content

This course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For a (not fully up-to-date) overview of courses offered in the area of mathematical finance, see Link.

Lecture notes

The course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.

Literature

Literature:

Michael U. Dothan, "Prices in Financial Markets", Oxford University Press

Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter

Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press

Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer

Dmitry Kramkov and Walter Schachermayer, "The asymptotic elasticity of utility functions and optimal investment in incomplete markets", Annals of Applied Probability 9 (1999), 904-950

Prerequisites / Notice

A related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In particular, it is not possible to earn credit points with one course for the Bachelor and with the other course for the Master degree.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	fostered
	Problem-solving	assessed
Social Competencies	Communication	fostered
	Cooperation and Teamwork	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	assessed
	Critical Thinking	assessed

401-3932-19L

Mathematics for New Technologies in Finance

formerly until FS22: Machine Learning in Finance

4 credits

Selection: Mathematical Physics, Theoretical Physics

J. Teichmann

Abstract

The course will deal with the following topics with rigorous proofs and many coding excursions: Universal approximation theorems, Stochastic gradient Descent, Deep
networks and wavelet analysis, Deep Hedging, Deep calibration,
Different network architectures, Reservoir Computing, Time series analysis by machine learning, Reinforcement learning, generative adversersial networks, Economic games.

Learning objective

Content

CATALOGUE DATA TO BE ADJUSTED

Prerequisites / Notice

Bachelor in mathematics, physics, economics or computer science.

401-3928-00L

Reinsurance Analytics

4 credits

P. Arbenz

Abstract

This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and models for extreme events such as natural or man-made catastrophes. The lecture covers reinsurance contracts, Experience and Exposure pricing, natural catastrophe modelling, solvency regulation, and insurance linked securities

Learning objective

This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes.
Topics covered include:
- Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business.
- Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models
- Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks
- Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context
- Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2
- Insurance linked securities: Alternative risk transfer techniques such as catastrophe bonds

Content

Lecture notes

Slides and lecture notes will be made available.

An excerpt of last year's lecture notes is available here: https://sites.google.com/site/philipparbenz/reinsuranceanalytics

Prerequisites / Notice

Basic knowledge in statistics, probability theory, and actuarial techniques

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Media and Digital Technologies	assessed
	Problem-solving	assessed
	Project Management	fostered
Social Competencies	Communication	assessed
	Cooperation and Teamwork	fostered
	Customer Orientation	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
	Sensitivity to Diversity	fostered
	Negotiation	fostered
Personal Competencies	Adaptability and Flexibility	assessed
	Creative Thinking	fostered
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered
	Self-awareness and Self-reflection	fostered
	Self-direction and Self-management	fostered

Number

Title

Type

ECTS

Hours

Lecturers

402-0206-00L

Quantum Mechanics II

10 credits

C. Anastasiou

Abstract

Many-body quantum physics rests on symmetry considerations that lead to two kinds of particles, fermions and bosons. Formal techniques include Hartree-Fock theory and second-quantization techniques, as well as quantum statistics with ensembles. Few- and many-body systems include atoms, molecules, the Fermi sea, elastic chains, radiation and its interaction with matter, and ideal quantum gases.

Learning objective

Basic command of few- and many-particle physics for fermions and bosons, including second quantisation and quantum statistical techniques. Understanding of elementary many-body systems such as atoms, molecules, the Fermi sea, electromagnetic radiation and its interaction with matter, ideal quantum gases and relativistic theories.

Content

The description of indistinguishable particles leads us to (exchange-) symmetrized wave functions for fermions and bosons. We discuss simple few-body problems (Helium atoms, hydrogen molecule) und proceed with a systematic description of fermionic many body problems (Hartree-Fock approximation, screening, correlations with applications on atomes and the Fermi sea). The second quantisation formalism allows for the compact description of the Fermi gas, of elastic strings (phonons), and the radiation field (photons). We study the interaction of radiation and matter and the associated phenomena of radiative decay, light scattering, and the Lamb shift. Quantum statistical description of ideal Bose and Fermi gases at finite temperatures concludes the program. If time permits, we will touch upon of relativistic one particle physics, the Klein-Gordon equation for spin-0 bosons and the Dirac equation describing spin-1/2 fermions.

Literature

G. Baym, Lectures on Quantum Mechanics (Benjamin, Menlo Park, California, 1969)
L.I. Schiff, Quantum Mechanics (Mc-Graw-Hill, New York, 1955)
A. Messiah, Quantum Mechanics I & II (North-Holland, Amsterdam, 1976)
E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1998)
C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics I & II (John Wiley, New York, 1977)
P.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (Mc Graw-Hill, New York, 1965)
A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua (Mc Graw-Hill, New York, 1980)
J.J. Sakurai, Modern Quantum Mechanics (Addison Wesley, Reading, 1994)
J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley)
F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley, New York, 1993)

Prerequisites / Notice

Basic knowledge of single-particle Quantum Mechanics

402-0844-00L

Quantum Field Theory II

UZH students are not allowed to register this course unit at ETH. They must book the corresponding module directly at UZH.

10 credits

Selection: Mathematical Optimization, Discrete Mathematics

A. Lazopoulos

Abstract

The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.

Learning objective

The goal of this course is to lay down the path integral formulation of quantum field theories and in particular to provide a solid basis for the study of non-abelian gauge theories and of the Standard Model

Content

The following topics will be covered:
- path integral quantization
- non-abelian gauge theories and their quantization
- systematics of renormalization, including BRST symmetries, Slavnov-Taylor Identities and the Callan-Symanzik equation
- the Goldstone theorem and the Higgs mechanism
- gauge theories with spontaneous symmetry breaking and their quantization
- renormalization of spontaneously broken gauge theories and quantum effective actions

Literature

M.E. Peskin and D.V. Schroeder, "An introduction to Quantum Field Theory", Perseus (1995).
S. Pokorski, "Gauge Field Theories" (2nd Edition), Cambridge Univ. Press (2000)
P. Ramond, "Field Theory: A Modern Primer" (2nd Edition), Westview Press (1990)
S. Weinberg, "The Quantum Theory of Fields" (Volume 2), CUP (1996).

Number

Title

Type

ECTS

Hours

Lecturers

401-3902-21L

Network & Integer Optimization: From Theory to Application

6 credits

R. Zenklusen

Abstract

This course covers various topics in Network and (Mixed-)Integer Optimization. It starts with a rigorous study of algorithmic techniques for some network optimization problems (with a focus on matching problems) and moves to key aspects of how to attack various optimization settings through well-designed (Mixed-)Integer Programming formulations.

Learning objective

Our goal is for students to both get a good foundational understanding of some key network algorithms and also to learn how to effectively employ (Mixed-)Integer Programming formulations, techniques, and solvers, to tackle a wide range of discrete optimization problems.

Content

Key topics include:
- Matching problems;
- Integer Programming techniques and models;
- Extended formulations and strong problem formulations;
- Solver techniques for (Mixed-)Integer Programs;
- Decomposition approaches.

Literature

- Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018.
- Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes.
- Vanderbeck François, Wolsey Laurence: Reformulations and Decomposition of Integer Programs. Chapter 13 in: 50 Years of Integer Programming 1958-2008. Springer, 2010.
- Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.

Prerequisites / Notice

Solid background in linear algebra. Preliminary knowledge of Linear Programming is ideal but not a strict requirement. Prior attendance of the course Linear & Combinatorial Optimization is a plus.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Problem-solving	assessed
Social Competencies	Communication	assessed
Personal Competencies	Creative Thinking	assessed

401-3904-22L

Convex Optimization

6 credits

A. A. Kurpisz

Abstract

Introduction to Convex Optimization with a focus on algorithms and the numerous applications of Convex Optimization.

Learning objective

The main goal of this course is to obtain a solid understanding of classical Convex Optimization techniques and their numerous applications, including in Data Science, Machine Learning, and, more generally, in science and engineering. Apart from building up a solid foundational understanding of Convex Optimization, students also get hands-on experience through regular coding exercises. This aims at providing a holistic view on the process of identifying, modeling, and solving a wide range of computational questions that can be cast as Convex Optimization problems.

Content

Key topics include:
- Introduction to Convex Optimization.
- Subclasses of Convex Optimization: Semidefinite Programming, Second-Order Cone Programming and Geometric Programming.
- Applications of Convex Optimization in science and engineering.
- Algorithms for Convex Optimization.

Lecture notes

A script will be provided.

Literature

- Boyd, S., \& Vandenberghe, L. (2004). Convex Optimization. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511804441

Prerequisites / Notice

Background in Linear Programming is recommended.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Media and Digital Technologies	fostered
	Problem-solving	assessed
	Project Management	fostered
Social Competencies	Communication	assessed
	Cooperation and Teamwork	fostered
	Customer Orientation	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
	Sensitivity to Diversity	fostered
	Negotiation	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	assessed
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered
	Self-awareness and Self-reflection	fostered
	Self-direction and Self-management	fostered

Selection: Theoretical Computer Science, Discrete Mathematics

In the Master's programme in Mathematics 401-3052-05L Graph Theory is eligible as an elective course, but only if 401-3052-10L Graph Theory isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.

Number

Title

Type

ECTS

Hours

Lecturers

263-4400-00L

Advanced Graph Algorithms and Optimization

10 credits

3V + 3U + 3A

R. Kyng, M. Probst

Abstract

This course will cover a number of advanced topics in optimization and graph algorithms.

Learning objective

The course will take students on a deep dive into modern approaches to
graph algorithms using convex optimization techniques.

By studying convex optimization through the lens of graph algorithms,
students should develop a deeper understanding of fundamental
phenomena in optimization.

The course will cover some traditional discrete approaches to various graph
problems, especially flow problems, and then contrast these approaches
with modern, asymptotically faster methods based on combining convex
optimization with spectral and combinatorial graph theory.

Content

Students should leave the course understanding key
concepts in optimization such as first and second-order optimization,
convex duality, multiplicative weights and dual-based methods,
acceleration, preconditioning, and non-Euclidean optimization.

Students will also be familiarized with central techniques in the
development of graph algorithms in the past 15 years, including graph
decomposition techniques, sparsification, oblivious routing, and
spectral and combinatorial preconditioning.

Prerequisites / Notice

This course is targeted toward masters and doctoral students with an
interest in theoretical computer science.

Students should be comfortable with design and analysis of algorithms, probability, and linear algebra.

Having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, but not formally required. If you are not
sure whether you're ready for this class or not, please consult the
instructor.

252-0408-00L

Cryptographic Protocols

6 credits

2V + 2U + 1A

M. Hirt

Abstract

In a cryptographic protocol, a set of parties wants to achieve some common goal, while some of the parties are dishonest. Most prominent example of a cryptographic protocol is multi-party computation, where the parties compute an arbitrary (but fixed) function of their inputs, while maintaining the secrecy of the inputs and the correctness of the outputs even if some of the parties try to cheat.

Learning objective

To know and understand a selection of cryptographic protocols and to
be able to analyze and prove their security and efficiency.

Content

The selection of considered protocols varies. Currently, we consider
multi-party computation, secret-sharing, broadcast and Byzantine
agreement. We look at both the synchronous and the asynchronous
communication model, and focus on simple protocols as well as on
highly-efficient protocols.

Lecture notes

We provide handouts of the slides. For some of the topics, we also
provide papers and/or lecture notes.

Prerequisites / Notice

A basic understanding of fundamental cryptographic concepts (as taught
for example in the course Information Security) is useful, but not
required.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	fostered
Personal Competencies	Creative Thinking	fostered
	Critical Thinking	fostered

263-4660-00L

Applied Cryptography

8 credits

3V + 2U + 2P

K. Paterson, F. Günther

Abstract

This course will introduce the basic primitives of cryptography, using rigorous syntax and game-based security definitions. The course will show how these primitives can be combined to build cryptographic protocols and systems.

Learning objective

The goal of the course is to put students' understanding of cryptography on sound foundations, to enable them to start to build well-designed cryptographic systems, and to expose them to some of the pitfalls that arise when doing so.

Content

Basic symmetric primitives (block ciphers, modes, hash functions); generic composition; AEAD; basic secure channels; basic public key primitives (encryption,signature, DH key exchange); ECC; randomness; applications.

Literature

Textbook: Boneh and Shoup, “A Graduate Course in Applied Cryptography”, http://toc.cryptobook.us/book.pdf.

Prerequisites / Notice

Students should have taken the D-INFK Bachelor's course “Information Security" (252-0211-00) or an alternative first course covering cryptography at a similar level. / In this course, we will use Moodle for content delivery: https://moodle-app2.let.ethz.ch/course/view.php?id=19644.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Media and Digital Technologies	assessed
Personal Competencies	Creative Thinking	fostered
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered

263-4656-00L

Digital Signatures

5 credits

2V + 2A

D. Hofheinz

Abstract

Digital signatures as one central cryptographic building block. Different security goals and security definitions for digital signatures, followed by a variety of popular and fundamental signature schemes with their security analyses.

Learning objective

The student knows a variety of techniques to construct and analyze the security of digital signature schemes. This includes modularity as a central tool of constructing secure schemes, and reductions as a central tool to proving the security of schemes.

Content

We will start with several definitions of security for signature schemes, and investigate the relations among them. We will proceed to generic (but inefficient) constructions of secure signatures, and then move on to a number of efficient schemes based on concrete computational hardness assumptions. On the way, we will get to know paradigms such as hash-then-sign, one-time signatures, and chameleon hashing as central tools to construct secure signatures.

Literature

Jonathan Katz, "Digital Signatures."

Prerequisites / Notice

Ideally, students will have taken the D-INFK Bachelors course "Information Security" or an equivalent course at Bachelors level.

Selection: Further Realms and Some UZH Courses

Number

Title

Type

ECTS

Hours

Lecturers

401-4498-23L

Advances in Optimal Transport and Stochastics

4 credits

G. Pammer, B. Acciaio

Abstract

We study recent developments of stochastic transport with applications to mathematical finance. In particular, we will cover weak transport, martingale transport, causal and adapted transport.

Learning objective

Understanding of the main results and tools from classical transport and from the different new kinds of transports; intuition behind the main concepts and understanding of the proofs of the main results; ability to apply tools from optimal transport for applications in mathematical finance.

Content

We start by recalling the main concepts and results from the classical optimal transport theory, providing intuition of the main ideas and understanding of the needed mathematical methods. We then focus on recent developments of stochastic transport with applications to mathematical finance. In particular, we will cover the following topics: weak transport (including the special cases of entropic transport and barycentric transport), martingale transport (especially in connection with model-independent finance and the Skorokhod Embedding problem), causal and adapted transport (also related to stability in mathematical finance, and with applications to filtration enlargement, equilibrium problems, quantification of arbitrage). We will motivate the introduction of these different kinds of optimal transport in order to deal with several problems especially in mathematical finance, as pricing and hedging in a model-independent framework, gauging the distance between financial models, accounting for model uncertainty.

Lecture notes

Lecture notes will be provided at the beginning of the semester

Prerequisites / Notice

Measure Theory, Probability and Stochastic Calculus (basic)

261-5110-00L

Optimization for Data Science

10 credits

3V + 2U + 4A

B. Gärtner, N. He

Abstract

This course provides an in-depth theoretical treatment of optimization methods that are relevant in data science.

Learning objective

Understanding the guarantees and limits of relevant optimization methods used in data science. Learning theoretical paradigms and techniques to deal with optimization problems arising in data science.

Content

This course provides an in-depth theoretical treatment of classical and modern optimization methods that are relevant in data science.

After a general discussion about the role that optimization has in the process of learning from data, we give an introduction to the theory of (convex) optimization. Based on this, we present and analyze algorithms in the following four categories: first-order methods (gradient and coordinate descent, Frank-Wolfe, subgradient and mirror descent, stochastic and incremental gradient methods); second-order methods (Newton and quasi Newton methods); non-convexity (local convergence, provable global convergence, cone programming, convex relaxations); min-max optimization (extragradient methods).

The emphasis is on the motivations and design principles behind the algorithms, on provable performance bounds, and on the mathematical tools and techniques to prove them. The goal is to equip students with a fundamental understanding about why optimization algorithms work, and what their limits are. This understanding will be of help in selecting suitable algorithms in a given application, but providing concrete practical guidance is not our focus.

Prerequisites / Notice

A solid background in analysis and linear algebra; some background in theoretical computer science (computational complexity, analysis of algorithms); the ability to understand and write mathematical proofs.

227-0434-10L

Mathematics of Information

8 credits

3V + 2U + 2A

H. Bölcskei

Abstract

The class focuses on mathematical aspects of

1. Information science: Sampling theorems, frame theory, compressed sensing, sparsity, super-resolution, spectrum-blind sampling, subspace algorithms, dimensionality reduction

2. Learning theory: Approximation theory, greedy algorithms, uniform laws of large numbers, Rademacher complexity, Vapnik-Chervonenkis dimension

Learning objective

The aim of the class is to familiarize the students with the most commonly used mathematical theories in data science, high-dimensional data analysis, and learning theory. The class consists of the lecture and exercise sessions with homework problems.

Content

Mathematics of Information

1. Signal representations: Frame theory, wavelets, Gabor expansions, sampling theorems, density theorems

2. Sparsity and compressed sensing: Sparse linear models, uncertainty relations in sparse signal recovery, super-resolution, spectrum-blind sampling, subspace algorithms (ESPRIT), estimation in the high-dimensional noisy case, Lasso

3. Dimensionality reduction: Random projections, the Johnson-Lindenstrauss Lemma

Mathematics of Learning

4. Approximation theory: Nonlinear approximation theory, best M-term approximation, greedy algorithms, fundamental limits on compressibility of signal classes, Kolmogorov-Tikhomirov epsilon-entropy of signal classes, optimal compression of signal classes

5. Uniform laws of large numbers: Rademacher complexity, Vapnik-Chervonenkis dimension, classes with polynomial discrimination

Lecture notes

Detailed lecture notes will be provided at the beginning of the semester.

Prerequisites / Notice

This course is aimed at students with a background in basic linear algebra, analysis, statistics, and probability.

We encourage students who are interested in mathematical data science to take both this course and "401-4944-20L Mathematics of Data Science" by Prof. A. Bandeira. The two courses are designed to be complementary.

H. Bölcskei and A. Bandeira

252-1424-00L

Models of Computation

6 credits

2V + 2U + 1A

M. Cook

Abstract

This course surveys many different models of computation: Turing Machines, Cellular Automata, Finite State Machines, Graph Automata, Circuits, Tilings, Lambda Calculus, Fractran, Chemical Reaction Networks, Hopfield Networks, String Rewriting Systems, Tag Systems, Diophantine Equations, Register Machines, Primitive Recursive Functions, and more.

Learning objective

The goal of this course is to become acquainted with a wide variety of models of computation, to understand how models help us to understand the modeled systems, and to be able to develop and analyze models appropriate for new systems.

Content

227-0432-00L

Learning, Classification and Compression

4 credits

E. Riegler

Abstract

The focus of the course is aligned to a theoretical approach of learning theory and classification and an introduction to lossy and lossless compression for general sets and measures. We will mainly focus on a probabilistic approach, where an underlying distribution must be learned/compressed. The concepts acquired in the course are of broad and general interest in data sciences.

Learning objective

After attending this lecture and participating in the exercise sessions, students will have acquired a working knowledge of learning theory, classification, and compression.

Content

1. Learning Theory
(a) Framework of Learning
(b) Hypothesis Spaces and Target Functions
(c) Reproducing Kernel Hilbert Spaces
(d) Bias-Variance Tradeoff
(e) Estimation of Sample and Approximation Error

2. Classification
(a) Binary Classifier
(b) Support Vector Machines (separable case)
(c) Support Vector Machines (nonseparable case)
(d) Kernel Trick

3. Lossy and Lossless Compression
(a) Basics of Compression
(b) Compressed Sensing for General Sets and Measures
(c) Quantization and Rate Distortion Theory for General Sets and Measures

Lecture notes

Detailed lecture notes will be provided.

Prerequisites / Notice

This course is aimed at students with a solid background in measure theory and linear algebra and basic knowledge in functional analysis.

401-3502-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

2 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3503-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

3 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-22L

Reading Course

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-02L

Reading Course (No. 2)

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-3504-03L

Reading Course (No. 3)

To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.

4 credits

Supervisors

Abstract

For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.

Learning objective

401-8822-23L

Introduction to the Statistical Mechanics of Lattice Systems (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT778

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.htm

6 credits

Electives (Direction Applied Mathematics MSc Only)

University lecturers

Abstract

Learning objective

Knowledge of mathematical techniques suitable for the study of classical lattice models describing phase transitions.

Content

Statistical mechanics was originally introduced to provide a microscopic justification of equilibrium thermodynamics, the physical theory of heat. In the last 70 years, it also developed into a well-established branch of mathematics and its ideas and methods have had an important impact on several other fields of mathematics, such as probability, analysis, geometry…
The goal of the course is to give an introduction to statistical mechanics from a mathematical point of view. Topics to be covered by the course are:
-) Ising model. The Ising model is one of the most important models in statistical mechanics. Introduced to describe the ferromagnetic phase transition, it is an ideal testing ground for new mathematical techniques because of its simplicity. We will use it to discuss the concepts of thermodynamic functions, thermodynamic limit (infinite volume limit), infinite volume states, and phase transition.
-) Cluster expansions. Cluster expansions are a powerful tool in the study of statistical mechanics that allow for the rigorous implementation of perturbative arguments. We will introduce a general framework for cluster expansions and afterward provide applications to the Ising model.
-) Depending on the background of the audience, the third part of the lecture will either be focusing on the construction of infinite volume Gibbs measures (approach by Dobrushin, Lanford, Ruelle (DLR)) or on Pirogov-Sinai theory. The former aims at constructing a probability measure (with the example of the Ising model in mind) that yields a more detailed description of states in the thermodynamic limit, and therefore of infinite systems. The latter is a general framework to establish the possible macroscopic behaviors of a class of statistical mechanics models that share some key features with the Ising model.

Literature

The course follows Chapters 3 (Ising model), 5 (Cluster expansion), 6 (Infinite volume Gibbs measures), and 7 (Pirogov-Sinai theory) in the book ``Statistical mechanics of lattice systems’’ by Sascha Friedli and Yvan Velenik, Cambridge University Press, Cambridge, 2018, that is available online (link will be provided). Handwritten lecture notes will also be available.

Prerequisites / Notice

Analysis, linear algebra. An introduction to probability theory is helpful but not required.

Electives from applied mathematics and further application-oriented fields that are only eligible for credits for the Master's degree in Applied Mathematics.

Number

Title

Type

ECTS

Hours

Lecturers

151-0530-00L

Nonlinear Dynamics and Chaos II

4 credits

G. Haller

Abstract

The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems

Learning objective

The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.

Content

I. The internal structure of chaos: symbolic dynamics, Bernoulli shift map, sub-shifts of finite type; chaos is numerical iterations.

II.Hamiltonian dynamical systems: conservation and recurrence, stability of fixed points, integrable systems, invariant tori, Liouville-Arnold-Jost Theorem, KAM theory.

III. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds, existence, persistence, and smoothness, applications.
IV. Geometric singular perturbation theory: slow manifolds and their stability, physical examples. V. Finite-time dynamical system; detecting Invariant manifolds and coherent structures in finite-time flows

Lecture notes

Handwritten instructor's notes and typed lecture notes will be downloadable from Moodle.

Literature

Books will be recommended in class

Prerequisites / Notice

Nonlinear Dynamics I (151-0532-00) or equivalent

Application Area

Only necessary and eligible for the Master degree in Applied Mathematics.
One of the application areas specified must be selected for the category Application Area for the Master degree in Applied Mathematics. At least 8 credits are required in the chosen application area. Credits from other application areas cannot be recognised for further application areas.

Atmospherical Physics

Number

Title

Type

ECTS

Hours

Lecturers

701-1216-00L

Weather and Climate Models

4 credits

C. Schär, D. Leutwyler, M. Wild

Abstract

The course provides an introduction to weather and climate models. It discusses how these models are built addressing both the dynamical core and the physical parameterizations, and it provides an overview of how these models are used in numerical weather prediction and climate research. As a tutorial, students conduct a term project and build a simple atmospheric model using the language PYTHON.

Learning objective

At the end of this course, students understand how weather and climate models are formulated from the governing physical principles, and how they are used for climate and weather prediction purposes.

Content

The course provides an introduction into the following themes: numerical methods (finite differences and spectral methods); adiabatic formulation of atmospheric models (vertical coordinates, hydrostatic approximation); parameterization of physical processes (e.g. clouds, convection, boundary layer, radiation); atmospheric data assimilation and weather prediction; predictability (chaos-theory, ensemble methods); climate models (coupled atmospheric, oceanic and biogeochemical models); climate prediction. Hands-on experience with simple models will be acquired in the tutorials.

Lecture notes

Slides and lecture notes will be made available at
Link

Literature

List of literature will be provided.

Prerequisites / Notice

Prerequisites: to follow this course, you need some basic background in atmospheric science, numerical methods (e.g., "Numerische Methoden in der Umweltphysik", 701-0461-00L) as well as experience in programming. Previous experience with PYTHON is useful but not required.

Biology

Number

Title

Type

ECTS

Hours

Lecturers

262-0200-00L

Bayesian Phylodynamics

4 credits

2G + 2A

T. Vaughan, T. Stadler

Abstract

How fast is the latest variant of COVID-19 spreading? How fast was Ebola spreading in West Africa? Where did these epidemics come from? How can we construct the phylogenetic tree of great apes, and did gene flow occur between different apes? At the end of the course, students will have designed, performed, presented, and discussed their own phylodynamic data analysis to answer such questions.

Learning objective

Attendees will extend their knowledge of Bayesian phylodynamics obtained in the “Computational Biology” class (636-0017-00L) and will learn how to apply this theory to real world data. The main theoretical concepts introduced are:
* Bayesian statistics
* Phylogenetic and phylodynamic models
* Markov Chain Monte Carlo methods
Attendees will apply these concepts to a number of applications yielding biological insight into:
* Epidemiology
* Pathogen evolution
* Macroevolution of species

Content

In the first part of the block course, we will present the theoretical concepts of Bayesian phylodynamics. This will involve both lectures and tutorials, during which students will gain experience in using the software package BEAST 2 to apply these theoretical concepts to empirical data. We use previously published datasets on e.g. Ebola, Zika, Yellow Fever, Apes, and Penguins for analysis. Examples of these practical tutorials are available on https://taming-the-beast.org/.
In the second part of the block course, students will choose a set of real genetic sequence data and possibly some non-genetic metadata. They will then design and conduct a research project in which they perform Bayesian phylogenetic analyses of their chosen data. A final written report on the research project will be submitted after the block course for grading

Lecture notes

All material will be available on https://taming-the-beast.org/.

Literature

The following books provide excellent background material:
• Drummond, A. & Bouckaert, R. 2015. Bayesian evolutionary analysis with BEAST.
• Yang, Z. 2014. Molecular Evolution: A Statistical Approach.
• Felsenstein, J. 2003. Inferring Phylogenies.
More detailed information is available on https://taming-the-beast.org/.

Prerequisites / Notice

This class builds upon the content which we teach in the Computational Biology class (636-0017-00L). Attendees must have either taken the Computational Biology class or acquired the content elsewhere.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Media and Digital Technologies	fostered
	Problem-solving	assessed
	Project Management	assessed
Social Competencies	Communication	assessed
	Cooperation and Teamwork	assessed
	Customer Orientation	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
	Sensitivity to Diversity	fostered
	Negotiation	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	assessed
	Critical Thinking	assessed
	Integrity and Work Ethics	fostered
	Self-awareness and Self-reflection	fostered
	Self-direction and Self-management	fostered

Control and Automation

Number

Title

Type

ECTS

Hours

Lecturers

151-0660-00L

Model Predictive Control

4 credits

E. Gallestey Alvarez, P. F. Al Hokayem

M. Zeilinger

Abstract

Model predictive control is a flexible paradigm that defines the control law as an optimization problem, enabling the specification of time-domain objectives, high performance control of complex multivariable systems and the ability to explicitly enforce constraints on system behavior. This course provides an introduction to the theory and practice of MPC and covers advanced topics.

Learning objective

Design and implement Model Predictive Controllers (MPC) for various system classes to provide high performance controllers with desired properties (stability, tracking, robustness,..) for constrained systems.

Content

- Review of required optimal control theory
- Basics on optimization
- Receding-horizon control (MPC) for constrained linear systems
- Theoretical properties of MPC: Constraint satisfaction and stability
- Computation: Explicit and online MPC
- Practical issues: Tracking and offset-free control of constrained systems, soft constraints
- Robust MPC: Robust constraint satisfaction
- Simulation-based project providing practical experience with MPC

Lecture notes

Script / lecture notes will be provided.

Prerequisites / Notice

One semester course on automatic control, Matlab, linear algebra.
Courses on signals and systems and system modeling are recommended. Important concepts to start the course: State-space modeling, basic concepts of stability, linear quadratic regulation / unconstrained optimal control.

Expected student activities: Participation in lectures, exercises and course project; homework (~2hrs/week).

227-0207-00L

Nonlinear Systems and Control

Prerequisite: Control Systems (227-0103-00L)

6 credits

Abstract

Introduction to the area of nonlinear systems and their control. Familiarization with tools for analysis of nonlinear systems. Discussion of the various nonlinear controller design methods and their applicability to real life problems.

Learning objective

On completion of the course, students understand the difference between linear and nonlinear systems, know the mathematical techniques for analysing these systems, and have learnt various methods for designing controllers accounting for their characteristics.

Course puts the student in the position to deploy nonlinear control techniques in real applications. Theory and exercises are combined for better understanding of the virtues and drawbacks present in the different methods.

Content

Virtually all practical control problems are of nonlinear nature. In some cases application of linear control methods leads to satisfactory controller performance. In many other cases however, only application of nonlinear analysis and control synthesis methods will guarantee achievement of the desired objectives.

During the past decades mature nonlinear controller design methods have been developed and have proven themselves in applications. After an introduction of the basic methods for analysing nonlinear systems, these methods will be introduced together with a critical discussion of their pros and cons. Along the course the students will be familiarized with the basic concepts of nonlinear control theory.

This course is designed as an introduction to the nonlinear control field and thus no prior knowledge of this area is required. The course builds, however, on a good knowledge of the basic concepts of linear control and mathematical analysis.

Lecture notes

An english manuscript will be made available on the course homepage during the course.

Literature

H.K. Khalil: Nonlinear Systems, Prentice Hall, 2001.

Prerequisites / Notice

Prerequisites: Linear Control Systems, or equivalent.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Problem-solving	assessed
Personal Competencies	Adaptability and Flexibility	assessed
	Creative Thinking	assessed
	Critical Thinking	assessed

151-0530-00L

Nonlinear Dynamics and Chaos II

4 credits

G. Haller

Abstract

The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems

Learning objective

The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.

Content

Lecture notes

Handwritten instructor's notes and typed lecture notes will be downloadable from Moodle.

Literature

Books will be recommended in class

Prerequisites / Notice

Nonlinear Dynamics I (151-0532-00) or equivalent

151-0566-00L

Recursive Estimation

4 credits

See: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/

R. D'Andrea

Abstract

Estimation of the state of a dynamic system based on a model and observations in a computationally efficient way.

Learning objective

Learn the basic recursive estimation methods and their underlying principles.

Content

Introduction to state estimation; probability review; Bayes' theorem; Bayesian tracking; extracting estimates from probability distributions; Kalman filter; extended Kalman filter; particle filter; observer-based control and the separation principle.

Lecture notes

Lecture notes available on course website: http://www.idsc.ethz.ch/education/lectures/recursive-estimation.html

Prerequisites / Notice

Requirements: Introductory probability theory and matrix-vector algebra.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Media and Digital Technologies	fostered
	Problem-solving	assessed
	Project Management	fostered
Social Competencies	Communication	fostered
	Cooperation and Teamwork	fostered
	Customer Orientation	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
	Sensitivity to Diversity	fostered
	Negotiation	fostered
Personal Competencies	Integrity and Work Ethics	fostered

Economics

Number

Title

Type

ECTS

Hours

Lecturers

363-0552-00L

Economic Growth and Resource Use

3 credits

E. Komarov

Abstract

The course deals with the factors that contribute to economic development. Throughout the course theoretical economic modelling will be used to discuss the effects of factors – such as land, human/physical capital, technology, fossil energy resources, and climate change – on economic growth and to draw conclusions for the future.

Learning objective

The general objective of the course is to provide students tools and intuition to:

i) think in a structured way – though economic modelling – about the factors that have lead to the different growth experiences among countries, and still shape our contemporary situation;
ii) assess and design policies on the basis of economic development;
iii) draw conclusions for the future of economic development, that take into account prevalent issues such as the scarcity of fossil energy resources and climate change.

Content

Why is economic growth worth studying? Which are the factors behind economic growth? What is the role of natural resources in shaping economic development? Is our finite planet able to support sustainable long-term economic growth? Economics aims at explaining human behaviour; how do we model it and how can we steer it for the better? How do you design an efficient economic policy for a sustainable future? What is sustainable anyway? These are some of the questions you will learn to answer in this course.

After spending the first lecture on overviewing the course, and the second lecture on building our mathematical and economic foundation, we begin with the three main modules that comprise this course.

The first module – called “Land and Economic Growth” – deals with the historical evolution of the factors behind economic development from the pre-industrial times to our modern growth experiences. By studying the history of economic growth, we understand change and how the society we live in came to be. In this module we will develop economic models that capture the transition from an era of miniscule economic growth that persisted for millennia before the industrial revolution – with land and human labour as the main inputs to economic activity – to our modern growth experience where the continuous improvement in technology and services is our status quo.

The second module – called “Non-Renewable Resources and Growth” – deals with the problem of optimal exploitation of non-renewable resources, as well as with the issue of “Resource Curse” – i.e., the observed negative relationship between economic development and resource abundance. Emerging in the 1970s due to two oil crises, the problem of the economy’s extreme dependence on fossil and depletable energy resources sparked a great deal of research to guide our way forward. Some important questions we will formally answer in this module are the following. How do we optimally exploit a given stock of a non-renewable resource? What affects the prices of non-renewable resources? If fossil energy sources – a (so far) important input to production – are getting ever depleted, is long-term growth possible? How do we explain the “Resource Curse” and what are the policies that allow a sustainable future in countries that suffer from such a curse?

The third module – called “Climate Change and Growth” – deals with the pressing problem of our changing climate. Greenhouse gas emissions – so far essential for economic activity – accumulate in the atmosphere and alter environmental patterns. This phenomenon – commonly known as climate change – is responsible for the increase in the frequency and the intensity of natural disasters, which damage our stocks of capital and put a drag on economic growth. To derive appropriate policies for a sustainable future, we will incorporate these aspects in workhorse models of the economics and finance literature. Students will learn how to derive and set the “correct” price on the use of polluting energy resources from the perspective of policy-makers. Additionally, pricing of climate change risks for financial markets is important, both for individual investors and central banks, as it is they who provide liquidity to firms to pursue their long-term growth targets. Accordingly, we will close the lecture with the pricing of climate change risks from an investor’s perspective.

Literature

The main reference of the course is the set of lecture notes; students will also be encouraged to read some influential academic articles dealing with the issues under study.

The course is self-contained and only material that was discussed in the lecture will be relevant for the exam.

Prerequisites / Notice

Knowledge of basic calculus (differentiation - integration) and basic statistics (e.g. what is an expectation; variance-covariance) is considered as a prerequisite. Elementary knowledge of dynamic systems analysis, optimal control theory and economic theory is a plus but not a prerequisite.

363-0514-00L

Energy Economics and Policy

It is recommended for students to have taken a course in introductory microeconomics. If not, they should be familiar with microeconomics as in, for example,"Microeconomics" by Mankiw & Taylor and the appendices 4 and 7 of the book "Microeconomics" by Pindyck & Rubinfeld.

3 credits

M. Filippini, S. Srinivasan

Abstract

An introduction to energy economics and policy that covers the following topics: energy demand, investment in energy efficiency, investment in renewables, energy markets, market failures and behavioral anomalies, market-based and non-market based energy and climate policy instruments in industrialized and developing countries.

Learning objective

The students will develop an understanding of economic principles and tools necessary to analyze energy issues and to understand energy and climate policy instruments. Emphasis will be put on empirical analysis of energy demand and supply, market failures, behavioral anomalies, energy and climate policy instruments in industrialized and developing countries, and investments in renewables and in energy-efficient technologies.

Content

The course provides an introduction to energy economics principles and policy applications. The first part of the course will introduce the microeconomic foundation of energy demand and supply as well as market failures and behavioral anomalies. In a second part, we introduce the concept of investment analysis (such as the NPV) in the context of renewable and energy-efficient technologies. In the last part, we use the previously introduced concepts to analyze energy policies: from a government perspective, we discuss the mechanisms and implications of market oriented and non-market oriented policy instruments as well as applications in developing countries.

Throughout the entire course, we combine the material with insights from current research in energy economics. This combination will enable students to understand standard scientific literature in the field of energy economics and policy. Moreover, the class aims to show students how to relate current issues in the energy and climate spheres that influence industrialized and developing countries to insights from energy economics and policy.

Course evaluation: at the end of the course, there will be a written exam covering the topics of the course.

Prerequisites / Notice

It is recommended for students to have taken a course in introductory microeconomics. If not, they should be familiar with microeconomics as in, for example, "Microeconomics" by Mankiw & Taylor and the appendices 4 and 7 of the book "Microeconomics" by Pindyck & Rubinfeld.

364-0576-00L

Advanced Sustainability Economics

PhD course, open for MSc students

3 credits

E. Komarov, C. Renoir

Abstract

The course covers current resource and sustainability economics, including ethical foundations of sustainability, intertemporal optimisation in capital-resource economies, sustainable use of non-renewable and renewable resources, pollution dynamics, population growth, and sectoral heterogeneity. A final part is on empirical contributions, e.g. the resource curse, energy prices, and the EKC.

Learning objective

Understanding of the current issues and economic methods in sustainability research; ability to solve typical problems like the calculation of the growth rate under environmental restriction with the help of appropriate model equations.

Please note that the course takes places in Zurichbergstrasse 18, which requires an ETH card to enter. We kindly ask Non-ETH students to inform
Clément Renoir if they would like to attend.

363-0575-00L

Economic Growth, Cycles and Policy

3 credits

H. Gersbach

Abstract

Learning objective

- Fundamental knowledge about the drivers of economic growth in the short and long run, key macroeconomic variables and observed patterns in developed countries

- Comprehensive understanding of core macroeconomic frameworks and thinking devices

Content

This intermediate course focuses on the core thinking devices and foundations in macroeconomics and monetary economics, and uses these devices to understand economic growth, business cycles, crises as well as how to conduct monetary and fiscal policies and policies to foster the stability of financial and economic systems. The course is structured in the following way:

Part I: Basics
- Introduction
- IS-LM Model in Closed Economy (Repetition)
- Schools of Thought
- Consumption and Investment
- The Solow Growth Model

Part II: Special Themes
- Money Holding, Inflation, and Monetary Policy
- Crises in Market Economies
- IS-LM Model and Open Economy
- Theories of exchange rate determination
- Technical Appendix

Lecture notes

Copies of the slides will be made available.

Literature

Chapters in
Manfred Gärtner (2009), Macroeconomics, Third Edition, Prentice Hall.
and selected chapters in other books and/or papers

Prerequisites / Notice

It is required that participants have attended the lecture "Principles of Macroeconomics" (363-0565-00L).

363-0515-00L

Decisions and Markets

3 credits

A. Bommier

Abstract

This course provides an introduction to microeconomics. The course emphasizes the conceptual foundations of microeconomics and contains concrete examples of their application.

Learning objective

The purpose of this course is to provide master students with an introduction to graduate-level microeconomics, particularly for students considering further graduate work in economics, business administration or management science. The course provides the fundamental concepts and tools for graduate courses in economics offered at ETH and UZH.

After completing this course:
- Students will be able to understand and use existing models to make predictions of consumer and firm behavior.
- Students understand the fundamental welfare theorems and will be able to analyze equilibria of markets with perfect and imperfect competition.
- Students will be able to analyze under which conditions market allocations are not efficient (market failure).

Content

Microeconomics is the branch of economics which studies the decision-making by an individual, household, firm, industry or level of government. The economic equilibrium is the result of agents' interactions. Microeconomics is an element of nearly every subfield in economic analysis today. This course introduces the fundamental frameworks which form the basis of many economic models.

Theory of the consumer:
- Consumer preferences and utility
- Budget sets and optimal choice
- Demand functions
- Labor supply and intertemporal choice
- Welfare economics

Theory of the producer:
- Technological constraints and the production function
- Cost minimization
- Profit maximization

Market structure:
- Perfectly competitive markets
- Monopoly behavior
- Duopoly behavior

General equilibrium analysis:
- Market equilibrium in an exchange economy

Lecture notes

The lecture will be based on lecture slides, which will be made available on Moodle.

Literature

The course is mostly based on the textbook by R. Serrano and A. Feldman: "A Short Course in Intermediate Microeconomics with Calculus" (Cambridge University Press, 2013). Another textbook of interest is "Intermediate Microeconomics: A Modern Approach" by H. Varian (Norton, 2014).
Exercises are available in the textbook by R. Serrano and A. Feldman ("A Short Course in Intermediate Microeconomics with Calculus", Cambridge University Press, 2013). More exercises can be found in the book "Workouts in Intermediate Microeconomics" by T. Bergstrom and H. Varian (Norton, 2010).

Prerequisites / Notice

The course is open to students who have completed an undergraduate course in economics principles and an undergraduate course in multivariate calculus.

Finance

Number

Title

Type

ECTS

Hours

Lecturers

401-8905-00L

Financial Engineering (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH as an incoming student.
UZH Module Code: 22MO0142

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

At most one of the two course units
401-8905-00L Financial Engineering (University of Zurich)
401-8908-00L Continuous Time Quantitative Finance (University of Zurich)
is eligible for credits.

6 credits

University lecturers

Abstract

This lecture is intended for students who would like to learn more on equity derivatives modelling and pricing.

Learning objective

Quantitative models for European option pricing (including stochastic
volatility and jump models), volatility and variance derivatives,
American and exotic options.

Content

After introducing fundamental
concepts of mathematical finance including no-arbitrage, portfolio
replication and risk-neutral measure, we will present the main models
that can be used for pricing and hedging European options e.g. Black-
Scholes model, stochastic and jump-diffusion models, and highlight their
assumptions and limitations. We will cover several types of derivatives
such as European and American options, Barrier options and Variance-
Swaps. Basic knowledge in probability theory and stochastic calculus is
required. Besides attending class, we strongly encourage students to
stay informed on financial matters, especially by reading daily
financial newspapers such as the Financial Times or the Wall Street
Journal.

Lecture notes

Script.

Prerequisites / Notice

Basic knowledge of probability theory and stochastic calculus.
Asset Pricing.

401-8915-00L

Advanced Financial Economics (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH as an incoming student.
UZH Module Code: 22MO0016

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

6 credits

University lecturers

Abstract

Portfolio Theory, CAPM, Financial Derivatives, Incomplete Markets, Corporate Finance, Behavioural Finance, Evolutionary Finance

Learning objective

Students should get familiar with the cornerstones of modern financial economics.

Prerequisites / Notice

This course replaces "Advanced Financial Economics" (MFOEC105), which will be discontinued. Students who have taken "Advanced Financial Economics" (MFOEC105) in the past, are not allowed to book this course "Advanced Financial Economics" (MFOEC206).

There will be a podcast for this lecture.

401-8916-00L

Advanced Corporate Finance II (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH as an incoming student.
UZH Module Code: 22MO0173

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

3 credits

University lecturers

Abstract

To provide the students with good understanding of the problems and issues in corporate finance.

Learning objective

To provide the students with good understanding of the problems and issues in corporate finance.

Content

The following topics are covered in this course: the role of information and incentives in determining the forms of financing a firm chooses; hedging; venture capital; initial public offerings; investment in very large projects; the setting up of a "bad" bank; the securitisation of commercial and industrial loans; the transfer of catastrophe risk to financial markets; agency in insurance; and dealing with a run on an insurance company.

Lecture notes

Literature

See: http://www.isb.uzh.ch/institut/staff/habib.michel/teaching/

Finance (only conditionally recognised)

Number

Title

Type

ECTS

Hours

Lecturers

401-8908-00L

Continuous Time Quantitative Finance (University of Zurich)

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH as an incoming student.
UZH Module Code: 22MO0125

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html

At most one of the two course units
401-8905-00L Financial Engineering (University of Zurich)
401-8908-00L Continuous Time Quantitative Finance (University of Zurich)
is eligible for credits.
For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.

3 credits

University lecturers

Abstract

American Options, Stochastic Volatility, Lévy Processes and Option Pricing, Exotic Options, Transaction Costs and Real Options.

Learning objective

The course focuses on the theoretical foundations of modern derivative pricing. It aims at deriving and explaining important option pricing models by relying on some mathematical tools of continuous time finance.
A particular focus on jump processes is given. The introduction of possible financial crashes is now essential in some models and a clear understanding of Poisson processes is therefore important. A standard background in stochastic calculus is required.

Content

Stochastic volatility models
Itô's formula and Girsanov theorem for jump-diffusion processes
The pricing of options in presence of possible discontinuities
Exotic options
Transaction costs

Lecture notes

See: http://www.isb.uzh.ch/institut/staff/chesney.marc/teaching/

Literature

See: http://www.isb.uzh.ch/institut/staff/chesney.marc/teaching/

Prerequisites / Notice

This course replaces "Continuous Time Quantitative Finance" (MFOEC108), which will be discontinued. Students who have taken "Continuous Time Quantitative Finance" (MFOEC108) in the past, are not allowed to book this course "Continuous Time Quantitative Finance" (MFOEC204).

Image Processing and Computer Vision

Number

Title

Type

ECTS

Hours

Lecturers

102-0617-01L

Methodologies for Image Processing of Remote Sensing Data

3 credits

I. Hajnsek, O. Frey, L. Huang

Abstract

The aim of this course is to get an overview of several methodologies/algorithms for analysis of different sensor specific information products. It is focused at students that like to deepen their knowledge and understanding of remote sensing for environmental applications.

Learning objective

The course is divided into two main parts, starting with a brief introduction to remote sensing imaging (4 lectures), and is followed by an introduction to different methodologies (8 lectures) for the quantitative estimation of bio-/geo-physical parameters. The main idea is to deepen the knowledge in remote sensing tools in order to be able to understand the information products, with respect to quality and accuracy.

Content

Each lecture will be composed of two parts:
Theory: During the first hour, we go trough the main concepts needed to understand the specific algorithm.
Practice: During the second hour, the student will test/develop the actual algorithm over some real datasets using Matlab. The student will not be asked to write all the code from scratch (especially during the first lectures), but we will provide some script with missing parts or pseudo-code. However, in the later lectures the student is supposed to build up some working libraries.

Lecture notes

Handouts for each topic will be provided.

Literature

Suggested readings:
T. M. Lillesand, R.W. Kiefer, J.W. Chipman, Remote Sensing and Image Interpretation, John Wiley & Sons Verlag, 2008
J. R. Jensen, Remote Sensing of the Environment: An Earth Resource Perspective, Prentice Hall Series in Geograpic Information Science, 2000

Prerequisites / Notice

At the end of the course (last course day) there will a written exam.

Competencies

Subject-specific Competencies	Concepts and Theories	fostered
	Techniques and Technologies	fostered
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	fostered
Personal Competencies	Critical Thinking	fostered

227-0391-00L

Medical Image Analysis

Basic knowledge of computer vision would be helpful.

3 credits

E. Konukoglu, E. Erdil, M. A. Reyes Aguirre

Abstract

It is the objective of this lecture to introduce the basic concepts used
in Medical Image Analysis. In particular the lecture focuses on shape
representation schemes, segmentation techniques, machine learning based predictive models and various image registration methods commonly used in Medical Image Analysis applications.

Learning objective

This lecture aims to give an overview of the basic concepts of Medical Image Analysis and its application areas.

Prerequisites / Notice

Prerequisites:
Basic concepts of mathematical analysis and linear algebra.

Preferred:
Basic knowledge of computer vision and machine learning would be helpful.

The course will be held in English.

227-0396-00L

EXCITE Interdisciplinary Summer School on Bio-Medical Imaging

The school admits 60 MSc or PhD students with backgrounds in biology, chemistry, mathematics, physics, computer science or engineering based on a selection process.

Students have to apply for acceptance. To apply a curriculum vitae and an application letter need to be submitted.
Further information can be found at: www.excite.ethz.ch.

4 credits

S. Kozerke, B. Menze, M. P. Wolf, U. Ziegler Lang

Abstract

Two-week summer school organized by EXCITE (Center for EXperimental & Clinical Imaging TEchnologies Zurich) on biological and medical imaging. The course covers X-ray imaging, magnetic resonance imaging, nuclear imaging, ultrasound imaging, optoacoustic imaging, infrared and optical microscopy, electron microscopy, image processing and analysis.

Learning objective

Students understand basic concepts and implementations of biological and medical imaging. Based on relative advantages and limitations of each method they can identify preferred procedures and applications. Common foundations and conceptual differences of the methods can be explained.

Content

Two-week summer school on biological and medical imaging. The course covers concepts and implementations of X-ray imaging, magnetic resonance imaging, nuclear imaging, ultrasound imaging, optoacoustic imaging, infrared and optical microscopy and electron microscopy. Multi-modal and multi-scale imaging and supporting technologies such as image analysis and modeling are discussed. Dedicated modules for physical and life scientists taking into account the various backgrounds are offered.

Lecture notes

Presentation slides, Web links

Prerequisites / Notice

The school admits 60 MSc or PhD students with backgrounds in biology, chemistry, mathematics, physics, computer science or engineering based on a selection process. To apply a curriculum vitae, a statement of purpose and applicants references need to be submitted. Further information can be found at: http://www.excite.ethz.ch/education/summer-school.html

Information and Communication Technology

Number

Title

Type

ECTS

Hours

Lecturers

227-0420-00L

Information Theory II

6 credits

A. Lapidoth, S. M. Moser

Abstract

This course builds on Information Theory I. It introduces additional topics in single-user communication, connections between Information Theory and Statistics, and Network Information Theory.

Learning objective

The course's objective is to introduce the students to additional information measures and to equip them with the tools that are needed to conduct research in Information Theory as it relates to Communication Networks and to Statistics.

Content

Sanov's Theorem, Rényi entropy and guessing, differential entropy, maximum entropy, the Gaussian channel, the entropy-power inequality, the broadcast channel, the multiple-access channel, Slepian-Wolf coding, the Gelfand-Pinsker problem, and Fisher information.

Lecture notes

n/a

Literature

T.M. Cover and J.A. Thomas, Elements of Information Theory, second edition, Wiley 2006

Prerequisites / Notice

Basic introductory course on Information Theory.

227-0104-00L

Communication and Detection Theory

6 credits

A. Lapidoth

Abstract

This course teaches the foundations of modern digital communications and detection theory. Topics include the geometry of the space of energy-limited signals; the baseband representation of passband signals, spectral efficiency and the Nyquist Criterion; the power and power spectral density of PAM and QAM; hypothesis testing; Gaussian stochastic processes; and detection in white Gaussian noise.

Learning objective

This is an introductory class to the field of wired and wireless communication. It offers a glimpse at classical analog modulation (AM, FM), but mainly focuses on aspects of modern digital communication, including modulation schemes, spectral efficiency, power budget analysis, block and convolu- tional codes, receiver design, and multi- accessing schemes such as TDMA, FDMA and Spread Spectrum.

Content

- Baseband representation of passband signals.
- Bandwidth and inner products in baseband and passband.
- The geometry of the space of energy-limited signals.
- The Sampling Theorem as an orthonormal expansion.
- Sampling passband signals.
- Pulse Amplitude Modulation (PAM): energy, power, and power spectral density.
- Nyquist Pulses.
- Quadrature Amplitude Modulation (QAM).
- Hypothesis testing.
- The Bhattacharyya Bound.
- The multivariate Gaussian distribution
- Gaussian stochastic processes.
- Detection in white Gaussian noise.

Lecture notes

n/a

Literature

A. Lapidoth, A Foundation in Digital Communication, Cambridge University Press, 2nd edition (2017)

227-0427-10L

Model-Based Estimation and Signal Analysis

6 credits

H.‑A. Loeliger

Abstract

The course develops a selection of topics pivoting around state space models, factor graphs, and pertinent algorithms for estimation, model fitting, and learning.

Learning objective

The course develops a selection of topics pivoting around state space methods, factor graphs, and pertinent algorithms:
- hidden-Markov models
- factor graphs and message passing algorithms
- linear state space models, Kalman filtering, and recursive least squares
- Gibbs sampling, particle filter
- recursive local polynomial fitting for signal analysis
- parameter learning by expectation maximization
- linear-model fitting beyond least squares: sparsity, Lp-fitting and regularization, jumps
- binary, M-level, and half-plane constraints in control and communications

Lecture notes

Prerequisites / Notice

Solid mathematical foundations (especially in probability, estimation, and linear algebra) as provided by the course "Introduction to Estimation and Machine Learning".

227-0438-00L

Wireless Communications

6 credits

Material Modelling and Simulation

C. Studer

Abstract

The course teaches the fundamentals of wireless communication as well as state-of-the-art technologies used in modern wireless systems. The main topics are wireless channels, data detection, multi-antenna and multi-user communication, information theory of wireless systems, and emerging technologies. The exercises cover theoretical aspects as well as modeling of wireless systems using MATLAB.

Learning objective

After attending the lectures, participating in the exercise sessions, and working on the homework problems (which include MATLAB coding assignments), the students will be able to:
• understand the key principles and trade-offs of modern wireless system design
• analyze wireless channels and existing wireless communication systems
• apply the fundamental principles to design new wireless communication systems
• create software-based simulation frameworks to model complex wireless systems

Content

This course focuses on the fundamentals of modern wireless communication systems. The course begins with the basics of wireless channels and discusses the main building blocks of modern wireless transceivers. The topics include:
• Wireless channels, multi-path propagation, and de/modulation
• Geometrical and statistical channel models
• Delay spread and coherence bandwidth; Doppler spread and coherence time
• Diversity techniques (time, frequency, space, and multi-user) and space-time coding
• Orthogonal frequency-division multiplexing (OFDM)
• Multi-antenna and multiple-input multiple-output (MIMO) technologies
• MIMO data detection and beamforming
• Multi-user (MU) communication
• Basic information theory for wireless channels
• Basic forward error correction schemes
• Emerging topics: millimeter-wave communication and massive MU-MIMO
The exercises cover theoretical aspects as well as the basics of software-based communication-system-modeling in MATLAB and Monte-Carlo simulation techniques.

Lecture notes

Lecture notes are written in English and will be provided at the beginning of semester.

Literature

A set of handouts covering digital communication basics and mathematical preliminaries will be available on Moodle. For further reading, we recommend the following books:
• D. Tse and P. Viswanath, “Fundamentals of Wireless Communication,” Cambridge University Press, 2005
• J. G. Proakis and M. Salehi, “Digital Communications,” McGraw-Hill, 2008, 5th Ed.
• T. M. Cover and J. A. Thomas, "Elements of Information Theory," Wiley, 1991

Prerequisites / Notice

This class will be taught in English. The oral exam will be in English. The oral exam will include questions on the topics covered in all the lectures, supplementary reading material, and exercises.
The prerequisites for this course are basic knowledge of digital communications, random processes, and detection theory.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	fostered
	Problem-solving	assessed

Machine Learning

Number

Title

Type

ECTS

Hours

Lecturers

263-4400-00L

Advanced Graph Algorithms and Optimization

10 credits

3V + 3U + 3A

R. Kyng, M. Probst

Abstract

This course will cover a number of advanced topics in optimization and graph algorithms.

Learning objective

Content

Prerequisites / Notice

261-5110-00L

Optimization for Data Science

10 credits

3V + 2U + 4A

B. Gärtner, N. He

Abstract

This course provides an in-depth theoretical treatment of optimization methods that are relevant in data science.

Learning objective

Content

Prerequisites / Notice

252-0526-00L

Statistical Learning Theory

8 credits

3V + 2U + 2A

J. M. Buhmann

Abstract

The course covers advanced methods of statistical learning:

- Variational methods and optimization.
- Deterministic annealing.
- Clustering for diverse types of data.
- Model validation by information theory.

Learning objective

The course surveys recent methods of statistical learning. The fundamentals of machine learning, as presented in the courses "Introduction to Machine Learning" and "Advanced Machine Learning", are expanded from the perspective of statistical learning.

Content

- Variational methods and optimization. We consider optimization approaches for problems where the optimizer is a probability distribution. We will discuss concepts like maximum entropy, information bottleneck, and deterministic annealing.

- Clustering. This is the problem of sorting data into groups without using training samples. We discuss alternative notions of "similarity" between data points and adequate optimization procedures.

- Model selection and validation. This refers to the question of how complex the chosen model should be. In particular, we present an information theoretic approach for model validation.

- Statistical physics models. We discuss approaches for approximately optimizing large systems, which originate in statistical physics (free energy minimization applied to spin glasses and other models). We also study sampling methods based on these models.

Lecture notes

A draft of a script will be provided. Lecture slides will be made available.

Literature

Hastie, Tibshirani, Friedman: The Elements of Statistical Learning, Springer, 2001.

L. Devroye, L. Gyorfi, and G. Lugosi: A probabilistic theory of pattern recognition. Springer, New York, 1996

Prerequisites / Notice

Knowledge of machine learning (introduction to machine learning and/or advanced machine learning)
Basic knowledge of statistics.

263-3710-00L

Machine Perception

8 credits

3V + 2U + 2A

O. Hilliges, J. Song

Abstract

Recent developments in neural networks have drastically advanced the performance of machine perception systems in a variety of areas including computer vision, robotics, and human shape modeling This course is a deep dive into deep learning algorithms and architectures with applications to a variety of perceptual and generative tasks.

Learning objective

Students will learn about fundamental aspects of modern deep learning approaches for perception and generation. Students will learn to implement, train and debug their own neural networks and gain a detailed understanding of cutting-edge research in learning-based computer vision, robotics, and shape modeling. The optional final project assignment will involve training a complex neural network architecture and applying it to a real-world dataset.

The core competency acquired through this course is a solid foundation in deep-learning algorithms to process and interpret human-centric signals. In particular, students should be able to develop systems that deal with the problem of recognizing people in images, detecting and describing body parts, inferring their spatial configuration, performing action/gesture recognition from still images or image sequences, also considering multi-modal data, among others.

Content

We will focus on teaching: how to set up the problem of machine perception, the learning algorithms, network architectures, and advanced deep learning concepts in particular probabilistic deep learning models.

The course covers the following main areas:
I) Foundations of deep learning.
II) Advanced topics like probabilistic generative modeling of data (latent variable models, generative adversarial networks, auto-regressive models, invertible neural networks, diffusion models).
III) Deep learning in computer vision, human-computer interaction, and robotics.

Specific topics include:
I) Introduction to Deep Learning:
a) Neural Networks and training (i.e., backpropagation)
b) Feedforward Networks
c) Timeseries modelling (RNN, GRU, LSTM)
d) Convolutional Neural Networks
II) Advanced topics:
a) Latent variable models (VAEs)
b) Generative adversarial networks (GANs)
c) Autoregressive models (PixelCNN, PixelRNN, TCN, Transformer)
d) Invertible Neural Networks / Normalizing Flows
e) Coordinate-based networks (neural implicit surfaces, NeRF)
f) Diffusion models
III) Applications in machine perception and computer vision:
a) Fully Convolutional architectures for dense per-pixel tasks (i.e., instance segmentation)
b) Pose estimation and other tasks involving human activity
c) Neural shape modeling (implicit surfaces, neural radiance fields)
d) Deep Reinforcement Learning and Applications in Physics-Based Behavior Modeling

Literature

Deep Learning
Book by Ian Goodfellow and Yoshua Bengio

Prerequisites / Notice

This is an advanced grad-level course that requires a background in machine learning. Students are expected to have a solid mathematical foundation, in particular in linear algebra, multivariate calculus, and probability. The course will focus on state-of-the-art research in deep learning and will not repeat the basics of machine learning

Please take note of the following conditions:
1) Students must have taken the exam in Machine Learning (252-0535-00) or have acquired equivalent knowledge
2) All practical exercises will require basic knowledge of Python and will use libraries such as Pytorch, scikit-learn, and scikit-image. We will provide introductions to Pytorch and other libraries that are needed but will not provide introductions to basic programming or Python.

The following courses are strongly recommended as prerequisites:
* "Visual Computing" or "Computer Vision"

The course will be assessed by a final written examination in English. No course materials or electronic devices can be used during the examination. Note that the examination will be based on the contents of the lectures, the associated reading materials, and the exercises.

The exam will be a 3-hour end-of-term exam and take place at the end of the teaching period.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	assessed
	Project Management	assessed
Social Competencies	Communication	fostered
	Cooperation and Teamwork	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	fostered
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered
	Self-direction and Self-management	fostered

Number

Title

Type

ECTS

Hours

Lecturers

327-2201-00L

Transport Phenomena II

5 credits

J. Vermant

Abstract

Numerical and analytical methods for real-world "Transport Phenomena"; atomistic understanding of transport properties based on kinetic theory and mesoscopic models; fundamentals, applications, and simulations

Learning objective

The teaching goals of this course are on five different levels:
(1) Deep understanding of fundamentals: kinetic theory, mesoscopic models, ...
(2) Ability to use the fundamental concepts in applications
(3) Insight into the role of boundary conditions
(4) Knowledge of a number of applications
(5) Flavor of numerical techniques: finite elements, lattice Boltzmann, ...

Content

Thermodynamics of Interfaces
Interfacial Balance Equations
Interfacial Force-Flux Relations
Polymer Processing
Transport Around a Sphere
Refreshing Topics in Equilibrium Statistical Mechanics
Kinetic Theory of Gases
Kinetic Theory of Polymeric Liquids
Transport in Biological Systems
Dynamic Light Scattering

Lecture notes

The course is based on the book D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018)

Literature

1. D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018)
2. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd Ed. (Wiley, 2001)
3. Deen,W. Analysis of Transport Phenomena, Oxford University Press, 2012
4. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287

Prerequisites / Notice

Complex numbers. Vector analysis (integrability; Gauss' divergence theorem). Laplace and Fourier transforms. Ordinary differential equations (basic ideas). Linear algebra (matrices; functions of matrices; eigenvectors and eigenvalues; eigenfunctions). Probability theory (Gaussian distributions; Poisson distributions; averages; moments; variances; random variables). Numerical mathematics (integration). Statistical thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms; Gibbs' phase rule; ergodicity; partition functions; Einstein's fluctuation theory). Linear irreversible thermodynamics (forces and fluxes; Fourier's, Newton's and Fick's laws for fluxes). Hydrodynamics (local equilibrium; balance equations for mass, momentum, energy and entropy). Programming and simulation techniques (Matlab, Monte Carlo simulations).

151-0515-00L

Continuum Mechanics 2

4 credits

E. Mazza, R. M. Hopf

Abstract

An introduction to finite deformation continuum mechanics and nonlinear material behavior. Coverage of basic tensor- manipulations and calculus, descriptions of kinematics, and balance laws . Discussion of invariance principles and mechanical response functions for elastic materials.

Learning objective

To provide a modern introduction to the foundations of continuum mechanics and prepare students for further studies in solid
mechanics and related disciplines.

Content

1. Tensors: algebra, linear operators
2. Tensors: calculus
3. Kinematics: motion, gradient, polar decomposition
4. Kinematics: strain
5. Kinematics: rates
6. Global Balance: mass, momentum
7. Stress: Cauchy's theorem
8. Stress: alternative measures
9. Invariance: observer
10. Material Response: elasticity

Lecture notes

None.

Literature

Recommended texts:
(1) Nonlinear solid mechanics, G.A. Holzapfel (2000).
(2) An introduction to continuum mechanics, M.B. Rubin (2003).

Quantum Chemistry

Number

Title

Type

ECTS

Hours

Lecturers

529-0474-00L

Quantum Chemistry

6 credits

M. Reiher, J. P. Unsleber, T. Weymuth

Abstract

Introduction into the basic concepts of electronic structure theory and into numerical methods of quantum chemistry. Exercise classes are designed to deepen the theory; practical case studies using quantum chemical software to provide a 'hands-on' expertise in applying these methods.

Learning objective

Nowadays, chemical research can be carried out in silico, an intellectual achievement for which Pople and Kohn have been awarded the Nobel prize of the year 1998. This lecture shows how that has been accomplished. It works out the many-particle theory of many-electron systems (atoms and molecules) and discusses its implementation into computer programs. A complete picture of quantum chemistry shall be provided that will allow students to carry out such calculations on molecules (for accompanying experimental work in the wet lab or as a basis for further study of the theory).

Content

Basic concepts of many-particle quantum mechanics. Derivation of the many-electron theory for atoms and molecules; starting with the harmonic approximation for the nuclear problem and with Hartree-Fock theory for the electronic problem to Moeller-Plesset perturbation theory and configuration interaction and to coupled cluster and multi-configurational approaches. Density functional theory. Case studies using quantum mechanical software.

Lecture notes

Hand-outs in German will be provided for each lecture (they are supplemented by (computer) examples that continuously illustrate how the theory works).

All information regarding this course, including links to the online streaming, will be available on this web page:
https://reiher.ethz.ch/courses-and-seminars/exercises/QC_2023.html

Literature

Textbooks on Quantum Chemistry:
F.L. Pilar, Elementary Quantum Chemistry, Dover Publications
I.N. Levine, Quantum Chemistry, Prentice Hall

Hartree-Fock in basis set representation:
A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill

Textbooks on Computational Chemistry:
F. Jensen, Introduction to Computational Chemistry, John Wiley & Sons
C.J. Cramer, Essentials of Computational Chemistry, John Wiley & Sons

Prerequisites / Notice

Basic knowledge in quantum mechanics (e.g. through course physical chemistry III - quantum mechanics) required

Systems Design

Number

Title

Type

ECTS

Hours

Lecturers

151-0530-00L

Nonlinear Dynamics and Chaos II

4 credits

G. Haller

Abstract

The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems

Learning objective

The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis.

Content

Lecture notes

Handwritten instructor's notes and typed lecture notes will be downloadable from Moodle.

Literature

Books will be recommended in class

Prerequisites / Notice

Nonlinear Dynamics I (151-0532-00) or equivalent

363-0588-00L

Complex Networks

4 credits

G. Casiraghi

Abstract

The course provides an overview of the methods and abstractions used in (i) the quantitative study of complex networks, (ii) empirical network analysis, (iii) the study of dynamical processes in networked systems, (iv) the analysis of robustness of networked systems, (v) the study of network evolution, and (vi) data mining techniques for networked data sets.

Learning objective

* the network approach to complex systems, where actors are represented as nodes and interactions are represented as links
* learn about structural properties of classes of networks
* learn about feedback mechanism in the formation of networks
* learn about statistical inference and data mining techniques for data on networked systems
* learn methods and abstractions used in the growing literature on complex networks

Content

Networks matter! This holds for social and economic systems, for technical infrastructures as well as for information systems. Increasingly, these networked systems are outside the control of a centralized authority but rather evolve in a distributed and self-organized way. How can we understand their evolution and what are the local processes that shape their global features? How does their topology influence dynamical processes like diffusion? And how can we characterize the importance of specific nodes?

This course provides a systematic answer to such questions, by developing methods and tools which can be applied to networks in diverse areas like infrastructure, communication, information systems, biology or (online) social networks. In a network approach, agents in such systems (like e.g. humans, computers, documents, power plants, biological or financial entities) are represented as nodes, whereas their interactions are represented as links.

The first part of the course, "Introduction to networks: basic and advanced metrics", describes how networks can be represented mathematically and how the properties of their link structures can be quantified empirically.

In a second part "Stochastic Models of Complex Networks" we address how analytical statements about crucial properties like connectedness or robustness can be made based on simple macroscopic stochastic models without knowing the details of a topology.

In the third part we address "Dynamical processes on complex networks". We show how a simple model for a random walk in networks can give insights into the authority of nodes, the efficiency of diffusion processes as well as the existence of community structures.

A fourth part "Network Optimisation and Inference" introduces models for the emergence of complex topological features which are due to stochastic optimization processes, as well as statistical methods to detect patterns in large data sets on networks.

In a fifth part, we address "Network Dynamics", introducing models for the emergence of complex features that are due to (i) feedback phenomena in simple network growth processes or (iii) order correlations in systems with highly dynamic links.

A final part "Research Trends" introduces recent research on the application of data mining and machine learning techniques to relational data.

Lecture notes

The lecture slides are provided as handouts - including notes and literature sources - to registered students only.
All material is to be found on Moodle.

Literature

See handouts. Specific literature is provided for download - for registered students, only.

Prerequisites / Notice

There are no pre-requisites for this course. Self-study tasks (to be solved analytically and by means of computer simulations) are provided as home work. Weekly exercises (45 min) are used to discuss selected solutions. Active participation in the exercises is strongly suggested for a successful completion of the final exam.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	assessed

363-0543-00L

Agent-Based Modelling of Social Systems

3 credits

G. Vaccario

Abstract

Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems. The course is based on formal models of agents and their interactions. Computer simulations using Python allow the quantitative analysis of a wide range of social phenomena, e.g. cooperation and competition, opinion dynamics, spatial interactions and behaviour in social networks.

Learning objective

A successful participant of this course is able to
- understand the rationale of agent-based models of social systems
- understand the relation between rules implemented at the individual level and the emerging behavior at the global level
- learn to choose appropriate model classes to characterize different social systems
- grasp the influence of agent heterogeneity on the model output
- efficiently implement agent-based models using Python and visualize the output

Content

This full-featured course on agent-based modeling (ABM) allows participants with no prior expertise to understand concepts, methods and tools of ABM, to apply them in their master or doctoral thesis. We focus on a formal description of agents and their interactions, to allow for a suitable implementation in computer simulations. Given certain rules for the agents, we are interested to model their collective dynamics on the systemic level.

Agent-based modeling is introduced as a bottom-up approach to understand the complex dynamics of social systems.
Agents represent the basic constituents of such systems. The are described by internal states or degrees of freedom (opinions, strategies, etc.), the ability to perceive and change their environment, and the ability to interact with other agents. Their individual (microscopic) actions and interactions with other agents, result in macroscopic (collective, system) dynamics with emergent properties, which we want to understand and to analyze.

The course is structured in three main parts. The first two parts introduce two main agent concepts - Boolean agents and Brownian agents, which differ in how the internal dynamics of agents is represented. Boolean agents are characterized by binary internal states, e.g. yes/no opinion, while Brownian agents can have a continuous spectrum of internal states, e.g. preferences and attitudes. The last part introduces models in which agents interact in physical space, e.g. migrate or move collectively.

Throughout the course, we will discuss a wide variety of application areas, such as:
- opinion dynamics and social influence,
- cooperation and competition,
- online social networks,
- systemic risk
- emotional influence and communication
- swarming behavior
- spatial competition

While the lectures focus on the theoretical foundations of agent-based modeling, weekly exercise classes provide practical skills. Using the Python programming language, the participants implement agent-based models in guided and in self-chosen projects, which they present and jointly discuss.

Lecture notes

The lecture slides will be available on the Moodle platform, for registered students only.

Literature

See handouts. Specific literature is provided for download, for registered students only.

Prerequisites / Notice

Participants of the course should have some background in mathematics and an interest in formal modeling and in computer simulations, and should be motivated to learn about social systems from a quantitative perspective.

Prior knowledge of Python is not necessary.

Self-study tasks are provided as home work for small teams (2-4 members).
Weekly exercises (45 min) are used to discuss the solutions and guide the students.

The examination will account for 70% of the grade and will be conducted electronically. The "closed book" rule applies: no books, no summaries, no lecture materials. The exam questions and answers will be only in English. The use of a paper-based dictionary is permitted.
The group project to be handed in at the beginning of July will count 30% to the final grade.

Theoretical Physics

In the Master's programme in Applied Mathematics 402-0204-00L Electrodynamics is eligible as a course unit in the application area Theoretical Physics, but only if 402-0224-00L Theoretical Physics wasn't or isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.

Number

Title

Type

ECTS

Hours

Lecturers

402-0812-00L

Computational Statistical Physics

8 credits

M. Krstic Marinkovic

Abstract

Computer simulation methods in statistical physics. Classical Monte-Carlo-simulations: finite-size scaling, cluster algorithms, histogram-methods, renormalization group. Application to Boltzmann machines. Simulation of non-equilibrium systems.

Molecular dynamics simulations: long range interactions, Ewald summation, discrete elements, parallelization.

Learning objective

The lecture will give a deeper insight into computer simulation methods in statistical physics. Thus, it is an ideal continuation of the lecture
"Introduction to Computational Physics" of the autumn semester. In the first part students learn to apply the following methods: Classical Monte Carlo-simulations, finite-size scaling, cluster algorithms, histogram-methods, renormalization group. Moreover, students learn about the application of statistical physics methods to Boltzmann machines and how to simulate non-equilibrium systems.

In the second part, students apply molecular dynamics simulation methods. This part includes long range interactions, Ewald summation and discrete elements.

Content

Computer simulation methods in statistical physics. Classical Monte-Carlo-simulations: finite-size scaling, cluster algorithms, histogram-methods, renormalization group. Application to Boltzmann machines. Simulation of non-equilibrium systems. Molecular dynamics simulations: long range interactions, Ewald summation, discrete elements, parallelization.

Lecture notes

Lecture notes and slides are available online and will be distributed if desired.

Literature

Literature recommendations and references are included in the lecture notes.

Prerequisites / Notice

Some basic knowledge about statistical physics, classical mechanics and computational methods is recommended.

Competencies

	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Problem-solving	assessed
	Project Management	fostered
Social Competencies	Communication	assessed
	Cooperation and Teamwork	fostered
	Leadership and Responsibility	fostered
	Sensitivity to Diversity	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	assessed
	Critical Thinking	assessed
	Integrity and Work Ethics	fostered
	Self-direction and Self-management	fostered

402-0810-00L

Computational Quantum Physics

Special Students UZH must book the module PHY522 directly at UZH.

8 credits

M. H. Fischer

Abstract

This course provides an introduction to simulation methods for quantum systems. Starting from the one-body problem, a special emphasis is on quantum many-body problems, where we cover both approximate methods (Hartree-Fock, density functional theory) and exact methods (exact diagonalization, matrix product states, and quantum Monte Carlo methods).

Learning objective

Through lectures and practical programming exercises, after this course:
Students are able to describe the difficulties of quantum mechanical simulations.
Students are able to explain the strengths and weaknesses of the methods covered.
Students are able to select an appropriate method for a given problem.
Students are able to implement basic versions of all algorithms discussed.

Lecture notes

A script for this lecture will be provided.

Literature

A list of additional references will be provided in the script.

Prerequisites / Notice

A basic knowledge of quantum mechanics, numerical tools (numerical differentiation and integration, linear solvers, eigensolvers, root solvers, optimization), and a programming language (for the teaching assignments, you are free to choose your preferred one).

402-0206-00L

Quantum Mechanics II

10 credits

C. Anastasiou

Abstract

Learning objective

Content

Literature

Prerequisites / Notice

Basic knowledge of single-particle Quantum Mechanics

402-0871-00L

Solid State Theory

UZH students are not allowed to register this course unit at ETH. They must book the module PHY411 directly at UZH.

10 credits

M. Sigrist

Abstract

The course is addressed to students in experimental and theoretical condensed matter physics and provides a theoretical introduction to a variety of important concepts used in this field.

Learning objective

The course provides a theoretical frame for the understanding of basic pinciples in solid state physics. Such a frame includes the topics of symmetries, band structures, many body interactions, Landau Fermi-liquid theory, and specific topics such as transport, Quantum Hall effect and magnetism. The exercises illustrate the various themes in the lecture and help to develop problem-solving skills. The student understands basic concepts in solid state physics and is able to solve simple problems. No diagrammatic tools will be used.

Content

The course is addressed to students in experimental and theoretical condensed matter physics and provides a theoretical introduction to a variety of important concepts used in this field. The following subjects will be covered: Symmetries and their handling via group theoretical concepts, electronic structure in crystals, insulators-semiconductors-metals, phonons, interaction effects, (un-)screened Fermi-liquids, linear response theory, collective modes, screening, transport in semiconductors and metals, magnetism, Mott-insulators, quantum-Hall effect.

Lecture notes

in English

402-0844-00L

Quantum Field Theory II

UZH students are not allowed to register this course unit at ETH. They must book the corresponding module directly at UZH.

10 credits

» Electives Theoretical Physics

A. Lazopoulos

Abstract

The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-abelian gauge theories.

Learning objective

Content

Literature

402-0394-00L

Theoretical Cosmology

10 credits

4V + 2U

L. Senatore

Abstract

This is the second of a two course series which starts with "General Relativity" and continues in the spring with "Theoretical Astrophysics and Cosmology", where the focus will be on applying general relativity to cosmology as well as developing the modern theory of structure formation in a cold dark matter Universe.

Learning objective

Learning the fundamentals of modern physical cosmology. This
entails understanding the physical principles behind the description
of the homogeneous Universe on large scales in the first part of the
course, and moving on to the inhomogeneous Universe model where
perturbation theory is used to study the development of structure
through gravitational instability in the second part of the course.
Modern notions of dark matter and dark energy will also be introduced and discussed.

Content

The course will cover the following topics:
- Homogeneous cosmology
- Thermal history of the universe, recombination, baryogenesis and nucleosynthesis
- Dark matter and Dark Energy
- Inflation
- Perturbation theory: Relativistic and Newtonian
- Model of structure formation and initial conditions from Inflation
- Cosmic microwave background anisotropies
- Spherical collapse and galaxy formation
- Large scale structure and cosmological probes

Lecture notes

In 2021, the lectures will be live-streamed online at ETH from the Room HPV G5 at the lecture hours. The recordings will be available at the ETH website. The detailed information will be provided by the course website and the SLACK channel.

Literature

Suggested textbooks:
H.Mo, F. Van den Bosch, S. White: Galaxy Formation and Evolution
S. Carroll: Space-Time and Geometry: An Introduction to General Relativity
S. Dodelson: Modern Cosmology
Secondary textbooks:
S. Weinberg: Gravitation and Cosmology
V. Mukhanov: Physical Foundations of Cosmology
E. W. Kolb and M. S. Turner: The Early Universe
N. Straumann: General relativity with applications to astrophysics
A. Liddle and D. Lyth: Cosmological Inflation and Large Scale Structure

Prerequisites / Notice

Knowledge of General Relativity is recommended.

Transportation Science

Number

Title

Type

ECTS

Hours

Lecturers

101-0478-00L

Survey Methods and Discrete Choice Analysis

6 credits

B. Schmid

Abstract

Comprehensive introduction to survey methods in transport planning and modeling of travel behavior, using advanced discrete choice models.

Learning objective

Enabling the student to understand and apply the various measurement approaches and models of travel behaviour research.

Content

Behavioral model and measurement; travel diary, design process, hypothetical markets, parameter estimation, econometrics, pattern of travel behaviour, market segments, simulation, advanced discrete choice models

Lecture notes

Various papers and notes are distributed during the course.

Literature

The course heavily builds on Train, K. E. (2009) Discrete Choice Methods with Simulation, Cambridge University Press.

Prerequisites / Notice

This introduction in survey methods and (advanced) discrete choice modelling requires basic programming knowledge in the statistical software R. Solid understanding of statistical modeling and econometrics is of advantage.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	fostered
	Media and Digital Technologies	fostered
	Problem-solving	assessed
	Project Management	fostered
Social Competencies	Communication	fostered
	Cooperation and Teamwork	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	fostered
	Critical Thinking	assessed
	Integrity and Work Ethics	fostered
	Self-direction and Self-management	fostered

Seminars and Semester Papers

Seminars

NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. For the seminars with pecial selection procedure everybody is waitlisted first when he/she tries to register for a seminar in myStudies.
Moreover: At most 2 mathematics seminars can be chosen per semester. In case you need to attend 3 seminars in this semester, please take contact with the Study Administration (email: studiensekretariat@math.ethz.ch ).

Number

Title

Type

ECTS

Hours

Lecturers

401-3160-23L

q-Series, Difference Equations and Computer Algebra

Number of participants limited to 12.

4 credits

G. Felder

Abstract

We will study q-series and their relation with difference calculus, combinatorics, classical analysis and number theory. We will also develop software to deal with q-series and possibly generate new conjectures.

Learning objective

Literature

George E. Andrews, q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, AMS 1985
Victor Kac, Pokman Cheung, Quantum Calculus, Springer 2002

401-3300-23L

Riemann Surfaces

Number of participants limited to 15.

4 credits

A. Cela

Abstract

Learning objective

The goal of the course is to learn about compact Riemann Surfaces, with main reference chapter 2 of Forster's book 'Lectures on Riemann Surfaces'. Time permitting and depending on the interest of the participants we will also cover more advanced results about Riemann Surfaces from Arbarello-Cornalba-Griffiths-Harris book 'Geometry of Algebraic curves' Vol.1.

Competencies

Subject-specific Competencies

Concepts and Theories

assessed

401-3530-23L

Introduction to Morse Theory

Number of participants limited to 24.

4 credits

Y. Kawamoto

Abstract

In this seminar, we will explore various applications of Morse theory, which studies the geometry/topology of manifolds through functions.

Learning objective

Students will learn some modern aspects of geometry and how to effectively communicate sophisticated mathematics.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	assessed
	Media and Digital Technologies	assessed
	Problem-solving	assessed
	Project Management	assessed
Social Competencies	Communication	assessed
	Cooperation and Teamwork	assessed
	Customer Orientation	assessed
	Leadership and Responsibility	assessed
	Self-presentation and Social Influence	assessed
	Sensitivity to Diversity	assessed
	Negotiation	assessed
Personal Competencies	Adaptability and Flexibility	assessed
	Creative Thinking	assessed
	Critical Thinking	assessed
	Integrity and Work Ethics	assessed
	Self-awareness and Self-reflection	assessed
	Self-direction and Self-management	assessed

401-4530-23L

Introduction to Hofer's Geometry

Number of participants limited to 12.

4 credits

J.‑P. Chassé

Abstract

Learning objective

Introduce the students to the construction of Floer homology in the simplest setting and making sure that the analytic foundations are well understood. If time permits, we will explore the complications to extending this construction to more complex settings.

Literature

M. Audin & M. Damian: Morse theory and Floer homology
D. McDuff & D. Salamon: Introduction to symplectic topology

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
	Techniques and Technologies	assessed
Method-specific Competencies	Analytical Competencies	fostered
	Problem-solving	assessed
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	fostered
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered
	Self-awareness and Self-reflection	fostered
	Self-direction and Self-management	fostered

401-3580-23L

Semiclassical Analysis

Number of participants limited to 28.

4 credits

S. Becker

Abstract

'Semiclassical Analysis' provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation.

Learning objective

Literature

Semiclassical Analysis. Maciej Zworski.

Spectral Asymptotics in the semi-classical limit. M. Dimassi, J. Sjöstrand.

Prerequisites / Notice

Prerequisites: Functional Analysis I

401-3810-23L

Spectral Theory of Schrödinger Operators

Number of participants limited to 26.

4 credits

S. Becker

Abstract

Essentially all differential operators are unbounded operators on Hilbert spaces. We shall cover the basic spectral theory of unbounded operators with a particular emphasis on applications in quantum mechanics. Starting with general functional analytic concepts, we shall discuss the most important classes of operators appearing in quantum mechanics which includes random and ergodic operators.

Learning objective

Literature

Mathematical Methods in Quantum Mechanics. Gerald Teschl (available on the author's website)

Schroedinger operators. Cycon, Froese, Kirsch, Simon

Random Operators: Disorder Effects on Quantum Spectra and Dynamics. Aizenman, Warzel.

Prerequisites / Notice

Prerequisites: Functional Analysis I

401-3650-23L

Numerical Analysis Seminar: Deep Neural Network Methods for PDEs

Number of Participants: limited to seven.
Participation by consent of instructor.

4 credits

C. Schwab

Abstract

The seminar will review recent _mathematical results_
on approximation power of deep neural networks (DNNs).
The focus will be on mathematical proof techniques to
obtain approximation rate estimates (in terms of neural network
size and connectivity) on various classes of input data
including, in particular, selected types of PDE solutions.

Learning objective

Content

Deep Neural Networks (DNNs) have recently attracted substantial
interest and attention due to outperforming the best established
techniques in a number of tasks (Chess, Go, Shogi,
autonomous driving, language translation, image classification, etc.).
In big data analysis, DNNs achieved remarkable performance
in computer vision, speech recognition and natural language processing.
In many cases, these successes have been achieved by
heuristic implementations combined
with massive compute power and training data.

For a (bird's eye) view, see
https://doi.org/10.1017/9781108860604
and, more mathematical and closer to the seminar theme,
https://doi.org/10.1109/TIT.2021.3062161

The seminar will review recent _mathematical results_
on approximation power of deep neural networks (DNNs).
The focus will be on mathematical proof techniques to
obtain approximation rate estimates (in terms of neural network
size and connectivity) on various classes of input data
including, in particular, selected types of PDE solutions.
Mathematical results support that DNNs can
equalize or outperform the best mathematical results
known to date.

Particular cases comprise:
high-dimensional parametric maps,
analytic and holomorphic maps,
maps containing multi-scale features which arise as solution classes from PDEs,
classes of maps which are invariant under group actions.

Format of the Seminar:
The seminar format will be oral student presentations, combined with written report.
Student presentations will be based on a recent research paper
selected in two meetings at the start of the semester.

Grading of the Seminar:
Passing grade will require
a) 1hr oral presentation _via Zoom_ with Q/A from the seminar group, in early May 2022
and
b) typed seminar report (``Ausarbeitung'') of several key aspects
of the paper under review.

Each seminar topic will allow expansion to a semester or a
master thesis in the MSc MATH or MSc Applied MATH.

Disclaimer:
The seminar will _not_ address recent developments in DNN software,
eg. TENSORFLOW, and algorithmic training heuristics, or
programming techniques for DNN training in various specific applications.

401-3650-72L

Rational Approximation and Interpolation

4 credits

R. Hiptmair

Abstract

The seminar covers theory and algorithms for rational interpolation based on classical and modern literature. The various topics have to be presented by groups of students.

Learning objective

Participants of the seminar should acquire familiarity with the theoretical properties of approximation by means of rational functions as well as knowledge about algorithms used for computing approximating or interpolating rational functions.

Content

The simplest and most widely used function system for approximation in computational mathematics are polynomials. They are ideally suited for smooth (analytic) functions. However, in many application we encounter functions with kinks and other kinds of singularities. In this case approximation by rational functions, that is, quotients of polynomials, may be vastly superior. This is why rational approximation and interpolation is receiving increased attention for the construction of surrogate models in model order reduction.

This seminar will study a number of research papers dealing with both theoretical and algorithmic aspects of rational approximation and interpolation.

Topics:

1. Best approximation by rational functions
2. Best rational approximation of x 7→ |x|
3. Meinardus conjecture
4. Approximation by composite rational functions
5. Rational interpolation and linearized least-squares
6. Padé approximationj
7. Vector fitting
8. The AAA algorithm for rational approximation
9. The RKFIT algorithm for non-linear rational approximation
10. Rational minimax approximation
11. Multivariate Padé approximation
12. Fast least-squares Padé approximation

Student groups will be decided and topics will be assigned during the preparatory meeting on March 1, 2023

Implementation and numerical experiments:

Quite a few of the topics are algorithmic in nature. Many of the related papers mention open
source implementations of the methods, mainly in MATLAB, often relying on the Chebfun
library. It is desirable that groups presenting an algorithmic topic also conduct numerical
experiments, those covered in the articles or others, and report their observations.

More information: https://people.math.ethz.ch/~ralfh/Seminars/RAP_23/SeminarRAP_FS23.pdf

Literature

See https://people.math.ethz.ch/~ralfh/Seminars/RAP_23/SeminarRAP_FS23.pdf

Prerequisites / Notice

Good skills in analysis are required as well as basic familiarity with numerical methods for interpolation and approximation with polynomials.

Preparatory meeting onj Wed March 1

Every presentation has to be done jointly by a group of 2-3 students with presenters selected at random. Every participant will have to present on 2-3 occasions.

See https://people.math.ethz.ch/~hiptmair/Seminars/RAP_23/SeminarRAP_FS23.pdf for more information.

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Media and Digital Technologies	assessed
	Problem-solving	assessed
Social Competencies	Communication	assessed
	Cooperation and Teamwork	assessed
Personal Competencies	Critical Thinking	assessed
	Self-awareness and Self-reflection	assessed

401-3940-23L

Student Seminar in Mathematics and Data: Random Matrices

Number of participants limited to 12.

4 credits

A. Sousa Bandeira, M. T. Boedihardjo

Abstract

This student seminar will go through the basics of random matrix theory.

Learning objective

Content

Each week a student will present some parts of papers/books assigned by the instructors.

Prerequisites / Notice

A working knowledge of basic Probability Theory and Linear Algebra is needed.

Competencies

Subject-specific Competencies	Concepts and Theories	fostered
	Techniques and Technologies	fostered
Method-specific Competencies	Analytical Competencies	fostered
	Decision-making	fostered
	Problem-solving	fostered
Social Competencies	Communication	fostered
	Cooperation and Teamwork	fostered
	Leadership and Responsibility	fostered
	Self-presentation and Social Influence	fostered
Personal Competencies	Adaptability and Flexibility	fostered
	Creative Thinking	fostered
	Critical Thinking	fostered
	Integrity and Work Ethics	fostered
	Self-awareness and Self-reflection	fostered
	Self-direction and Self-management	fostered

401-3600-23L

Student Seminar in Probability Theory

Limited number of participants.
Registration to the seminar will only be effective once confirmed by email from the organizers.

4 credits

J. Bertoin, V. Tassion, W. Werner

Abstract

Learning objective

401-3620-22L

Student Seminar in Statistics: Causality

Number of participants limited to 76.
Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics. Also offered in the Master Programmes Statistics resp. Data Science.

4 credits

P. L. Bühlmann, N. Meinshausen

Abstract

Causality is dealing with fundamental questions about cause and effect. The student seminar covers statistical and mathematical aspects of causality ranging from fundamental formalization of concepts to practical algorithms and methods.

Learning objective

The participants of the seminar acquire knowledge about: concepts and formalization of statistical causality; methods, algorithms and corresponding assumptions for inferring causal relations from data; causal analysis in practice based on real data.

Prerequisites / Notice

Basic course in probability and statistics.

401-3900-16L

Advanced Topics in Discrete Optimization

Number of participants limited to 12.

4 credits

R. Zenklusen, D. E. K. Hershkowitz, R. Santiago Torres

Abstract

In this seminar we will discuss selected topics in discrete optimization. The main focus is on mostly recent research papers in the field of Combinatorial Optimization.

Learning objective

The goal of the seminar is twofold. First, we aim at improving students' presentation and communication skills. In particular, students are to present a research paper to their peers and the instructors in a clear and understandable way. Second, students learn a selection of recent cutting-edge approaches in the field of Combinatorial Optimization by attending the other students' talks. A very active participation in the seminar helps students to build up the necessary skills for parsing and digesting advanced technical texts on a significantly higher complexity level than usual textbooks.

A key goal is that students prepare their presentations in a concise and accessible way to make sure that other participants get a clear idea of the presented results and techniques.

Students intending to do a project in optimization are strongly encouraged to participate.

Content

The selected topics will cover various classical and modern results in Combinatorial Optimization.

Contrary to prior years, a very significant component of the seminar will be interactive discussions where active participation of the students is required.

Literature

The learning material will be in the form of scientific papers.

Prerequisites / Notice

Requirements: We expect students to have a thorough understanding of topics covered in the course "Linear & Combinatorial Optimization".

Competencies

Subject-specific Competencies	Concepts and Theories	assessed
Method-specific Competencies	Analytical Competencies	assessed
	Decision-making	fostered
Social Competencies	Communication	assessed

263-4203-00L

Geometry: Combinatorics and Algorithms

The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar.

2 credits

B. Gärtner, M. Hoffmann, E. Welzl, P. Schnider

Abstract

This seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent.

Learning objective

Each student is expected to read, understand, and elaborate on a selected research paper. To this end, (s)he should give a 45-min. presentation about the paper. The process includes

* getting an overview of the related literature;
* understanding and working out the background/motivation:
why and where are the questions addressed relevant?
* understanding the contents of the paper in all details;
* selecting parts suitable for the presentation;
* presenting the selected parts in such a way that an audience
with some basic background in geometry and graph theory can easily understand and appreciate it.

Content

This seminar is held once a year and complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent. The seminar is a good preparation for a master, diploma, or semester thesis in the area.

Prerequisites / Notice

Prerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required.

Semester Papers

There are several course units "Semester Paper" that are all equivalent. If, during your studies, you write several semester papers, choose among the different numbers in order to be able to obtain credits again.

Number

Title

Type

ECTS

Hours

Lecturers

401-3750-01L

Semester Paper

Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see https://math.ethz.ch/intranet/students/theses.html

8 credits

11A

Supervisors

Abstract

Semester Papers help to deepen the students' knowledge of a specific subject area. Students are offered a selection of topics. These papers serve to develop the students' ability for independent mathematical work as well as to enhance skills in presenting mathematical results in writing.

Learning objective

Prerequisites / Notice

401-3750-02L

Semester Paper (No. 2)

Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see https://math.ethz.ch/intranet/students/theses.html

8 credits

11A

Supervisors

Abstract

Learning objective

Prerequisites / Notice

401-3750-03L

Semester Paper (No. 3)

Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see https://math.ethz.ch/intranet/students/theses.html

8 credits

11A

Supervisors

Abstract

Learning objective

Prerequisites / Notice

Science in Perspective

Two credits are needed from the "Science in Perspective" programme with language courses excluded if three credits from language courses have already been recognised for the Bachelor's degree.
see Link (Eight credits must be acquired in this category: normally six during the Bachelor’s degree programme, and two during the Master’s degree programme. A maximum of three credits from language courses from the range of the Language Center of the University of Zurich and ETH Zurich may be recognised. In addition, only advanced courses (level B2 upwards) in the European languages English, French, Italian and Spanish are recognised. German language courses are recognised from level C2 upwards.)

» see Science in Perspective: Language Courses ETH/UZH

» see Science in Perspective: Type A: Enhancement of Reflection Capability

Master's Thesis

Number

Title

Type

ECTS

Hours

Lecturers

401-2000-00L

Scientific Works in Mathematics

Target audience:
Third year Bachelor students;
Master students who cannot document to have received an adequate training in working scientifically.

0 credits

D. Possamaï

Abstract

Introduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)

Learning objective

Learn the basic standards of scientific works in mathematics.

Content

- Types of mathematical works
- Publication standards in pure and applied mathematics
- Data handling
- Ethical issues
- Citation guidelines

Lecture notes

Moodle of the Mathematics Library: https://moodle-app2.let.ethz.ch/course/view.php?id=519

Prerequisites / Notice

Directive Link

401-2000-01L

Lunch Sessions – Thesis Basics for Mathematics Students

Details and registration for the optional MathBib training course: https://www.math.ethz.ch/mathbib-schulungen

0 credits

Speakers

Abstract

Optional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library.

Learning objective

401-4990-00L

Master's Thesis

Only students who fulfil the following criteria are allowed to begin with their Master's thesis:
a. successful completion of the Bachelor's programme;
b. fulfilling of any additional requirements necessary to gain admission to the Master's programme.

Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see https://math.ethz.ch/intranet/students/theses.html

30 credits

57D

Supervisors

Abstract

The master's thesis concludes the study programme. Writing up the master's thesis allows students to independently produce a major piece of work on a mathematical topic. It generally involves consulting the literature, solving any ensuing problems, and putting together the results in writing.

Learning objective

401-4990-10L

Master's Thesis

30 credits

57D

Supervisors

Abstract

Learning objective

Additional Courses

Number

Title

Type

ECTS

Hours

Lecturers

401-5000-00L

Zurich Colloquium in Mathematics

0 credits

R. Abgrall, M. Iacobelli, A. Iozzi, S. Mishra, R. Pandharipande, A. Sousa Bandeira, University lecturers

Abstract

The lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians.

Learning objective

401-5990-00L

Zurich Graduate Colloquium

0 credits

0.5K

A. Iozzi, University lecturers

Abstract

The Graduate Colloquium is an informal seminar aimed at graduate students and postdocs whose purpose is to provide a forum for communicating one's interests and thoughts in mathematics.

Learning objective

401-4530-00L

Geometry Graduate Colloquium

0 credits

Speakers

Abstract

Talks by doctoral students in the broad area of geometry for doctoral students and master students.

Learning objective

401-5110-00L

Number Theory Seminar

0 credits

Ö. Imamoglu, E. Kowalski, R. Pink, G. Wüstholz

Abstract

Research colloquium

Learning objective

Talks on various topics of current research.

Content

Research seminar in algebra, number theory and geometry. This seminar is aimed in particular to members of the research groups in these areas and their graduate students.

401-5350-00L

Analysis Seminar

0 credits

F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, T. Ilmanen, T. Rivière, J. Serra, University lecturers

Abstract

Research colloquium

Learning objective

Content

Research seminar in Analysis

401-5370-00L

Ergodic Theory and Dynamical Systems

0 credits

M. Akka Ginosar, M. Einsiedler, University lecturers

Abstract

Research colloquium

Learning objective

401-5530-00L

Geometry Seminar

0 credits

M. Einsiedler, P. Feller, A. Iozzi, U. Lang, University lecturers

Abstract

Research colloquium

Learning objective

401-5580-00L

Symplectic Geometry Seminar

0 credits

P. Biran, A. Cannas da Silva

Abstract

Research colloquium

Learning objective

401-5330-00L

Talks in Mathematical Physics

0 credits

A. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, P. Hintz, T. H. Willwacher

Abstract

Research colloquium

Learning objective

Content

Forschungsseminar mit wechselnden Themen aus dem Gebiet der mathematischen Physik.

401-5650-00L

Zurich Colloquium in Applied and Computational Mathematics

0 credits

R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, S. Mishra, S. Sauter, C. Schwab

Abstract

Research colloquium

Learning objective

401-5600-00L

Seminar on Stochastic Processes

0 credits

J. Bertoin, A. Nikeghbali, B. D. Schlein, V. Tassion, W. Werner

Abstract

Research colloquium

Learning objective

401-5620-00L

Research Seminar on Statistics

0 credits

P. L. Bühlmann, N. Meinshausen, S. van de Geer, R. Furrer, L. Held, T. Hothorn, D. Kozbur, J. Peters, A. Sousa Bandeira, M. Wolf

Abstract

Research colloquium

Learning objective

401-5640-00L

ZüKoSt: Seminar on Applied Statistics

0 credits

M. Kalisch, F. Balabdaoui, P. L. Bühlmann, R. Furrer, L. Held, T. Hothorn, M. Mächler, L. Meier, N. Meinshausen, J. Peters, M. Robinson, A. Sousa Bandeira, C. Strobl, S. van de Geer

Abstract

5 to 6 talks on applied statistics.

Learning objective

Kennenlernen von statistischen Methoden in ihrer Anwendung in verschiedenen Gebieten, besonders in Naturwissenschaft, Technik und Medizin.

Content

In 5-6 Einzelvorträgen pro Semester werden Methoden der Statistik einzeln oder überblicksartig vorgestellt, oder es werden Probleme und Problemtypen aus einzelnen Anwendungsgebieten besprochen.
3 bis 4 der Vorträge stehen in der Regel unter einem Semesterthema.

Lecture notes

Bei manchen Vorträgen werden Unterlagen verteilt.
Eine Zusammenfassung ist kurz vor den Vorträgen im Internet unter http://stat.ethz.ch/talks/zukost abrufbar.
Ankündigunen der Vorträge werden auf Wunsch zugesandt.

Prerequisites / Notice

Dies ist keine Vorlesung. Es wird keine Prüfung durchgeführt, und es werden keine Kreditpunkte vergeben.
Nach besonderem Programm. Koordinator M. Kalisch, Tel. 044 632 3435
Lehrsprache ist Englisch oder Deutsch je nach ReferentIn.
Course language is English or German and may depend on the speaker.

401-5680-00L

Foundations of Data Science Seminar

0 credits

P. L. Bühlmann, H. Bölcskei, A. Sousa Bandeira, S. van de Geer, F. Yang

Abstract

Research colloquium

Learning objective

401-5660-00L

DACO Seminar

0 credits

A. Sousa Bandeira, R. Weismantel, R. Zenklusen

Abstract

Research colloquium

Learning objective

401-5910-00L

Talks in Financial and Insurance Mathematics

https://www.math.ethz.ch/imsf/courses/talks-in-imsf.html

0 credits

B. Acciaio, P. Cheridito, D. Possamaï, M. Schweizer, J. Teichmann, M. V. Wüthrich

Abstract

Research colloquium

Learning objective

Introduction to current research topics in "Insurance Mathematics and Stochastic Finance".

Content

402-0101-00L

The Zurich Physics Colloquium

0 credits

S. Huber, University lecturers

Abstract

Research colloquium

Learning objective

The goal of this event is to bring you closer to current day research in all fields of physics. In each semester we have a set of distinguished speakers covering the full range of topics in physics. As a participating student should learn how to follow a research talk. In particular, you should be able to extract key points from a colloquium where you don't necessarily understand every detail that is presented.

251-0100-00L

Computer Science Colloquium

0 credits

Lecturers

Abstract

Invited talks on the entire spectrum of Computer Science. External guests are welcome. A detailed program is published at the beginning of every semester.

Learning objective

Renowned international computer scientists take the floor at our distinguished colloquium series, to present topics across all areas of computer science. The colloquium series is held every semester and includes inaugural and farewell lectures of the department's professors. Outside attendance is welcome.

Content

Renowned international computer scientists take the floor at our distinguished colloquium series, to present topics across all areas of computer science.

252-4202-00L

Seminar in Theoretical Computer Science

Course Units for Additional Admission Requirements

2 credits

E. Welzl, B. Gärtner, M. Hoffmann, J. Lengler, A. Steger, D. Steurer, B. Sudakov

Abstract

Presentation of recent publications in theoretical computer science, including results by diploma, masters and doctoral candidates.

Learning objective

To get an overview of current research in the areas covered by the involved research groups. To present results from the literature.

Prerequisites / Notice

This seminar takes place as part of the joint research seminar of several theory groups. Intended participation is for students with excellent performance only. Formal restriction is: prior successful participation in a master level seminar in theoretical computer science.

The courses below are only available for MSc students with additional admission requirements.

Number

Title

Type

ECTS

Hours

Lecturers

406-2303-AAL

Complex Analysis

Enrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement.

Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit.

6 credits

E. Kowalski

Abstract

Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem.

Learning objective

Literature

L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co.

B. Palka: "An introduction to complex function theory."
Undergraduate Texts in Mathematics. Springer-Verlag, 1991.

R.Remmert: Theory of Complex Functions.. Springer Verlag

E.Hille: Analytic Function Theory. AMS Chelsea Publication

Prerequisites / Notice

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.

401-2003-AAL

Algebra I

Does not take place this semester.
Enrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement.

Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit.

7 credits

15R

not available

Abstract

Learning objective

Prerequisites / Notice

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.

406-2004-AAL

Algebra II

6 credits

R. Pink

Abstract

Galois theory and related topics.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.

Learning objective

Introduction to fundamentals of field extensions, Galois theory, and related topics.

Content

The main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals.

Literature

Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1,
Graduate Studies in Mathematics,Volume 165
American Mathematical Society

Galois Theory is the topic treated in Chapter A5.

Prerequisites / Notice

Algebra I, in Rotman's book this corresponds to the topics treated in the Chapters A3 and A4.

401-2283-AAL

Analysis III (Measure Theory)

6 credits

not available

Abstract

Learning objective

401-2464-AAL

Analysis IV (Fourier Theory and Hilbert Spaces)

6 credits

not available

Abstract

Learning objective

401-2465-AAL

Analysis III and IV (Measure Theory / Fourier Theory and Hilbert Spaces)

12 credits

26R

not available

Abstract

Learning objective

406-2554-AAL

Topology

7 credits

15R

P. Feller

Abstract

Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.

Learning objective

An introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.

Lecture notes

See lecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-2554-00L/

Literature

James Munkres: Topology

Prerequisites / Notice

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.

406-2604-AAL

Probability and Statistics

8 credits

17R

F. Balabdaoui

Abstract

- Probability spaces
- Discrete models, Randiom walk
- Conditional probabilities, independence
- Continuous models
- Limit theorems

- Methods of moments
- Maximum likelihood estimation
- Hypothesis testing
- Confidence intervals
- Introductory Bayesian statistics
- Linear regression model

Learning objective

The first part of the course gives an overview of the main concepts needed to understand probability theory (sample spaces, discrete models, random walk, contiuous models and limit theorems such as the Laws of Large Numbers and the Central limit theorem). It will be based on the German script
"Wahrscheinlichkeitsrechnung und Statistik".

The second part covers some fundamental results of mathematical statistics including estimation methods, hypothesis testing as well as the linear regression model. For this part, we will use the script "Statistics for Mathematics". Both scripts are available at

https://www.stat.math.ethz.ch/~fadouab/

Lecture notes

(*) Wahrscheinlichkeitsrechnung und Statistik

(*) Statistics for Mathematics

Both scripts can be found at

https://www.stat.math.ethz.ch/~fadouab/

Literature

A. DasGupta, Fundamentals of Probability: A First Course, Springer (2010)

R. Berger and G. Casella, Statistical Inference, Duxbury Press (1990)

J. A. Rice, Mathematical Statistics and Data Analysis, Wadsworth, second edition (1995)

H.-O. Georgii, Stochastik, de Gruyter, 5. Auflage (2015)

A. Irle, Wahrscheinlichkeitstheorie und Statistik, Teubner (2001)

401-2334-AAL

Mathematical Methods of Physics II

6 credits

not available

Abstract

Learning objective

406-2005-AAL

Algebra I and II