Search result: Catalogue data in Autumn Semester 2020

Mathematics Master Information
Core Courses
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3225-00LIntroduction to Lie Groups Information
Self-service registration for this course unit in myStudies has been closed.
W8 credits4GA. Iozzi
AbstractTopological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.
ObjectiveThe goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.
LiteratureA. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)
A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
H. Samelson: "Notes on Lie algebras" (Springer, '90)
S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)
Prerequisites / NoticeTopology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.

Course webpage: Link
401-3001-61LAlgebraic Topology I Information W8 credits4GP. Biran
AbstractThis is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include:
singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms.
Objective
Literature1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.


2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
Link

See also:
Link


3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeYou should know the basics of point-set topology.

Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology").

Some knowledge of differential geometry and differential topology is useful but not strictly necessary.

Some (elementary) group theory and algebra will also be needed.
401-3145-70LAlgebraic Geometry I
Registration for this course unit has been closed.
W10 credits4V + 1UP. Yang
AbstractThis course is an introduction to Algebraic Geometry (algebraic varieties).
ObjectiveLearning Algebraic Geometry.
LiteraturePrimary reference:
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publ., 1969.

Secondary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra, Springer 2013.
* D. Eisenbud: Commutative algebra. With a view towards algebraic geometry, GTM 150, Springer Verlag, 1995.
* H. Matsumura, Commutative ring theory, Cambridge University Press 1989.
* N. Bourbaki, Commutative Algebra.

Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry, Link
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link

"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry, Link

Further readings:
* Günter Harder: Algebraic Geometry 1 & 2
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Prerequisites / NoticeLinear Algebra
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
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NumberTitleTypeECTSHoursLecturers
401-3651-00LNumerical Analysis for Elliptic and Parabolic Partial Differential Equations Information
Course audience at ETH:
3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course
"Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.
W10 credits4V + 1UC. Schwab
AbstractThis course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
ObjectiveParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
ContentThe course will address the mathematical analysis of numerical solution methods
for linear and nonlinear elliptic and parabolic partial differential equations.
Functional analytic and algebraic (De Rham complex) tools will be provided.
Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed.

Particular attention will be placed on developing mathematical foundations
(Regularity, Approximation theory) for a-priori convergence rate analysis.
A-posteriori error analysis and mathematical proofs of adaptivity and optimality
will be covered.
Implementations for model problems in MATLAB and python will illustrate the
theory.

A selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
LiteratureSUPPLEMENTARY Literature (core material will be in lecture notes)


Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp.

A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer,
1st Ed. 2004, 2nd Ed. 2015.

R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013

Additional Literature:
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)

Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp.

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]

V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).
Prerequisites / NoticePractical exercises based on MATLAB

Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations
401-3621-00LFundamentals of Mathematical Statistics Information W10 credits4V + 1US. van de Geer
AbstractThe course covers the basics of inferential statistics.
Objective
401-3622-00LStatistical Modelling Information W8 credits4GP. L. Bühlmann, M. Mächler
AbstractIn regression, the dependency of a random response variable on other variables is examined. We consider the theory of linear regression with one or more covariates, high-dimensional linear models, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory.
ObjectiveIntroduction into theory and practice of a broad and popular area of statistics, from a modern viewpoint.
ContentIn der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Hoch-dimensionale Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression. Durchrechnung und Diskussion von Anwendungsbeispielen.
Lecture notesLecture notes
Prerequisites / NoticeThis is the course unit with former course title "Regression".
Credits cannot be recognised for both courses 401-3622-00L Statistical Modelling and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden).
401-4889-00LMathematical Finance Information W11 credits4V + 2UJ. Teichmann
AbstractAdvanced course on mathematical finance:
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- additional topics
ObjectiveAdvanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes)
ContentThis is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.

Topics include
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- and probably others
Lecture notesThe course is based on different parts from different books as well as on original research literature.

Lecture notes will not be available.
Literature(will be updated later)
Prerequisites / NoticePrerequisites are the standard courses
- Probability Theory (for which lecture notes are available)
- Brownian Motion and Stochastic Calculus (for which lecture notes are available)
Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts.

This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
401-3901-00LMathematical OptimizationW11 credits4V + 2UR. Zenklusen
AbstractMathematical treatment of diverse optimization techniques.
ObjectiveThe goal of this course is to get a thorough understanding of various classical mathematical optimization techniques with an emphasis on polyhedral approaches. In particular, we want students to develop a good understanding of some important problem classes in the field, of structural mathematical results linked to these problems, and of solution approaches based on this structural understanding.
ContentKey topics include:
- Linear programming and polyhedra;
- Flows and cuts;
- Combinatorial optimization problems and techniques;
- Equivalence between optimization and separation;
- Brief introduction to Integer Programming.
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.
- Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
- Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.
Prerequisites / NoticeSolid background in linear algebra.
Bachelor Core Courses: Pure Mathematics
Further restrictions apply, but in particular:
401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme.
Analogously for:
401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II
401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II
401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry
For the category assignment take contact with the Study Administration Office (Link) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3461-00LFunctional Analysis I Restricted registration - show details
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits.
E-10 credits4V + 1UA. Carlotto
AbstractBaire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces.
ObjectiveAcquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps.
LiteratureRecommended references include the following:

Michael Struwe: "Funktionalanalysis I" (Skript available at Link)

Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011.

Peter D. Lax: "Functional analysis". Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.

Elias M. Stein and Rami Shakarchi: "Functional analysis" (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.

Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017.

Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
Prerequisites / NoticeSolid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with topology and measure theory, in part. Lebesgue integration and L^p spaces).
401-3531-00LDifferential Geometry I
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits.
E-10 credits4V + 1UW. Merry
AbstractThis will be an introductory course in differential geometry.

Topics covered include:

- Smooth manifolds, submanifolds, vector fields,
- Lie groups, homogeneous spaces,
- Vector bundles, tensor fields, differential forms,
- Integration on manifolds and the de Rham theorem,
- Principal bundles.
Objective
LiteratureThere are many excellent textbooks on differential geometry. A friendly and readable book that covers everything in Differential Geometry I is:

John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.

A more advanced (and far less friendly) series of books that covers everything in both Differential Geometry I and II is:

S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volumes I and II (1963, 1969) Wiley.
Bachelor Core Courses: Applied Mathematics ...
Further restrictions apply, but in particular:
401-3601-00L Probability Theory can only be recognised for the Master Programme if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme.
402-0205-00L Quantum Mechanics I is eligible as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) 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 (Link) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3601-00LProbability Theory Information
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (Link) after having received the credits.
E-10 credits4V + 1UA.‑S. Sznitman
AbstractBasics of probability theory and the theory of stochastic processes in discrete time
ObjectiveThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
ContentThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Lecture notesavailable in electronic form.
LiteratureR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
402-0205-00LQuantum Mechanics I Information W10 credits3V + 2UG. M. Graf
AbstractIntroduction to quantum theory: Wave mechanics, Schrödinger equation, angular momentum, central force problems, potential scattering, spin. General structure: Hilbert space, states, observables, equation of motion, density matrix, symmetries, Schrödinger and Heisenberg picture. Approximate methods:
perturbation theory, variational approach, quasi-classics.
ObjectiveIntroduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems.
ContentThe beginnings of quantum theory with Planck, Einstein and Bohr; wave and matrix mechanics; the formalism of quantum mechanics (states and observables, Hilbert spaces and operators), the measurement process, symmetries (translation, rotations), quantum mechanics in one dimension (bound states, scattering problems, tunnel effect, resonances) as well as in three (central force problems, potential scattering), perturbation theory, variational methods, angular momentum and spin; relationship of QM to classical physics; possibly composite systems and entanglement.
Lecture notesAuf Moodle, in deutscher Sprache
LiteratureG. Baym, Lectures on Quantum Mechanics
E. Merzbacher, Quantum Mechanics
L.I. Schiff, Quantum Mechanics
R. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals
J.J. Sakurai: Modern Quantum Mechanics
A. Messiah: Quantum Mechanics I
S. Weinberg: Lectures on Quantum Mechanics
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-3119-70Lp-Adic NumbersW4 credits2VP. Bengoechea Duro
AbstractThis course is an introduction to the p-adic numbers. We will see how the field of p-adic numbers Q_p is build. We will explore the (strange) topology and the arithmetic of Q_p, as well as some elementary analytic concepts such as functions, continuity, integrals, etc. We will explain an algebraic and an analytic reasons of interest for the existence of p-adic numbers.
Objective
Content- Absolute values on Q and Completions
- Topology and Arithmetic of Q_p, p-adic Integers
- Equations over p-adic numbers and Hensel's Lemma
- Local-global principle
- Hasse-Minkowski's Theorem on binary quadratic forms
- Elementary Analysis in Q_p
- the p-adic Riemann zeta function
Literature"p-adic Numbers. An Introduction", Fernando Q. Gouvea (Springer)
"p-adic Numbers, p-adic Analysis, and Zeta-Functions", Neal Koblitz (Springer)
"p-adic numbers and Diophantine equations", Yuri Bilu (online notes 2013)
Prerequisites / NoticeThe courses Topology, Measure and Integration, Algebra I/II are required prerequisites.
401-3059-00LCombinatorics II
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents 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.
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-3533-70LDifferential Geometry IIIW4 credits2VU. Lang
AbstractTopics in Riemannian geometry in the large: the structure of complete, non-compact Riemannian manifolds of non-negative sectional curvature, including Perelman's (1994) proof of the Cheeger-Gromoll soul conjecture; the Besson-Courtois-Gallot barycenter method (1996) and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces.
Objective
401-4531-66LTopics in Rigidity Theory Information W6 credits3VM. Burger
AbstractThe aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups.
ObjectiveUnderstand the basic techniques of rigidity theory.
ContentThis course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are:
- Describe all its proper quotients.
- Classify all its finite dimensional linear representations.
- More generally, can this group act by diffeomorphisms on "small" manifolds like the circle?
- Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure?
In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction.
Literature- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984.
- D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv
- Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage.
- M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online.
Prerequisites / NoticeFor this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques.
401-4141-70LCurves, Jacobians, and Modern Abel-Jacobi Theory Information W6 credits3VR. Pandharipande
Abstract
Objective
401-3057-00LFinite Geometries IIW4 credits2GN. Hungerbühler
AbstractFinite 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.
ObjectiveFinite 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.
ContentFinite 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.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4355-70LElliptic Regularity TheoryW8 credits4VM. Struwe
AbstractWe extend the theory developed in Functional Analysis II in various directions, including variants of the maximum principle, Harnack's inequality, L^p-theory, and systems. Certain limit cases will be discussed. Examples, including the harmonic map system, will illustrate the use of these methods.
Objective
LiteratureGiaquinta, Mariano: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

Gilbarg, David; Trudinger, Neil S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2001.

Further references will be given in the lectures.
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3502-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W2 credits4ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3503-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W3 credits6ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 credits9ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-0000-00LCommunication in MathematicsW2 credits1VW. Merry
AbstractDon't hide your Next Great Theorem behind bad writing.

This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX.
ObjectiveKnowing how to present written mathematics in a structured and clear manner.
ContentTopics covered include:

- Language conventions and common errors.
- How to write a thesis (more generally, a mathematics paper).
- How to use LaTeX.
- How to write a personal statement for Masters and PhD applications.
Lecture notesFull lecture notes will be made available on my website:

Link
Prerequisites / NoticeThere are no formal mathematical prerequisites.
Electives: Applied Mathematics and Further Application-Oriented Fields
¬
Selection: Numerical Analysis
NumberTitleTypeECTSHoursLecturers
401-4657-00LNumerical Analysis of Stochastic Ordinary Differential Equations Information Restricted registration - show details
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
W6 credits3V + 1UD. Salimova
AbstractCourse on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables.
ObjectiveThe aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.
ContentGeneration of random numbers
Monte Carlo methods for the numerical integration of random variables
Stochastic processes and Brownian motion
Stochastic ordinary differential equations (SODEs)
Numerical approximations of SODEs
Applications to computational finance: Option valuation
Lecture notesThere will be English, typed lecture notes for registered participants in the course.
LiteratureP. Glassermann:
Monte Carlo Methods in Financial Engineering.
Springer-Verlag, New York, 2004.

P. E. Kloeden and E. Platen:
Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 1992.
Prerequisites / NoticePrerequisites:

Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.

a) mandatory courses:
Elementary Probability,
Probability Theory I.

b) recommended courses:
Stochastic Processes.

Start of lectures: Wednesday, September 16, 2020.
401-4785-00LMathematical and Computational Methods in PhotonicsW8 credits4GH. Ammari
AbstractThe aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces
ObjectiveThe field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications.

The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength.

Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures.

The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions.

In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials.
401-4427-70LRepresentation Theory in Signal AnalysisW4 credits2VF. Bartolucci
AbstractThe scope of the course is to give an introduction to the theory of unitary representations of locally compact groups with a particular regard to the applications of this theory in signal analysis.
Objective
ContentThe scope of the course is to give an introduction to the theory of
unitary representations of locally compact groups with a particular regard to
the applications of this theory in signal analysis. The course starts with an
overview of the measure theory on locally compact groups. Then, the fundamental
definitions and results in representation theory are presented (irreducible
unitary representations, Schur’s lemma, voice transforms, square-integrable
representations, reproducing formulae). We conclude the course showing that
some of the most important transforms in applied harmonic analysis such as the
Gabor transform, the wavelet transform and the shearlet transform are related
to square-integrable unitary representations.
Prerequisites / NoticePrerequisites: measure theory, topology, functional analysis, operator
theory, Fourier analysis
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-4607-70LA Medley of Advanced ProbabilityW4 credits2VW. Werner
AbstractWe will review various topics of probability theory, with the goal to provide a short self-contained introduction to each of them, and try to describe the type of ideas and techniques that are used.
Exact topics will include (small bits of) Lévy processes, continuous-state branching processes, large deviation theory, large random matrices.
ObjectiveThe goal is for each of the topics that will be covered to provide:
- A general introduction to the subject
- An example of one of the main statements, and some of the ideas that go into the proof
- A detailed proof of one statement
Prerequisites / NoticePrerequisites: Martingales, Markov chains, Brownian motion, stochastic calculus.
401-3628-14LBayesian Statistics
Does not take place this semester.
W4 credits2V
AbstractIntroduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.
ObjectiveStudents understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.
ContentTopics that we will discuss are:

Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
Lecture notesA script will be available in English.
LiteratureChristian Robert, The Bayesian Choice, 2nd edition, Springer 2007.

A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).

Additional references will be given in the course.
Prerequisites / NoticeFamiliarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.
401-0625-01LApplied Analysis of Variance and Experimental DesignW5 credits2V + 1UL. Meier
AbstractPrinciples of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.
ObjectiveParticipants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R.
ContentPrinciples of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power.
LiteratureG. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000.
Prerequisites / NoticeThe exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held.
401-0649-00LApplied Statistical RegressionW5 credits2V + 1UM. Dettling
AbstractThis course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis.
ObjectiveThe students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling.
ContentThe course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies.

The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data.
Lecture notesA script will be available.
LiteratureFaraway (2005): Linear Models with R
Faraway (2006): Extending the Linear Model with R
Draper & Smith (1998): Applied Regression Analysis
Fox (2008): Applied Regression Analysis and GLMs
Montgomery et al. (2006): Introduction to Linear Regression Analysis
Prerequisites / NoticeThe exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held.

In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Statistical Modelling" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit.
401-4521-70LGeometric Tomography - Uniqueness, Statistical Reconstruction and Algorithms Information Restricted registration - show details W4 credits2VJ. Hörrmann
AbstractSelf-contained course on the theoretical aspects of the reconstruction of geometric objects from tomographic projection and section data.
ObjectiveIntroduction to geometric tomography and understanding of various theoretical aspects of reconstruction problems.
ContentThe problem of reconstruction of an object from geometric information like X-ray data is a classical inverse problem on the overlap between applied mathematics, statistics, computer science and electrical engineering. We focus on various aspects of the problem in the case of prior shape information on the reconstruction object. We will answer questions on uniqueness of the reconstruction and also cover statistical and algorithmic aspects.
LiteratureR. Gardner: Geometric Tomography
F. Natterer: The Mathematics of Computerized Tomography
A. Rieder: Keine Probleme mit inversen Problemen
Prerequisites / NoticeA sound mathematical background in geometry, analysis and probability is required though a repetition of relevant material will be included. The ability to understand and write mathematical proofs is mandatory.
401-4607-59LPercolation Theory Information W4 credits2VV. Tassion
AbstractAn introduction to the percolation theory.
ObjectivePercolation theory has many applications and is one of the most famous model to
describe phase transition phenomena in physics. One reason for this success is
the variety of mathematical tools, which allows for a precise and rigorous
description of the models. The objective of this course is to gain familiarity
with the methods of the percolation theory and to learn some of its important
results. The students will develop their background and intuition in
probability, and the course is particularly recommended to students with
additional interests in physics or graph theory.
ContentDefinition of percolation. Standard tools: FKG, BK inequalities, Mixing
property, Russo's formula. Sharpness of the phase transition. Correlation
length and interpretations. Uniqueness of the infinite cluster. Critical
percolation in dimension 2. Supercritical percolation in dimension d>2,
Grimmett-Marstrand Theorem and consequences.
LiteratureB. Bollobas, O. Riordan: Percolation, CUP 2006
G. Grimmett: Percolation 2ed, Springer 1999
Prerequisites / NoticePreliminaries:
401-2604-00L Probability and Statistics (mandatory)
401-3601-00L Probability Theory (recommended)
401-4619-67LAdvanced Topics in Computational Statistics
Does not take place this semester.
W4 credits2Vnot available
AbstractThis lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.
ObjectiveStudents learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.
ContentThe main focus will be on graphical models in various forms:
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
Prerequisites / NoticeWe assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.
401-3627-00LHigh-Dimensional Statistics
Does not take place this semester.
W4 credits2VP. L. Bühlmann
Abstract"High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed.
ObjectiveKnowledge of methods and basic theory for high-dimensional statistical inference
ContentLasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling
LiteraturePeter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag.
ISBN 978-3-642-20191-2.
Prerequisites / NoticeKnowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).
401-4623-00LTime Series AnalysisW6 credits3GF. Balabdaoui
AbstractThe course offers an introduction into analyzing times series, that is observations which occur in time. The material will cover Stationary Models, ARMA processes, Spectral Analysis, Forecasting, Nonstationary Models, ARIMA Models and an introduction to GARCH models.
ObjectiveThe goal of the course is to have a a good overview of the different types of time series and the approaches used in their statistical analysis.
ContentThis course treats modeling and analysis of time series, that is random variables which change in time. As opposed to the i.i.d. framework, the main feature exibited by time series is the dependence between successive observations.

The key topics which will be covered as:

Stationarity
Autocorrelation
Trend estimation
Elimination of seasonality
Spectral analysis, spectral densities
Forecasting
ARMA, ARIMA, Introduction into GARCH models
LiteratureThe main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis
Prerequisites / NoticeBasic knowledge in probability and statistics
401-3612-00LStochastic SimulationW5 credits3GF. Sigrist
AbstractThis course introduces statistical Monte Carlo methods. This includes applications of stochastic simulation in various fields (statistics, statistical mechanics, operations research, financial mathematics), generating uniform and arbitrary random variables (incl. rejection and importance sampling), the accuracy of methods, variance reduction, quasi-Monte Carlo, and Markov chain Monte Carlo.
ObjectiveStudents know the stochastic simulation methods introduced in this course. Students understand and can explain these methods, show how they are related to each other, know their weaknesses and strengths, apply them in practice, and proof key results.
ContentExamples of simulations in different fields (statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (including rejection sampling and importance sampling), simulation of multivariate normal variables and stochastic differential equations. The accuracy of Monte Carlo methods. Methods for variance reduction and quasi-Monte Carlo. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). Algorithms introduced in the course are illustrated with the statistical software R.
Lecture notesA script will be available in English.
LiteratureP. Glasserman, Monte Carlo Methods in Financial Engineering.
Springer 2004.

B. D. Ripley. Stochastic Simulation. Wiley, 1987.

Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
Prerequisites / NoticeIt is assumed that students have had an introduction to probability theory and statistics (random variables, joint and conditional distributions, law of large numbers, central limit theorem, basics of measure theory).

The course resources (including script, slides, exercises) will be provided via the Moodle online learning platform.
Selection: Financial and Insurance Mathematics
In the Master's programmes in Mathematics resp. Applied Mathematics 401-3913-01L Mathematical Foundations for Finance is eligible as an elective course, but only if 401-3888-00L Introduction to Mathematical Finance 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 (Link) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3925-00LNon-Life Insurance: Mathematics and Statistics Information W8 credits4V + 1UM. V. Wüthrich
AbstractThe lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial science. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models and neural networks, credibility theory, claims reserving and solvency.
ObjectiveThe student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations.
ContentThe following topics are treated:
Collective Risk Modeling
Individual Claim Size Modeling
Approximations for Compound Distributions
Ruin Theory in Discrete Time
Premium Calculation Principles
Tariffication
Generalized Linear Models and Neural Networks
Bayesian Models and Credibility Theory
Claims Reserving
Solvency Considerations
Lecture notesM. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics
Link
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

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

Prerequisites: knowledge of probability theory, statistics and applied stochastic processes.
401-3922-00LLife Insurance MathematicsW4 credits2VM. Koller
AbstractThe classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated.
Objective
401-3928-00LReinsurance AnalyticsW4 credits2VP. Antal, P. Arbenz
AbstractThis 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
ObjectiveThis 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
ContentThis 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
Lecture notesSlides and lecture notes will be made available.

An excerpt of last year's lecture notes is available here: Link
Prerequisites / NoticeBasic knowledge in statistics, probability theory, and actuarial techniques
401-3927-00LMathematical Modelling in Life InsuranceW4 credits2VT. J. Peter
AbstractIn life insurance, it is essential to have adequate mortality tables, be it for reserving or pricing purposes. The course provides the tools necessary to create mortality tables from scratch. Additionally, we study various guarantees embedded in life insurance products and learn to price them with the help of stochastic models.
ObjectiveThe course's objective is to provide the students with the understanding and the tools to create mortality tables on their own.
Additionally, students should learn to price embedded options in life insurance. Aside of the mere application of specific models, they should develop an intuition for the various drivers of the value of these options.
ContentFollowing main topics are covered:

1. Guarantees and options embedded in life insurance products.
- Stochastic valuation of participating contracts
- Stochastic valuation of Unit Linked contracts
2. Mortality Tables:
- Determining raw mortality rates
- Smoothing techniques: Whittaker-Henderson, smoothing splines,...
- Trends in mortality rates
- Stochastic mortality model due to Lee and Carter
- Neural Network extension of the Lee-Carter model
- Integration of safety margins
Lecture notesLectures notes and slides will be provided
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

The course counts towards the diploma of "Aktuar SAV".

Good knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful.
Selection: Mathematical Physics, Theoretical Physics
NumberTitleTypeECTSHoursLecturers
402-0843-00LQuantum Field Theory I
Special Students UZH must book the module PHY551 directly at UZH.
W10 credits4V + 2UC. Anastasiou
AbstractThis course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity.
Topics include:
- Relativistic quantum mechanics
- Quantisation of bosonic and fermionic fields
- Interactions in perturbation theory
- Scattering processes and decays
- Elementary processes in QED
- Radiative corrections
ObjectiveThe goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics.
402-0861-00LStatistical PhysicsW10 credits4V + 2UG. Blatter
AbstractThe lecture focuses on classical and quantum statistical physics. Various techniques, cumulant expansion, path integrals, and specific systems are discussed: Fermions, photons/phonons, Bosons, magnetism, van der Waals gas. Phase transitions are studied in mean field theory (Weiss, Landau). Including fluctuations leads to critical phenomena, scaling, and the renormalization group.
ObjectiveThis lecture gives an introduction into the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.
ContentThermodynamics, three laws of thermodynamics, thermodynamic potentials, phenomenology of phase transitions.
Classical statistical physics: micro-canonical-, canonical-, and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons, statistical interaction.
Techniques: variational approach, cumulant expansion, path integral formulation.
Degenerate fermions: Fermi gas, electrons in magnetic field.
Bosons: photons and phonons, Bose-Einstein condensation.
Magnetism: Ising-, XY-, Heisenberg models, Weiss mean-field theory.
Van der Waals gas-liquid transition in mean field theory.
General mean-field (Landau) theory of phase transitions, first- and second order, tricritical point.
Fluctuations: field theory approach, Gauss theory, self-consistent field, Ginzburg criterion.
Critical phenomena: scaling theory, universality.
Renormalization group: general theory and applications to spin models (real space RG), phi^4 theory (k-space RG), Kosterlitz-Thouless theory.
Lecture notesLecture notes available in English.
LiteratureNo specific book is used for the course. Relevant literature will be given in the course.
402-0830-00LGeneral Relativity Information
Special Students UZH must book the module PHY511 directly at UZH.
W10 credits4V + 2UR. Renner
AbstractIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations of the theory as well as the underlying physical principles and concepts. It covers selected applications, such as the Schwarzschild solution and gravitational waves.
ObjectiveBasic understanding of general relativity, its mathematical foundations (in particular the relevant aspects of differential geometry), and some of the phenomena it predicts (with a focus on black holes).
ContentIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations, such as differentiable manifolds, the Riemannian and Lorentzian metric, connections, and curvature. It discusses the underlying physical principles, e.g., the equivalence principle, and concepts, such as curved spacetime and the energy-momentum tensor. The course covers some basic applications and special cases, including the Newtonian limit, post-Newtonian expansions, the Schwarzschild solution, light deflection, and gravitational waves.
LiteratureSuggested textbooks:

C. Misner, K, Thorne and J. Wheeler: Gravitation
S. Carroll - Spacetime and Geometry: An Introduction to General Relativity
R. Wald - General Relativity
S. Weinberg - Gravitation and Cosmology
402-0897-00LIntroduction to String TheoryW6 credits2V + 1UM. Gaberdiel
AbstractThis course is an introduction to string theory. It will mainly concentrate on the bosonic string and its quantisation in flat space.
ObjectiveThe objective of this course is to motivate the subject of string theory, exploring the important role it has played in the development of modern theoretical and mathematical physics. The goal of the course is to give a pedagogical introduction to the bosonic string in flat space.
ContentI. Introduction
II. The classical relativistic string
III. Light-cone quantisation
IV. Covariant quantisation
V. Closed strings and T-duality
VI. String interactions
LiteratureLecture notes:

String Theory - D. Tong
Link

Lectures on String Theory - G. Arutyunov
Link

Books:

Superstring Theory - M. Green, J. Schwarz and E. Witten (two volumes, CUP, 1988)
Volume 1: Introduction
Volume 2: Loop Amplitudes, Anomalies and Phenomenology

String Theory - J. Polchinski (two volumes, CUP, 1998)
Volume 1: An Introduction to the Bosonic String
Volume 2: Superstring Theory and Beyond
Errata: Link

Basic Concepts of String Theory - R. Blumenhagen, D. Lüst and S. Theisen (Springer-Verlag, 2013)

A First Course in String Theory - B. Zwiebach (CUP, 2009)
Selection: Mathematical Optimization, Discrete Mathematics
NumberTitleTypeECTSHoursLecturers
401-3054-14LProbabilistic Methods in Combinatorics Information W6 credits2V + 1UB. Sudakov
AbstractThis course provides a gentle introduction to the Probabilistic Method, with an emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems.
Objective
ContentThe topics covered in the class will include (but are not limited to): linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness.
Literature- The Probabilistic Method, by N. Alon and J. H. Spencer, 3rd Edition, Wiley, 2008.
- Random Graphs, by B. Bollobás, 2nd Edition, Cambridge University Press, 2001.
- Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000.
- Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002.
Auswahl: Theoretical Computer Science, Discrete Mathematics
NumberTitleTypeECTSHoursLecturers
263-4500-00LAdvanced Algorithms Information W9 credits3V + 2U + 3AM. Ghaffari
AbstractThis is a graduate-level course on algorithm design (and analysis). It covers a range of topics and techniques in approximation algorithms, sketching and streaming algorithms, and online algorithms.
ObjectiveThis course familiarizes the students with some of the main tools and techniques in modern subareas of algorithm design.
ContentThe lectures will cover a range of topics, tentatively including the following: graph sparsifications while preserving cuts or distances, various approximation algorithms techniques and concepts, metric embeddings and probabilistic tree embeddings, online algorithms, multiplicative weight updates, streaming algorithms, sketching algorithms, and derandomization.
Lecture notesLink
Prerequisites / NoticeThis course is designed for masters and doctoral students and it especially targets those interested in theoretical computer science, but it should also be accessible to last-year bachelor students.

Sufficient comfort with both (A) Algorithm Design & Analysis and (B) Probability & Concentrations. E.g., having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, though not required formally. If you are not sure whether you're ready for this class or not, please consult the instructor.
252-1425-00LGeometry: Combinatorics and Algorithms Information W8 credits3V + 2U + 2AB. Gärtner, E. Welzl, M. Hoffmann, M. Wettstein
AbstractGeometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?)
ObjectiveThe goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains.
In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project.
ContentPlanar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations.
Lecture notesyes
LiteratureMark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008.
Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011.
Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004.
Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002.
Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004.
Prerequisites / NoticePrerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH.
Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area.
252-0417-00LRandomized Algorithms and Probabilistic Methods
Does not take place this semester.
W10 credits3V + 2U + 4AA. Steger
AbstractLas Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks
ObjectiveAfter this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas.
ContentRandomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas.
Lecture notesYes.
Literature- Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995)
- Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005)
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
227-0423-00LNeural Network Theory Information W4 credits2V + 1UH. Bölcskei
AbstractThe class focuses on fundamental mathematical aspects of neural networks with an emphasis on deep networks: Universal approximation theorems, basics of approximation theory, fundamental limits of deep neural network learning, geometry of decision surfaces, capacity of separating surfaces, dimension measures relevant for generalization, VC dimension of neural networks.
ObjectiveAfter attending this lecture, participating in the exercise sessions, and working on the homework problem sets, students will have acquired a working knowledge of the mathematical foundations of (deep) neural networks.
Content1. Universal approximation with single- and multi-layer networks

2. Introduction to approximation theory: Fundamental limits on compressibility of signal classes, Kolmogorov epsilon-entropy of signal classes, non-linear approximation theory

3. Fundamental limits of deep neural network learning

4. Geometry of decision surfaces

5. Separating capacity of nonlinear decision surfaces

6. Dimension measures: Pseudo-dimension, fat-shattering dimension, Vapnik-Chervonenkis (VC) dimension

7. Dimensions of neural networks

8. Generalization error in neural network learning
Lecture notesDetailed lecture notes will be provided.
Prerequisites / NoticeThis course is aimed at students with a strong mathematical background in general, and in linear algebra, analysis, and probability theory in particular.
227-0445-10LMathematical Methods of Signal Processing Information W6 credits4GH. G. Feichtinger
AbstractThis course offers a mathematical correct but still non-technical description of key objects relevant for signal processing, such as Dirac
measures, Dirac combs, various function spaces (like L^2), impulse response, transfer function, Gabor expansion, and so on. The approach is based on properties of "Feichtinger's algebra". MATLAB routines will serve as illustration.
ObjectiveThe aim of the class to familiarize the participants with the idea of generalized functions (usual called distributions), and to provide a (novel approach) to a theory of mild distributions, which cannot be found in books so far (the course will contribute to the development of such a book). From the physical point of view, such an object is something, which can be measured or captured by (linear) measurements, such as an audio signal. The Harmonic Analysis perspective is, that the Fourier transform and time-frequency transforms are possible over any locally compact group. Engineers talk about discrete or continuous, periodic and non-periodic signals. Hence, a unified approach to these settings and a discussion of their interconnection (e.g. approximately computing the Fourier transform of a function using the DFT) is at the heart of this course.
ContentMathematical Foundations of Signal Processing:

0. Recalling (on and off) concepts from linear algebra (e.g. linear mappings, etc.) and introducing concepts from basic linear functional analysis (Hilbert spaces, Banach spaces)

1. Translation invariant systems and convolution, elementary functional analytic approach;

2. Pure frequencies and the Fourier transform, convolution theorem

3. The subalgebra L1(Rd) of integrable functions (without Lebesgue integration), Riemann Lebesgue Lemma

4. Plancherels Theorem, L2(Rd) and basic Hilbert space theory, unitary mappings

5. Short-time Fourier transform, the Feichtinger algebra S0(Rd) as algebra of test functions

6. The dual space of mild distributions, relationship to tempered distributions (for this familiar); various characterization

7. Gabor expansions of signals, characterization of smoothness and decay, Gabor frames and Riesz bases;

8. Transition from continuous to discrete variables, from periodic to the non-periodic case;

9. The kernel theorem, as the continuous analogue of matrix representations;

10. Sobolev spaces (describing smoothness) and weighted spaces;

11. Spreading representation and Kohn-Nirenberg representation of operators;

12. Gabor multipliers and approximation of slowly varying systems;

13. As time permits: the idea of generalized stochastic processes

14. Further subjects as demanded by the audience can be covered on demand.


Detailed lecture notes will be provided. This material will become part of an on-going book-project, which has many facets.
Lecture notesThis material will be regularly updated and posted at the lecturer's homepage, at Link

There will be also a dedicated WEB page at Link (to be installed in the near future).
Prerequisites / NoticeWe encourage students who are interested in mathematics, but also students of physics or mathematics who want to learn about application of modern methods from functional analysis to their sciences, especially those who are interested to understand what the connections between the continuous and the discrete world are (from continuous functions or images to samples or pixels, and back).

Hans G. Feichtinger (Link)

For any kind of questions concerning this course please contact the lecturer. He will be in Zurich most of the time, even if the course has to be held offline. It will start by October 1st 2020 only.
401-3502-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W2 credits4ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3503-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W3 credits6ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 credits9ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-0000-00LCommunication in MathematicsW2 credits1VW. Merry
AbstractDon't hide your Next Great Theorem behind bad writing.

This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX.
ObjectiveKnowing how to present written mathematics in a structured and clear manner.
ContentTopics covered include:

- Language conventions and common errors.
- How to write a thesis (more generally, a mathematics paper).
- How to use LaTeX.
- How to write a personal statement for Masters and PhD applications.
Lecture notesFull lecture notes will be made available on my website:

Link
Prerequisites / NoticeThere are no formal mathematical prerequisites.
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.
Atmospherical Physics
NumberTitleTypeECTSHoursLecturers
701-1221-00LDynamics of Large-Scale Atmospheric Flow Information W4 credits2V + 1UH. Wernli, L. Papritz
AbstractThis lecture course is about the fundamental aspects of the dynamics of extratropical weather systems (quasi-geostropic dynamics, potential vorticity, Rossby waves, baroclinic instability). The fundamental concepts are formally introduced, quantitatively applied and illustrated with examples from the real atmosphere. Exercises (quantitative and qualitative) form an essential part of the course.
ObjectiveUnderstanding the dynamics of large-scale atmospheric flow
ContentDynamical Meteorology is concerned with the dynamical processes of the
earth's atmosphere. The fundamental equations of motion in the atmosphere will be discussed along with the dynamics and interactions of synoptic system - i.e. the low and high pressure systems that determine our weather. The motion of such systems can be understood in terms of quasi-geostrophic theory. The lecture course provides a derivation of the mathematical basis along with some interpretations and applications of the concept.
Lecture notesDynamics of large-scale atmospheric flow
Literature- Holton J.R., An introduction to Dynamic Meteorogy. Academic Press, fourth edition 2004,
- Pichler H., Dynamik der Atmosphäre, Bibliographisches Institut, 456 pp. 1997
Prerequisites / NoticePhysics I, II, Environmental Fluid Dynamics
Biology
NumberTitleTypeECTSHoursLecturers
551-0015-00LBiology IW2 credits2VE. Hafen, E. Dufresne
AbstractThe lecture Biology I, together with the lecture Biology II in the following summer semester, is a basic, introductory course into Biology for Students of Materials Sciences and other students with biology as subsidiary subject.
ObjectiveThe goal of this course is to give the students a basic understanding of the molecules that build a cell and make it function, and the basic principles of metabolism and molecular genetics.
ContentDie folgenden Kapitelnummern beziehen sich auf das der Vorlesung zugrundeliegende Lehrbuch "Biology" (Campbell & Rees, 10th edition, 2015)
Kapitel 1-4 des Lehrbuchs werden als Grundwissen vorausgesetzt

1. Aufbau der Zelle

Kapitel 5: Struktur und Funktion biologischer Makromoleküle
Kapitel 6: Eine Tour durch die Zelle
Kaptiel 7: Membranstruktur und-funktion
Kapitel 8: Einführung in den Stoffwechsel
Kapitel 9: Zelluläre Atmung und Speicherung chemischer Energie
Kapitel 10: Photosynthese
Kapitel 12: Der Zellzyklus
Kapitel 17: Vom Gen zum Protein

2. Allgemeine Genetik

Kapitel 13: Meiose und Reproduktionszyklen
Kapitel 14: Mendel'sche Genetik
Kapitel 15: Die chromosomale Basis der Vererbung
Kapitel 16: Die molekulare Grundlage der Vererbung
Kapitel 18: Genetik von Bakterien und Viren
Kapitel 46: Tierische Reproduktion

Grundlagen des Stoffwechsels und eines Überblicks über molekulare Genetik
Lecture notesDer Vorlesungsstoff ist sehr nahe am Lehrbuch gehalten, Skripte werden ggf. durch die Dozenten zur Verfügung gestellt.
LiteratureDas folgende Lehrbuch ist Grundlage für die Vorlesungen Biologie I und II:

„Biology“, Campbell and Rees, 10th Edition, 2015, Pearson/Benjamin Cummings, ISBN 978-3-8632-6725-4
Prerequisites / NoticeZur Vorlesung Biologie I gibt es während der Prüfungssessionen eine einstündige, schriftliche Prüfung. Die Vorlesung Biologie II wird separat geprüft.
636-0017-00LComputational BiologyW6 credits3G + 2AT. Stadler, T. Vaughan
AbstractThe aim of the course is to provide up-to-date knowledge on how we can study biological processes using genetic sequencing data. Computational algorithms extracting biological information from genetic sequence data are discussed, and statistical tools to understand this information in detail are introduced.
ObjectiveAttendees will learn which information is contained in genetic sequencing data and how to extract information from this data using computational tools. The main concepts introduced are:
* stochastic models in molecular evolution
* phylogenetic & phylodynamic inference
* maximum likelihood and Bayesian statistics
Attendees will apply these concepts to a number of applications yielding biological insight into:
* epidemiology
* pathogen evolution
* macroevolution of species
ContentThe course consists of four parts. We first introduce modern genetic sequencing technology, and algorithms to obtain sequence alignments from the output of the sequencers. We then present methods for direct alignment analysis using approaches such as BLAST and GWAS. Second, we introduce mechanisms and concepts of molecular evolution, i.e. we discuss how genetic sequences change over time. Third, we employ evolutionary concepts to infer ancestral relationships between organisms based on their genetic sequences, i.e. we discuss methods to infer genealogies and phylogenies. Lastly, we introduce the field of phylodynamics, the aim of which is to understand and quantify population dynamic processes (such as transmission in epidemiology or speciation & extinction in macroevolution) based on a phylogeny. Throughout the class, the models and methods are illustrated on different datasets giving insight into the epidemiology and evolution of a range of infectious diseases (e.g. HIV, HCV, influenza, Ebola). Applications of the methods to the field of macroevolution provide insight into the evolution and ecology of different species clades. Students will be trained in the algorithms and their application both on paper and in silico as part of the exercises.
Lecture notesLecture slides will be available on moodle.
LiteratureThe course is not based on any of the textbooks below, but they are excellent choices as accompanying material:
* Yang, Z. 2006. Computational Molecular Evolution.
* Felsenstein, J. 2004. Inferring Phylogenies.
* Semple, C. & Steel, M. 2003. Phylogenetics.
* Drummond, A. & Bouckaert, R. 2015. Bayesian evolutionary analysis with BEAST.
Prerequisites / NoticeBasic knowledge in linear algebra, analysis, and statistics will be helpful. Programming in R will be required for the project work (compulsory continuous performance assessments). We provide an R tutorial and help sessions during the first two weeks of class to learn the required skills. However, in case you do not have any previous experience with R, we strongly recommend to get familiar with R prior to the semester start. For the D-BSSE students, we highly recommend the voluntary course „Introduction to Programming“, which takes place at D-BSSE from Wednesday, September 12 to Friday, September 14, i.e. BEFORE the official semester starting date Link
For the Zurich-based students without R experience, we recommend the R course Link, or working through the script provided as part of this R course.
636-0007-00LComputational Systems Biology Information W6 credits3V + 2UJ. Stelling
AbstractStudy of fundamental concepts, models and computational methods for the analysis of complex biological networks. Topics: Systems approaches in biology, biology and reaction network fundamentals, modeling and simulation approaches (topological, probabilistic, stoichiometric, qualitative, linear / nonlinear ODEs, stochastic), and systems analysis (complexity reduction, stability, identification).
ObjectiveThe aim of this course is to provide an introductory overview of mathematical and computational methods for the modeling, simulation and analysis of biological networks.
ContentBiology has witnessed an unprecedented increase in experimental data and, correspondingly, an increased need for computational methods to analyze this data. The explosion of sequenced genomes, and subsequently, of bioinformatics methods for the storage, analysis and comparison of genetic sequences provides a prominent example. Recently, however, an additional area of research, captured by the label "Systems Biology", focuses on how networks, which are more than the mere sum of their parts' properties, establish biological functions. This is essentially a task of reverse engineering. The aim of this course is to provide an introductory overview of corresponding computational methods for the modeling, simulation and analysis of biological networks. We will start with an introduction into the basic units, functions and design principles that are relevant for biology at the level of individual cells. Making extensive use of example systems, the course will then focus on methods and algorithms that allow for the investigation of biological networks with increasing detail. These include (i) graph theoretical approaches for revealing large-scale network organization, (ii) probabilistic (Bayesian) network representations, (iii) structural network analysis based on reaction stoichiometries, (iv) qualitative methods for dynamic modeling and simulation (Boolean and piece-wise linear approaches), (v) mechanistic modeling using ordinary differential equations (ODEs) and finally (vi) stochastic simulation methods.
Lecture notesLink
LiteratureU. Alon, An introduction to systems biology. Chapman & Hall / CRC, 2006.

Z. Szallasi et al. (eds.), System modeling in cellular biology. MIT Press, 2010.

B. Ingalls, Mathematical modeling in systems biology: an introduction. MIT Press, 2013
636-0009-00LEvolutionary DynamicsW6 credits2V + 1U + 2AN. Beerenwinkel
AbstractEvolutionary dynamics is concerned with the mathematical principles according to which life has evolved. This course offers an introduction to mathematical modeling of evolution, including deterministic and stochastic models.
ObjectiveThe goal of this course is to understand and to appreciate mathematical models and computational methods that provide insight into the evolutionary process.
ContentEvolution is the one theory that encompasses all of biology. It provides a single, unifying concept to understand the living systems that we observe today. We will introduce several types of mathematical models of evolution to describe gene frequency changes over time in the context of different biological systems, focusing on asexual populations. Viruses and cancer cells provide the most prominent examples of such systems and they are at the same time of great biomedical interest. The course will cover some classical mathematical population genetics and population dynamics, and also introduce several new approaches. This is reflected in a diverse set of mathematical concepts which make their appearance throughout the course, all of which are introduced from scratch. Topics covered include the quasispecies equation, evolution of HIV, evolutionary game theory, birth-death processes, evolutionary stability, evolutionary graph theory, somatic evolution of cancer, stochastic tunneling, cell differentiation, hematopoietic tumor stem cells, genetic progression of cancer and the speed of adaptation, diffusion theory, fitness landscapes, neutral networks, branching processes, evolutionary escape, and epistasis.
Lecture notesNo.
Literature- Evolutionary Dynamics. Martin A. Nowak. The Belknap Press of Harvard University Press, 2006.
- Evolutionary Theory: Mathematical and Conceptual Foundations. Sean H. Rice. Sinauer Associates, Inc., 2004.
Prerequisites / NoticePrerequisites: Basic mathematics (linear algebra, calculus, probability)
Control and Automation
NumberTitleTypeECTSHoursLecturers
151-0563-01LDynamic Programming and Optimal Control Information W4 credits2V + 1UR. D'Andrea
AbstractIntroduction to Dynamic Programming and Optimal Control.
ObjectiveCovers the fundamental concepts of Dynamic Programming & Optimal Control.
ContentDynamic Programming Algorithm; Deterministic Systems and Shortest Path Problems; Infinite Horizon Problems, Bellman Equation; Deterministic Continuous-Time Optimal Control.
LiteratureDynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. I, 3rd edition, 2005, 558 pages, hardcover.
Prerequisites / NoticeRequirements: Knowledge of advanced calculus, introductory probability theory, and matrix-vector algebra.
Economics
NumberTitleTypeECTSHoursLecturers
401-3929-00LFinancial Risk Management in Social and Pension Insurance Information W4 credits2VP. Blum
AbstractInvestment returns are an important source of funding for social and pension insurance, and financial risk is an important threat to stability. We study short-term and long-term financial risk and its interplay with other risk factors, and we develop methods for the measurement and management of financial risk and return in an asset/liability context with the goal of assuring sustainable funding.
ObjectiveUnderstand the basic asset-liability framework: essential principles and properties of social and pension insurance; cash flow matching, duration matching, valuation portfolio and loose coupling; the notion of financial risk; long-term vs. short-term risk; coherent measures of risk.

Understand the conditions for sustainable funding: derivation of required returns; interplay between return levels, contribution levels and other parameters; influence of guaranteed benefits.

Understand the notion of risk-taking capability: capital process as a random walk; measures of long-term risk and relation to capital; short-term solvency vs. long-term stability; effect of embedded options and guarantees; interplay between required return and risk-taking capability.

Be able to study empirical properties of financial assets: the Normal hypothesis and the deviations from it; statistical tools for investigating relevant risk and return properties of financial assets; time aggregation properties; be able to conduct analysis of real data for the most important asset classes.

Understand and be able to carry out portfolio construction: the concept of diversification; limitations to diversification; correlation breakdown; incorporation of constraints; sensitivities and shortcomings of optimized portfolios.

Understand and interpret the asset-liability interplay: the optimized portfolio in the asset-liability framework; short-term risk vs. long-term risk; the influence of constraints; feasible and non-feasible solutions; practical considerations.

Understand and be able to address essential problems in asset / liability management, e.g. optimal risk / return positioning, optimal discount rate, target value for funding ratio or turnaround issues.

Have an overall view: see the big picture of what asset returns can and cannot contribute to social security; be aware of the most relevant outcomes; know the role of the actuary in the financial risk management process.
ContentFor pension insurance and other forms of social insurance, investment returns are an important source of funding. In order to earn these returns, substantial financial risks must be taken, and these risks represent an important threat to financial stability, in the long term and in the short term.

Risk and return of financial assets cannot be separated from one another and, hence, asset management and risk management cannot be separated either. Managing financial risk in social and pension insurance is, therefore, the task of reconciling the contradictory dimensions of

1. Required return for a sustainable funding of the institution,
2. Risk-taking capability of the institution,
3. Returns available from financial assets in the market,
4. Risks incurred by investing in these assets.

This task must be accomplished under a number of constraints. Financial risk management in social insurance also means reconciling the long time horizon of the promised insurance benefits with the short time horizon of financial markets and financial risk.

It is not the goal of this lecture to provide the students with any cookbook recipes that can readily be applied without further reflection. The goal is rather to enable the students to develop their own understanding of the problems and possible solutions associated with the management of financial risks in social and pension insurance.

To this end, a rigorous intellectual framework will be developed and a powerful set of mathematical tools from the fields of actuarial mathematics and quantitative risk management will be applied. When analyzing the properties of financial assets, an empirical viewpoint will be taken using statistical tools and considering real-world data.
Lecture notesExtensive handouts will be provided. Moreover, practical examples and data sets in Excel and R will be made available.
Prerequisites / NoticeSolid base knowledge of probability and statistics is indispensable. Specialized concepts from financial and insurance mathematics as well as quantitative risk management will be introduced in the lecture as needed, but some prior knowledge in some of these areas would be an advantage.

This course counts towards the diploma of "Aktuar SAV".

The exams ONLY take place during the official ETH examination period.
363-0537-00LResource and Environmental EconomicsW3 credits2GL. Bretschger
AbstractRelationship between economy and environment, market failures, external effects and public goods, contingent valuation, internalisation of externalities, economics of non-renewable resources, economics of renewable resources, environmental cost-benefit analysis, sustainability economics, and international resource and environmental problems.
ObjectiveA successful completion of the course will enable a thorough understanding of the basic questions and methods of resource and environmental economics and the ability to solve typical problems using appropriate tools consisting of concise verbal explanations, diagrams or mathematical expressions. Concrete goals are first of all the acquisition of knowledge about the main questions of resource and environmental economics and about the foundation of the theory with different normative concepts in terms of efficiency and fairness. Secondly, students should be able to deal with environmental externalities and internalisation through appropriate policies or private negotiations, including knowledge of the available policy instruments and their relative strengths and weaknesses. Thirdly, the course will allow for in-depth economic analysis of renewable and non-renewable resources, including the role of stock constraints, regeneration functions, market power, property rights and the impact of technology. A fourth objective is to successfully use the well-known tool of cost-benefit analysis for environmental policy problems, which requires knowledge of the benefits of an improved natural environment. The last two objectives of the course are the acquisition of sufficient knowledge about the economics of sustainability and the application of environmental economic theory and policy at international level, e.g. to the problem of climate change.
ContentThe course covers all the interactions between the economy and the natural environment. It introduces and explains basic welfare concepts and market failure; external effects, public goods, and environmental policy; the measurement of externalities and contingent valuation; the economics of non-renewable resources, renewable resources, cost-benefit-analysis, sustainability concepts; international aspects of resource and environmental problems; selected examples and case studies. After a general introduction to resource and environmental economics, highlighting its importace and the main issues, the course explains the normative basis, utilitarianism, and fairness according to different principles. Pollution externalities are a deep core topic of the lecture. We explain the governmental internalisation of externalities as well as the private internalisation of externalities (Coase theorem). Furthermore, the issues of free rider problems and public goods, efficient levels of pollution, tax vs. permits, and command and control instruments add to a thorough analysis of environmental policy. Turning to resource supply, the lecture first looks at empirical data on non-renewable natural resources and then develops the optimal price development (Hotelling-rule). It deals with the effects of explorations, new technologies, and market power. When treating the renewable resources, we look at biological growth functions, optimal harvesting of renewable resources, and the overuse of open-access resources. A next topic is cost-benefit analysis with the environment, requiring measuring environmental benefits and measuring costs. In the chapter on sustainability, the course covers concepts of sustainability, conflicts with optimality, and indicators of sustainability. In a final chapter, we consider international environmental problems and in particular climate change and climate policy.
LiteraturePerman, R., Ma, Y., McGilvray, J, Common, M.: "Natural Resource & Environmental Economics", 4th edition, 2011, Harlow, UK: Pearson Education
363-0503-00LPrinciples of Microeconomics
GESS (Science in Perspective): This lecture is for MSc students only. BSc students register for 363-1109-00L Einführung in die Mikroökonomie.
W3 credits2GM. Filippini
AbstractThe course introduces basic principles, problems and approaches of microeconomics. This provides the students with reflective and contextual knowledge on how societies use scarce resources to produce goods and services and ensure a (fair) distribution.
ObjectiveThe learning objectives of the course are:

(1) Students must be able to discuss basic principles, problems and approaches in microeconomics. (2) Students can analyse and explain simple economic principles in a market using supply and demand graphs. (3) Students can contrast different market structures and describe firm and consumer behaviour. (4) Students can identify market failures such as externalities related to market activities and illustrate how these affect the economy as a whole. (5) Students can also recognize behavioural failures within a market and discuss basic concepts related to behavioural economics. (6) Students can apply simple mathematical concepts on economic problems.
ContentThe resources on our planet are finite. The discipline of microeconomics therefore deals with the question of how society can use scarce resources to produce goods and services and ensure a (fair) distribution. In particular, microeconomics deals with the behaviour of consumers and firms in different market forms. Economic considerations and discussions are not part of classical engineering and science study programme. Thus, the goal of the lecture "Principles of Microeconomics" is to teach students how economic thinking and argumentation works. The course should help the students to look at the contents of their own studies from a different perspective and to be able to critically reflect on economic problems discussed in the society.

Topics covered by the course are:

- Supply and demand
- Consumer demand: neoclassical and behavioural perspective
- Cost of production: neoclassical and behavioural perspective
- Welfare economics, deadweight losses
- Governmental policies
- Market failures, common resources and public goods
- Public sector, tax system
- Market forms (competitive, monopolistic, monopolistic competitive, oligopolistic)
- International trade
Lecture notesLecture notes, exercises and reference material can be downloaded from Moodle.
LiteratureN. Gregory Mankiw and Mark P. Taylor (2020), "Economics", 5th edition, South-Western Cengage Learning.
The book can also be used for the course 'Principles of Macroeconomics' (Sturm)

For students taking only the course 'Principles of Microeconomics' there is a shorter version of the same book:
N. Gregory Mankiw and Mark P. Taylor (2020), "Microeconomics", 5th edition, South-Western Cengage Learning.

Complementary:
R. Pindyck and D. Rubinfeld (2018), "Microeconomics", 9th edition, Pearson Education.
Prerequisites / NoticeGESS (Science in Perspective): This lecture is for MSc students only. BSc students register for 363-1109-00L Einführung in die Mikroökonomie.
363-0565-00LPrinciples of MacroeconomicsW3 credits2VJ.‑E. Sturm
AbstractThis course examines the behaviour of macroeconomic variables, such as gross domestic product, unemployment and inflation rates. It tries to answer questions like: How can we explain fluctuations of national economic activity? What can economic policy do against unemployment and inflation?
ObjectiveThis lecture will introduce the fundamentals of macroeconomic theory and explain their relevance to every-day economic problems.
ContentThis course helps you understand the world in which you live. There are many questions about the macroeconomy that might spark your curiosity. Why are living standards so meagre in many African countries? Why do some countries have high rates of inflation while others have stable prices? Why have some European countries adopted a common currency? These are just a few of the questions that this course will help you answer.
Furthermore, this course will give you a better understanding of the potential and limits of economic policy. As a voter, you help choose the policies that guide the allocation of society's resources. When deciding which policies to support, you may find yourself asking various questions about economics. What are the burdens associated with alternative forms of taxation? What are the effects of free trade with other countries? How does the government budget deficit affect the economy? These and similar questions are always on the minds of policy makers.
Lecture notesThe course webpage (to be found at Link) contains announcements, course information and lecture slides.
LiteratureThe set-up of the course will closely follow the book of
N. Gregory Mankiw and Mark P. Taylor (2020), Economics, Cengage Learning, Fifth Edition.

Besides this textbook, the slides, lecture notes and problem sets will cover the content of the lecture and the exam questions.
363-1021-00LMonetary PolicyW3 credits2VJ.‑E. Sturm, A. Rathke
AbstractThe main aim of this course is to analyse the goals of monetary policy and to review the instruments available to central banks in order to pursue these goals. It will focus on the transmission mechanisms of monetary policy and the differences between monetary policy rules and discretionary policy. It will also make connections between theoretical economic concepts and current real world issues.
ObjectiveThis lecture will introduce the fundamentals of monetary economics and explain the working and impact of monetary policy. The main aim of this course is to describe and analyze the goals of monetary policy and to review the instruments available to central banks in order to pursue these goals. It will focus on the transmission mechanisms of monetary policy, the effectiveness of monetary policy actions, the differences between monetary policy rules and discretionary policy, as well as in institutional issues concerning central banks, transparency of monetary authorities and monetary policy in a monetary union framework. Moreover, we discuss the implementation of monetary policy in practice and the design of optimal policy.
ContentFor the functioning of today’s economy, central banks and their policies play an important role. Monetary policy is the policy adopted by the monetary authority of a country, the central bank. The central bank controls either the interest rate payable on very short-term borrowing or the money supply, often targeting inflation or the interest rate to ensure price stability and general trust in the currency. This monetary policy course looks into today’s major questions related to policies of central banks. It provides insights into the monetary policy process using core economic principles and real-world examples.
Lecture notesThe course webpage (to be found at Link) contains announcements, course information and lecture slides.
LiteratureThe course will be based on chapters of:
Mishkin, Frederic S. (2018), The Economics of Money, Banking and Financial Markets, 12th edition, Pearson. ISBN 9780134733821
Prerequisites / NoticeBasic knowledge in international economics and a good background in macroeconomics.
Finance
NumberTitleTypeECTSHoursLecturers
401-8905-00LFinancial Engineering (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MFOEC200

Mind the enrolment deadlines at UZH:
Link
W6 credits4GUniversity lecturers
AbstractThis lecture is intended for students who would like to learn more on equity derivatives modelling and pricing.
ObjectiveQuantitative models for European option pricing (including stochastic
volatility and jump models), volatility and variance derivatives,
American and exotic options.
ContentAfter 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 notesScript.
Prerequisites / NoticeBasic knowledge of probability theory and stochastic calculus.
Asset Pricing.
401-8913-00LAdvanced Corporate Finance I (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MOEC0455

Mind the enrolment deadlines at UZH:
Link
W6 credits4GUniversity lecturers
AbstractThis course develops and refines tools for evaluating investments (capital budgeting), capital structure, and corporate securities. The course seeks to deepen students' understanding of the link between corporate finance theory and practice.
ObjectiveThis course develops and refines tools for evaluating investments
(capital budgeting), capital structure, and corporate securities. With
respect to capital structure, we start with the famous Miller and
Modigliani irrelevance proposition and then move on to study the effects
of taxes, bankruptcy costs, information asymmetries between firms and
the capital markets, and agency costs. In this context, we will also
study how leverage affects some central financial ratios that are often
used in practice to assess firms and their stock. Other topics include
corporate cash holdings, the use and pricing of convertible bonds, and
risk management. The latter two topics involve option pricing. With
respect to capital budgeting, the course pays special attention to tax
effects in valuation, including in the estimation of the cost of
capital. We will also study payout policy (dividends and share
repurchases). The course seeks to deepen students' understanding of the
link between corporate finance theory and practice. Various cases will
be assigned to help reach this objective.
ContentTopics covered
1. Capital structure: Perfect markets and irrelevance
2. Risk, leverage, taxes, and the cost of capital
3. Leverage and financial ratios
4. Payout policy: Dividends and share repurchases
5. Capital structure: Taxes and bankruptcy costs
6. Capital structure: Information asymmetries, agency costs, cash holdings
7. Valuation: DCF, adjusted present value and WACC
8. Valuation using options
9. The use and pricing of convertible bonds
10. Corporate risk management
Prerequisites / NoticeThis course replaces "Advanced Corporate Finance I" (MOEC0288), which will be discontinued from HS16.
Image Processing and Computer Vision
NumberTitleTypeECTSHoursLecturers
227-0447-00LImage Analysis and Computer Vision Information W6 credits3V + 1UL. Van Gool, E. Konukoglu, F. Yu
AbstractLight and perception. Digital image formation. Image enhancement and feature extraction. Unitary transformations. Color and texture. Image segmentation. Motion extraction and tracking. 3D data extraction. Invariant features. Specific object recognition and object class recognition. Deep learning and Convolutional Neural Networks.
ObjectiveOverview of the most important concepts of image formation, perception and analysis, and Computer Vision. Gaining own experience through practical computer and programming exercises.
ContentThis course aims at offering a self-contained account of computer vision and its underlying concepts, including the recent use of deep learning.
The first part starts with an overview of existing and emerging applications that need computer vision. It shows that the realm of image processing is no longer restricted to the factory floor, but is entering several fields of our daily life. First the interaction of light with matter is considered. The most important hardware components such as cameras and illumination sources are also discussed. The course then turns to image discretization, necessary to process images by computer.
The next part describes necessary pre-processing steps, that enhance image quality and/or detect specific features. Linear and non-linear filters are introduced for that purpose. The course will continue by analyzing procedures allowing to extract additional types of basic information from multiple images, with motion and 3D shape as two important examples. Finally, approaches for the recognition of specific objects as well as object classes will be discussed and analyzed. A major part at the end is devoted to deep learning and AI-based approaches to image analysis. Its main focus is on object recognition, but also other examples of image processing using deep neural nets are given.
Lecture notesCourse material Script, computer demonstrations, exercises and problem solutions
Prerequisites / NoticePrerequisites:
Basic concepts of mathematical analysis and linear algebra. The computer exercises are based on Python and Linux.
The course language is English.
Information and Communication Technology
NumberTitleTypeECTSHoursLecturers
227-0105-00LIntroduction to Estimation and Machine Learning Restricted registration - show details W6 credits4GH.‑A. Loeliger
AbstractMathematical basics of estimation and machine learning, with a view towards applications in signal processing.
ObjectiveStudents master the basic mathematical concepts and algorithms of estimation and machine learning.
ContentReview of probability theory;
basics of statistical estimation;
least squares and linear learning;
Hilbert spaces;
Gaussian random variables;
singular-value decomposition;
kernel methods, neural networks, and more
Lecture notesLecture notes will be handed out as the course progresses.
Prerequisites / Noticesolid basics in linear algebra and probability theory
227-0101-00LDiscrete-Time and Statistical Signal Processing Information W6 credits4GH.‑A. Loeliger
AbstractThe course introduces some fundamental topics of digital signal processing with a bias towards applications in communications: discrete-time linear filters, inverse filters and equalization, DFT, discrete-time stochastic processes, elements of detection theory and estimation theory, LMMSE estimation and LMMSE filtering, LMS algorithm, Viterbi algorithm.
ObjectiveThe course introduces some fundamental topics of digital signal processing with a bias towards applications in communications. The two main themes are linearity and probability. In the first part of the course, we deepen our understanding of discrete-time linear filters. In the second part of the course, we review the basics of probability theory and discrete-time stochastic processes. We then discuss some basic concepts of detection theory and estimation theory, as well as some practical methods including LMMSE estimation and LMMSE filtering, the LMS algorithm, and the Viterbi algorithm. A recurrent theme throughout the course is the stable and robust "inversion" of a linear filter.
Content1. Discrete-time linear systems and filters:
state-space realizations, z-transform and spectrum,
decimation and interpolation, digital filter design,
stable realizations and robust inversion.

2. The discrete Fourier transform and its use for digital filtering.

3. The statistical perspective:
probability, random variables, discrete-time stochastic processes;
detection and estimation: MAP, ML, Bayesian MMSE, LMMSE;
Wiener filter, LMS adaptive filter, Viterbi algorithm.
Lecture notesLecture Notes
227-0417-00LInformation Theory IW6 credits4GA. Lapidoth
AbstractThis course covers the basic concepts of information theory and of communication theory. Topics covered include the entropy rate of a source, mutual information, typical sequences, the asymptotic equi-partition property, Huffman coding, channel capacity, the channel coding theorem, the source-channel separation theorem, and feedback capacity.
ObjectiveThe fundamentals of Information Theory including Shannon's source coding and channel coding theorems
ContentThe entropy rate of a source, Typical sequences, the asymptotic equi-partition property, the source coding theorem, Huffman coding, Arithmetic coding, channel capacity, the channel coding theorem, the source-channel separation theorem, feedback capacity
LiteratureT.M. Cover and J. Thomas, Elements of Information Theory (second edition)
Material Modelling and Simulation
NumberTitleTypeECTSHoursLecturers
327-1201-00LTransport Phenomena IW5 credits4GJ. Vermant
AbstractPhenomenological approach to "Transport Phenomena" based on balance equations supplemented by thermodynamic considerations to formulate the undetermined fluxes in the local species mass, momentum, and energy balance equations; Solutions of a few selected problems relevant to materials science and engineering.
ObjectiveThe teaching goals of this course are on five different levels:
(1) Deep understanding of fundamentals: local balance equations, constitutive equations for fluxes, entropy balance, interfaces, idea of dimensionless numbers and scaling, ...
(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 and finite differences.
ContentPart 1 Approach to Transport Phenomena
Diffusion Equation
Refreshing Topics in Equilibrium Thermodynamics
Balance Equations
Forces and Fluxes
Applications
1. Measuring Transport Coefficients
2. Pressure-Driven Flows and Heat exchange
Lecture notesThe course is based on the book D. C. Venerus and H. C. Öttinger, A Modern Course in Transport Phenomena (Cambridge University Press, 2018) and slides are presented
Literature1. 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. L.G. Leal, Advanced Transport Phenomena (Oxford University Press, 2011)
4. W. M. Deen, Analysis of Transport Phenomena (Oxford University Press, 1998)
5. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287
Prerequisites / NoticeComplex 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). Equilibrium thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms). Maxwell equations. Programming and simulation techniques (Matlab, Monte Carlo simulations).
Quantum Chemistry
NumberTitleTypeECTSHoursLecturers
529-0003-01LAdvanced Quantum ChemistryW6 credits3GM. Reiher, A. Baiardi
AbstractAdvanced, but fundamental topics central to the understanding of theory in chemistry and for solving actual chemical problems with a computer.
Examples are:
* Operators derived from principles of relativistic quantum mechanics
* Relativistic effects + methods of relativistic quantum chemistry
* Open-shell molecules + spin-density functional theory
* New electron-correlation theories
ObjectiveThe aim of the course is to provide an in-depth knowledge of theory and method development in theoretical chemistry. It will be shown that this is necessary in order to be able to solve actual chemical problems on a computer with quantum chemical methods.

The relativistic re-derivation of all concepts known from (nonrelativistic) quantum mechanics and quantum-chemistry lectures will finally explain the form of all operators in the molecular Hamiltonian - usually postulated rather than deduced. From this, we derive operators needed for molecular spectroscopy (like those required by magnetic resonance spectroscopy). Implications of other assumptions in standard non-relativistic quantum chemistry shall be analyzed and understood, too. Examples are the Born-Oppenheimer approximation and the expansion of the electronic wave function in a set of pre-defined many-electron basis functions (Slater determinants). Overcoming these concepts, which are so natural to the theory of chemistry, will provide deeper insights into many-particle quantum mechanics. Also revisiting the workhorse of quantum chemistry, namely density functional theory, with an emphasis on open-shell electronic structures (radicals, transition-metal complexes) will contribute to this endeavor. It will be shown how these insights allow us to make more accurate predictions in chemistry in practice - at the frontier of research in theoretical chemistry.
Content1) Introductory lecture: basics of quantum mechanics and quantum chemistry
2) Einstein's special theory of relativity and the (classical) electromagnetic interaction of two charged particles
3) Klein-Gordon and Dirac equation; the Dirac hydrogen atom
4) Numerical methods based on the Dirac-Fock-Coulomb Hamiltonian, two-component and scalar relativistic Hamiltonians
5) Response theory and molecular properties, derivation of property operators, Breit-Pauli-Hamiltonian
6) Relativistic effects in chemistry and the emergence of spin
7) Spin in density functional theory
8) New electron-correlation theories: Tensor network and matrix product states, the density matrix renormalization group
Lecture notesA set of detailed lecture notes will be provided, which will cover the whole course. Please navigate to the lecture material starting here: Link
Literature1) M. Reiher, A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH, 2014, 2nd edition
2) F. Schwabl: Quantenmechanik für Fortgeschrittene (QM II), Springer-Verlag, 1997
[english version available: F. Schwabl, Advanced Quantum Mechanics]
3) R. McWeeny: Methods of Molecular Quantum Mechanics, Academic Press, 1992
4) C. R. Jacob, M. Reiher, Spin in Density-Functional Theory, Int. J. Quantum Chem. 112 (2012) 3661
Link
5) K. H. Marti, M. Reiher, New Electron Correlation Theories for Transition Metal Chemistry, Phys. Chem. Chem. Phys. 13 (2011) 6750
Link
6) K.H. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in Quantum Chemistry, Z. Phys. Chem. 224 (2010) 583
Link
7) E. Mátyus, J. Hutter, U. Müller-Herold, M. Reiher, On the emergence of molecular structure, Phys. Rev. A 83 2011, 052512
Link

Note also the standard textbooks:
A) A. Szabo, N.S. Ostlund. Verlag, Dover Publications
B) I. N. Levine, Quantum Chemistry, Pearson
C) T. Helgaker, P. Jorgensen, J. Olsen: Molecular Electronic-Structure Theory, Wiley, 2000
D) R.G. Parr, W. Yang: Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1994
E) R.M. Dreizler, E.K.U. Gross: Density Functional Theory, Springer-Verlag, 1990
Prerequisites / NoticeStrongly recommended (preparatory) courses are: quantum mechanics and quantum chemistry
Simulation of Semiconductor Devices
NumberTitleTypeECTSHoursLecturers
227-0157-00LSemiconductor Devices: Physical Bases and Simulation Information W4 credits3GA. Schenk
AbstractThe course addresses the physical principles of modern semiconductor devices and the foundations of their modeling and numerical simulation. Necessary basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided. Computer simulations of the most important devices and of interesting physical effects supplement the lectures.
ObjectiveThe course aims at the understanding of the principle physics of modern semiconductor devices, of the foundations in the physical modeling of transport and its numerical simulation. During the course also basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided.
ContentThe main topics are: transport models for semiconductor devices (quantum transport, Boltzmann equation, drift-diffusion model, hydrodynamic model), physical characterization of silicon (intrinsic properties, scattering processes), mobility of cold and hot carriers, recombination (Shockley-Read-Hall statistics, Auger recombination), impact ionization, metal-semiconductor contact, metal-insulator-semiconductor structure, and heterojunctions.
The exercises are focussed on the theory and the basic understanding of the operation of special devices, as single-electron transistor, resonant tunneling diode, pn-diode, bipolar transistor, MOSFET, and laser. Numerical simulations of such devices are performed with an advanced simulation package (Sentaurus-Synopsys). This enables to understand the physical effects by means of computer experiments.
Lecture notesThe script (in book style) can be downloaded from: Link
LiteratureThe script (in book style) is sufficient. Further reading will be recommended in the lecture.
Prerequisites / NoticeQualifications: Physics I+II, Semiconductor devices (4. semester).
227-0158-00LSemiconductor Devices: Transport Theory and Monte Carlo Simulation Information
Does not take place this semester.
The course was offered for the last time in HS19.
W4 credits2G
AbstractThe lecture combines quasi-ballistic transport theory with application to realistic devices
of current and future CMOS technology.
All aspects such as quantum mechanics, phonon scattering or Monte Carlo techniques to
solve the Boltzmann equation are introduced. In the exercises advanced devices such
as FinFETs and nanosheets are simulated.
ObjectiveThe aim of the course is a fundamental understanding of the derivation of the Boltzmann
equation and its solution by Monte Carlo methods. The practical aspect is to become
familiar with technology computer-aided design (TCAD) and perform simulations of
advanced CMOS devices.
ContentThe covered topics include:
- quantum mechanics and second quantization,
- band structure calculation including the pseudopotential method
- phonons
- derivation of the Boltzmann equation including scattering in the Markov limit
- stochastic Monte Carlo techniques to solve the Boltzmann equation
- TCAD environment and geometry generation
- Stationary bulk Monte Carlo simulation of velocity-field curves
- Transient Monte Carlo simulation for quasi-ballistic velocity overshoot
- Monte Carlo device simulation of FinFETs and nanosheets
Lecture notesLecture notes (in German)
LiteratureFurther reading will be recommended in the lecture.
Prerequisites / NoticeKnowledge of quantum mechanics is not required. Basic knowledge of semiconductor
physics is useful, but not necessary.
Systems Design
NumberTitleTypeECTSHoursLecturers
363-0541-00LSystems Dynamics and ComplexityW3 credits3GF. Schweitzer
AbstractFinding solutions: what is complexity, problem solving cycle.

Implementing solutions: project management, critical path method, quality control feedback loop.

Controlling solutions: Vensim software, feedback cycles, control parameters, instabilities, chaos, oscillations and cycles, supply and demand, production functions, investment and consumption
ObjectiveA successful participant of the course is able to:
- understand why most real problems are not simple, but require solution methods that go beyond algorithmic and mathematical approaches
- apply the problem solving cycle as a systematic approach to identify problems and their solutions
- calculate project schedules according to the critical path method
- setup and run systems dynamics models by means of the Vensim software
- identify feedback cycles and reasons for unintended systems behavior
- analyse the stability of nonlinear dynamical systems and apply this to macroeconomic dynamics
ContentWhy are problems not simple? Why do some systems behave in an unintended way? How can we model and control their dynamics? The course provides answers to these questions by using a broad range of methods encompassing systems oriented management, classical systems dynamics, nonlinear dynamics and macroeconomic modeling.
The course is structured along three main tasks:
1. Finding solutions
2. Implementing solutions
3. Controlling solutions

PART 1 introduces complexity as a system immanent property that cannot be simplified. It introduces the problem solving cycle, used in systems oriented management, as an approach to structure problems and to find solutions.

PART 2 discusses selected problems of project management when implementing solutions. Methods for identifying the critical path of subtasks in a project and for calculating the allocation of resources are provided. The role of quality control as an additional feedback loop and the consequences of small changes are discussed.

PART 3, by far the largest part of the course, provides more insight into the dynamics of existing systems. Examples come from biology (population dynamics), management (inventory modeling, technology adoption, production systems) and economics (supply and demand, investment and consumption). For systems dynamics models, the software program VENSIM is used to evaluate the dynamics. For economic models analytical approaches, also used in nonlinear dynamics and control theory, are applied. These together provide a systematic understanding of the role of feedback loops and instabilities in the dynamics of systems. Emphasis is on oscillating phenomena, such as business cycles and other life cycles.

Weekly self-study tasks are used to apply the concepts introduced in the lectures and to come to grips with the software program VENSIM.
Another objective of the self-study tasks is to practice efficient communication of such concepts.
These are provided as home work and two of these will be graded (see "Prerequisites").
Lecture notesThe lecture slides are provided as handouts - including notes and literature sources - to registered students only. All material is to be found on the Moodle platform. More details during the first lecture
Theoretical Physics
In the Master's programme in Applied Mathematics 402-0205-00L Quantum Mechanics I 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 (Link) after having received the credits.
NumberTitleTypeECTSHoursLecturers
402-0809-00LIntroduction to Computational PhysicsW8 credits2V + 2UA. Adelmann
AbstractThis course offers an introduction to computer simulation methods for physics problems and their implementation on PCs and super computers. The covered topics include classical equations of motion, partial differential equations (wave equation, diffusion equation, Maxwell's equations), Monte Carlo simulations, percolation, phase transitions, and complex networks.
ObjectiveStudents learn to apply the following methods: Random number generators, Determination of percolation critical exponents, numerical solution of problems from classical mechanics and electrodynamics, canonical Monte-Carlo simulations to numerically analyze magnetic systems. Students also learn how to implement their own numerical frameworks and how to use existing libraries to solve physical problems. In addition, students learn to distinguish between different numerical methods to apply them to solve a given physical problem.
ContentIntroduction to computer simulation methods for physics problems. Models from classical mechanics, electrodynamics and statistical mechanics as well as some interdisciplinary applications are used to introduce the most important object-oriented programming methods for numerical simulations (typically in C++). Furthermore, an overview of existing software libraries for numerical simulations is presented.
Lecture notesLecture notes and slides are available online and will be distributed if desired.
LiteratureLiterature recommendations and references are included in the lecture notes.
Prerequisites / NoticeLecture and exercise lessons in english, exams in German or in English
402-2203-01LClassical Mechanics Information Restricted registration - show details W7 credits4V + 2UN. Beisert
AbstractA conceptual introduction to theoretical physics: Newtonian mechanics, central force problem, oscillations, Lagrangian mechanics, symmetries and conservation laws, spinning top, relativistic space-time structure, particles in an electromagnetic field, Hamiltonian mechanics, canonical transformations, integrable systems, Hamilton-Jacobi equation.
ObjectiveFundamental understanding of the description of Mechanics in the Lagrangian and Hamiltonian formulation. Detailed understanding of important applications, in particular, the Kepler problem, the physics of rigid bodies (spinning top) and of oscillatory systems.
402-0861-00LStatistical PhysicsW10 credits4V + 2UG. Blatter
AbstractThe lecture focuses on classical and quantum statistical physics. Various techniques, cumulant expansion, path integrals, and specific systems are discussed: Fermions, photons/phonons, Bosons, magnetism, van der Waals gas. Phase transitions are studied in mean field theory (Weiss, Landau). Including fluctuations leads to critical phenomena, scaling, and the renormalization group.
ObjectiveThis lecture gives an introduction into the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.
ContentThermodynamics, three laws of thermodynamics, thermodynamic potentials, phenomenology of phase transitions.
Classical statistical physics: micro-canonical-, canonical-, and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons, statistical interaction.
Techniques: variational approach, cumulant expansion, path integral formulation.
Degenerate fermions: Fermi gas, electrons in magnetic field.
Bosons: photons and phonons, Bose-Einstein condensation.
Magnetism: Ising-, XY-, Heisenberg models, Weiss mean-field theory.
Van der Waals gas-liquid transition in mean field theory.
General mean-field (Landau) theory of phase transitions, first- and second order, tricritical point.
Fluctuations: field theory approach, Gauss theory, self-consistent field, Ginzburg criterion.
Critical phenomena: scaling theory, universality.
Renormalization group: general theory and applications to spin models (real space RG), phi^4 theory (k-space RG), Kosterlitz-Thouless theory.
Lecture notesLecture notes available in English.
LiteratureNo specific book is used for the course. Relevant literature will be given in the course.
402-0843-00LQuantum Field Theory I
Special Students UZH must book the module PHY551 directly at UZH.
W10 credits4V + 2UC. Anastasiou
AbstractThis course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity.
Topics include:
- Relativistic quantum mechanics
- Quantisation of bosonic and fermionic fields
- Interactions in perturbation theory
- Scattering processes and decays
- Elementary processes in QED
- Radiative corrections
ObjectiveThe goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics.
402-0830-00LGeneral Relativity Information
Special Students UZH must book the module PHY511 directly at UZH.
W10 credits4V + 2UR. Renner
AbstractIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations of the theory as well as the underlying physical principles and concepts. It covers selected applications, such as the Schwarzschild solution and gravitational waves.
ObjectiveBasic understanding of general relativity, its mathematical foundations (in particular the relevant aspects of differential geometry), and some of the phenomena it predicts (with a focus on black holes).
ContentIntroduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations, such as differentiable manifolds, the Riemannian and Lorentzian metric, connections, and curvature. It discusses the underlying physical principles, e.g., the equivalence principle, and concepts, such as curved spacetime and the energy-momentum tensor. The course covers some basic applications and special cases, including the Newtonian limit, post-Newtonian expansions, the Schwarzschild solution, light deflection, and gravitational waves.
LiteratureSuggested textbooks:

C. Misner, K, Thorne and J. Wheeler: Gravitation
S. Carroll - Spacetime and Geometry: An Introduction to General Relativity
R. Wald - General Relativity
S. Weinberg - Gravitation and Cosmology
» Electives Theoretical Physics
Transportation Science
NumberTitleTypeECTSHoursLecturers
101-0417-00LTransport Planning MethodsW6 credits4GA. Erath Rusterholtz, M. van Eggermond
AbstractThe course provides the necessary knowledge to develop models supporting and also evaluating the solution of given planning problems.
The course is composed of a lecture part, providing the theoretical knowledge, and an applied part in which students develop their own models in order to evaluate a transport project/ policy by means of cost-benefit analysis.
Objective- Knowledge and understanding of statistical methods and algorithms commonly used in transport planning
- Comprehend the reasoning and capabilities of transport models
- Ability to independently develop a transport model able to solve / answer planning problem
- Getting familiar with cost-benefit analysis as a decision-making supporting tool
ContentThe course provides the necessary knowledge to develop models supporting the solution of given planning problems and also introduces cost-benefit analysis as a decision-making tool. Examples of such planning problems are the estimation of traffic volumes, prediction of estimated utilization of new public transport lines, and evaluation of effects (e.g. change in emissions of a city) triggered by building new infrastructure and changes to operational regulations.

To cope with that, the problem is divided into sub-problems, which are solved using various statistical models (e.g. regression, discrete choice analysis) and algorithms (e.g. iterative proportional fitting, shortest path algorithms, method of successive averages).

The course is composed of a lecture part, providing the theoretical knowledge, and an applied part in which students develop their own models in order to evaluate a transport project/ policy by means of cost-benefit analysis. Interim lab session take place regularly to guide and support students with the applied part of the course.
Lecture notesMoodle platform (enrollment needed)
LiteratureWillumsen, P. and J. de D. Ortuzar (2003) Modelling Transport, Wiley, Chichester.

Cascetta, E. (2001) Transportation Systems Engineering: Theory and Methods, Kluwer Academic Publishers, Dordrecht.

Sheffi, Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice Hall, Englewood Cliffs.

Schnabel, W. and D. Lohse (1997) Verkehrsplanung, 2. edn., vol. 2 of Grundlagen der Strassenverkehrstechnik und der Verkehrsplanung, Verlag für Bauwesen, Berlin.

McCarthy, P.S. (2001) Transportation Economics: A case study approach, Blackwell, Oxford.
Seminars and Semester Papers
Seminars
This semester, many seminars have a waiting list with special selection procedure. If no other criteria apply, a definitive registration will be granted first of all to students who haven't got another seminar registration. Here is the best procedure for dealing with two waiting lists: first choose your preferred seminar and afterwards choose an alternative seminar.
NumberTitleTypeECTSHoursLecturers
401-3180-61LCategories and Derived Functors Restricted registration - show details
Mathematics Bachelor or Master programmes with preference for Mathematics Bachelor 5th semester
W4 credits2SR. Pink
AbstractCategories, functors, natural transformations, limits, colimits, adjoint functors, additive & abelian categories, exact sequences, diagram lemmas, injectives, projectives, Mitchell's embedding theorem, complexes, homology, derived functors, acyclic resolutions, Tor, Ext, Yoneda-Ext, spectral sequence of filtered or double complexes & composite functors, group cohomology, derived functor of limits
Objective
401-3110-70LStudent Seminar in Number Theory: Elliptic Curves Restricted registration - show details
Number of participants limited to 23.
W4 credits2SM. Schwagenscheidt
AbstractSeminar on the foundations of the theory of Elliptic Curves.
ObjectiveThe participants learn the basics about elliptic curves, which will enable them to write a Bachelor's or Master's thesis in number theory. In addition to a talk, the writing of a short manuscript in latex will be required.
ContentWe first study the basic properties of elliptic curves, such as the group law. Then we will proceed to study elliptic curves over the rationals and the question whether it has rational or integral points. One of the main goal of the seminar is the proof of the Mordell-Weil theorem, which states that the set of rational points of a rational elliptic curve is a finitely generated abelian group. Using the theory of elliptic functions we will show that an elliptic curve over the complex numbers can be viewed as a torus. As an outlook, we will sketch several deep results and conjectures about elliptic curves, such as Wiles' Modularity Theorem, which played an important role in the proof of Fermat's Last Theorem, and such as the Birch and Swinnerton-Dyer Conjecture.
LiteratureKnapp: Elliptic Curves
Koecher, Krieg: Elliptische Funktionen und Modulformen
Milne: Elliptic Curves
Silverman: The Arithmetic of Elliptic Curves
Silverman, Tate: Rational Points on Elliptic Curves
Prerequisites / NoticeBasic knowledge of Algebra and Complex Analysis will be helpful.
401-3420-70LTopics in Harmonic Analysis Restricted registration - show details
Number of participants limited to 20
W4 credits2SF. Da Lio, L. Kobel-Keller
AbstractThe aim of this seminar about harmonic analysis is to study the most important and most classical topics in that field, e.g. maximal functions, Marcinkiewicz interpolation, Fourier theory, distribution theory, singular integrals and Calderon-Zygmund theory.
After an introduction delivered by the two organisers, each week participants will give a seminar talk (usually in groups of two).
ObjectiveThe students will learn on one hand the most important concept in harmonic analysis and on the other hand improve their presentations skills (by delivering a seminar talk).
LiteratureThe main references are:
E. Stein: "Singular integrals and differentiability properties of functions "
E. Stein, G. Weiss: "Introduction to Fourier analysis on Euclidean spaces"
L. Grafakos: "Modern Fourier Analysis" & "Classical Fourier Analysis"
401-3650-68LNumerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Restricted registration - show details
Number of participants limited to 6. Consent of Instructor needed.
W4 credits2SC. Schwab
AbstractThe 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.
Objective
ContentPresentation of the Seminar:
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
Link
and, more mathematical and closer to the seminar theme,
Link

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 with Q/A from the seminar group 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.
401-4660-70LRobustness of Deep Neural Networks Restricted registration - show details
Number of participants limited to 40
W4 credits2SR. Alaifari
AbstractWhile deep neural networks have been very successfully employed in classification problems, their stability properties remain still unclear. In particular, the presence of so-called adversarial examples has demonstrated that state-of-the-art networks are extremely vulnerable to small perturbations in the data.
ObjectiveIn this seminar, we will consider the state-of-the-art in adversarial attacks and defenses.
Prerequisites / NoticeParticipants should already be familiar with the principles of deep neural networks. The course will also include programming that will require knowledge in using either PyTorch or Tensorflow.
401-3640-70LVolume Integral Equations: Theory and Numerics Restricted registration - show details
Number of participants limited to 10.
W4 credits2SR. Hiptmair
AbstractThe seminar covers recent research articles on the theory and numerical treatment of volume integral equations for acoustic and electromagnetic scattering problems.
ObjectiveBeside conveying knowledge about the functional analytic method for analyzing integral equations and a range of numerical methods, the seminar is also meant to practise scientific presentation skills.
ContentTopics (based on research articles)
1. VIE for acoustic scattering
2.The Operator Equations of Lippmann-Schwinger Type for Acoustic and Elec-
tromagnetic Scattering Problems in L2
3. VIE for electromagnetic scattering at dielectric bodies
4. Fast Solvers for the Lippmann-Schwinger equation
5. Numerical Solution of the Lippmann–Schwinger Equation by “Approx-
imate Approximations”
6. Higher-order Fourier approximation in scattering by two-dimensional, inho-
mogeneous media
7. Fast convolution with free-space Green’s functions
8. Fast numerical solution of the electromagnetic medium scattering problem
9. VIE Methods for Time-Harmonic Solutions of Maxwell’s Equations: Discretization, Spectrum and Preconditioning
10. The Discrete Dipole Approximation: an overview and recent developments
401-3920-17LNumerical Analysis Seminar: Mathematics for Biomimetics Restricted registration - show details
Number of participants limited to 8.
W4 credits2SH. Ammari, A. Vanel
AbstractThe aim of this seminar is to explore how we can learn from Nature to provide new approaches to solving some of the most challenging problems in sensing systems and materials science.

An emphasis will be put on the mathematical foundation of bio-inspired perception algorithms in electrolocation and echolocation.
Objective
401-3620-70LStudent Seminar in Statistics: Multiple Testing for Modern Data Science Restricted registration - show details
Number of participants limited to 24

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.
W4 credits2SM. Löffler, A. Taeb
AbstractThe course encompasses a review of approaches to multiple testing.
ObjectiveThe students understand the relevance of multiple testing in modern applications. Further, they learn about two commonly used measures -- namely family-wise-error-rate (FWER) and false discovery rate (FDR) -- and approaches to control for them.
ContentIn modern statistical applications it is often desired to perform thousands of statistical tests simultaneously. Performing a test at a desired level (e.g. 0.05) for each variable separately will result in many false positives. In science this is known as the ‘reproducibility crisis’.
In this seminar we will review and discuss approaches to deal with this issue. First, we will consider the strong notion of FWER and how to control it via Bonferroni correction, permutation tests, step-up and hierarchical procedures or Tukey’s higher criticism. In the second part of the seminar we will investigate the less conservative FDR, discussing the classical Benjamini-Hochberg procedure, as well as more modern methods such as Knockoffs and Bayesian approaches. Throughout, we highlight the utility of discussed methods for real world applications.
LiteratureLecture 1: Bonferroni and Simes
Link Link
Lecture 2: Permutation tests
Link Link
Lecture 3: Hierarchical testing
Link
Link
Link
Lecture 4: Higher criticism
Methodology: Link and for theoretical reference Link
Application: Link and for more reference
Link
Lecture 5: Benjamini-Hochberg (BH) with martingales
Link, Link
Lecture 6: FDR control under dependence
Link
Link
Lecture 7: Empirical null distribution
Link
Link
Lecture 8: Bayes FDR methods
Link
Link
Lecture 9: SLOPE
Link
Link
Lecture 10: Knockoffs
Link
Link
Lecture 11: Generalization of FWER and connections to FDR
Link
Link
Lecture 12: Exploratory testing
Link
Link
Prerequisites / NoticeEvery lecture will consist of an oral presentation highlighting key ideas of selected papers by a pair of students. Another two students will be responsible for asking questions during the presentation and providing a discussion of the pros+cons of the papers at the end. Finally, an additional two students are responsible for giving an evaluation on the quality of the presentations/discussions and provide constructive feedback for improvement.
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.
NumberTitleTypeECTSHoursLecturers
401-3750-01LSemester Paper Restricted registration - show details
Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see Link
W8 credits11ASupervisors
AbstractSemester 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.
Objective
Prerequisites / NoticeThere 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.
401-3750-02LSemester Paper (No. 2) Restricted registration - show details
Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see Link
W8 credits11ASupervisors
AbstractSemester 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.
Objective
Prerequisites / NoticeThere 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.
401-3750-03LSemester Paper (No. 3) Restricted registration - show details
Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see Link
W8 credits11ASupervisors
AbstractSemester 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.
Objective
Prerequisites / NoticeThere 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.
GESS 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
» Recommended Science in Perspective (Type B) for D-MATH.
Master's Thesis
NumberTitleTypeECTSHoursLecturers
401-2000-00LScientific Works in Mathematics
Target audience:
Third year Bachelor students;
Master students who cannot document to have received an adequate training in working scientifically.
O0 creditsM. Burger, E. Kowalski
AbstractIntroduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)
ObjectiveLearn 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
Prerequisites / NoticeDirective Link
401-2000-01LLunch Sessions – Thesis Basics for Mathematics Students
Details and registration for the optional MathBib training course: Link
Z0 creditsSpeakers
AbstractOptional MathBib training course
Objective
401-4990-00LMaster's Thesis Restricted registration - show details
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 Link
O30 credits57DSupervisors
AbstractThe 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.
Objective
Additional Courses
NumberTitleTypeECTSHoursLecturers
401-5000-00LZurich Colloquium in Mathematics Information E-0 creditsR. Abgrall, A. Bandeira, M. Iacobelli, A. Iozzi, S. Mishra, R. Pandharipande, University lecturers
AbstractThe 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.
Objective
401-5990-00LZurich Graduate Colloquium Information E-0 credits1KA. Iozzi, University lecturers
AbstractThe 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.
Objective
401-4530-00LGeometry Graduate Colloquium Information E-0 credits1KSpeakers
Abstract
Objective
401-5110-00LNumber Theory Seminar Information E-0 credits1KÖ. Imamoglu, P. S. Jossen, E. Kowalski, P. D. Nelson, R. Pink, G. Wüstholz
AbstractResearch colloquium
Objective
401-5350-00LAnalysis Seminar Information E-0 credits1KM. Struwe, A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, T. Ilmanen, L. Kobel-Keller, University lecturers
AbstractResearch colloquium
Objective
401-5370-00LErgodic Theory and Dynamical Systems Information E-0 credits1KM. Akka Ginosar, M. Einsiedler, University lecturers
AbstractResearch colloquium
Objective
401-5530-00LGeometry Seminar Information E-0 credits1KM. Burger, M. Einsiedler, P. Feller, A. Iozzi, U. Lang, University lecturers
AbstractResearch colloquium
Objective
401-5580-00LSymplectic Geometry Seminar Information E-0 credits2KP. Biran, A. Cannas da Silva
AbstractResearch colloquium
Objective
401-5330-00LTalks in Mathematical Physics Information E-0 credits1KA. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, T. H. Willwacher, University lecturers
AbstractResearch colloquium
Objective
401-5650-00LZurich Colloquium in Applied and Computational Mathematics Information E-0 credits1KR. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, S. Mishra, S. Sauter, C. Schwab
AbstractResearch colloquium
Objective
401-5600-00LSeminar on Stochastic Processes Information
Does not take place this semester.
E-0 credits1KJ. Bertoin, A. Nikeghbali, B. D. Schlein, A.‑S. Sznitman, V. Tassion, W. Werner
AbstractResearch colloquium
Objective
401-5620-00LResearch Seminar on Statistics Information E-0 credits1KP. L. Bühlmann, M. H. Maathuis, N. Meinshausen, S. van de Geer, A. Bandeira, R. Furrer, L. Held, T. Hothorn, D. Kozbur, C. Uhler, M. Wolf
AbstractResearch colloquium
Objective
401-5640-00LZüKoSt: Seminar on Applied Statistics Information E-0 credits1KM. Kalisch, A. Bandeira, P. L. Bühlmann, R. Furrer, L. Held, T. Hothorn, M. H. Maathuis, M. Mächler, L. Meier, M. Robinson, C. Strobl, C. Uhler, S. van de Geer
AbstractAbout 5 talks on applied statistics.
ObjectiveSee how statistical methods are applied in practice.
ContentThere will be about 5 talks on how statistical methods are applied in practice.
Prerequisites / NoticeThis is no lecture. There is no exam and no credit points will be awarded. The current program can be found on the web:
Link
Course language is English or German and may depend on the speaker.
401-5680-00LFoundations of Data Science Seminar Information E-0 creditsP. L. Bühlmann, A. Bandeira, H. Bölcskei, J. M. Buhmann, T. Hofmann, A. Krause, A. Lapidoth, H.‑A. Loeliger, M. H. Maathuis, G. Rätsch, C. Uhler, S. van de Geer, F. Yang
AbstractResearch colloquium
Objective
401-5660-00LMath and Data (MAD+) Information E-0 credits1KA. Bandeira, external organisers
AbstractResearch colloquium
Objective
401-5910-00LTalks in Financial and Insurance Mathematics Information E-0 credits1KB. Acciaio, P. Cheridito, D. Possamaï, M. Schweizer, J. Teichmann, M. V. Wüthrich
AbstractResearch colloquium
Objective
ContentRegular research talks on various topics in mathematical finance and actuarial mathematics
401-5900-00LOptimization Seminar Information E-0 credits1KA. Bandeira, R. Weismantel, R. Zenklusen
AbstractLectures on current topics in optimization
ObjectiveExpose graduate students to ongoing research acitivites (including applications) in the domain of otimization.
ContentThis seminar is a forum for researchers interested in optimization theory and its applications. Speakers are expected to stimulate discussions on theoretical and applied aspects of optimization and related subjects. The focus is on efficient algorithms for continuous and discrete optimization problems, complexity analysis of algorithms and associated decision problems, approximation algorithms, mathematical modeling and solution procedures for real-world optimization problems in science, engineering, industries, public sectors etc.
401-5960-00LColloquium on Mathematics, Computer Science, and Education Information
Does not take place this semester.
Subject didactics for mathematics and computer science teachers.
E-0 creditsN. Hungerbühler, J. Hromkovic
AbstractDidactics colloquium
Objective
402-0101-00LThe Zurich Physics Colloquium Information E-0 credits1KS. Huber, A. Refregier, University lecturers
AbstractResearch colloquium
Objective
402-0800-00LThe Zurich Theoretical Physics Colloquium Information E-0 credits1KO. Zilberberg, University lecturers
AbstractResearch colloquium
ObjectiveThe Zurich Theoretical Physics Colloquium is jointly organized by the University of Zurich and ETH Zurich. Its mission is to bring both students and faculty with diverse interests in theoretical physics together. Leading experts explain the basic questions in their field of research and communicate the fascination for their work.
251-0100-00LComputer Science Colloquium Information E-0 credits2KLecturers
AbstractInvited talks, covering the entire scope of computer science. External Listeners are welcome at no charge. A detailed schedule is published at the beginning of each semester.
ObjectiveTop international computer scientists take the floor at the distinguished computer science colloquium. Our guest speakers present impacting topics across various areas of the discipline. The colloquium series is held every semester and also includes inaugural and farewell lectures of the department's professors. The colloquium is a noteworthy event for all graduate students. Outside attendance is equally welcome.
ContentEingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
Course Units for Additional Admission Requirements
The courses below are only available for MSc students with additional admission requirements.
NumberTitleTypeECTSHoursLecturers
406-2004-AALAlgebra II
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.
E-5 credits11RR. Pink
AbstractGalois theory and related topics.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
ObjectiveIntroduction to fundamentals of field extensions, Galois theory, and related topics.
ContentThe 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.
LiteratureJoseph 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 / NoticeAlgebra I, in Rotman's book this corresponds to the topics treated in the Chapters A3 and A4.
406-2005-AALAlgebra I and II
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.
E-12 credits26RR. Pink
AbstractIntroduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
Objective
ContentBasic notions and examples of groups;
Subgroups, Quotient groups and Homomorphisms,
Group actions and applications

Basic notions and examples of rings;
Ring Homomorphisms,
ideals, and quotient rings, rings of fractions
Euclidean domains, Principal ideal domains, Unique factorization
domains

Basic notions and examples of fields;
Field extensions, Algebraic extensions, Classical straight edge and compass constructions

Fundamentals of Galois theory
Representation theory of finite groups and algebras
LiteratureJoseph J. Rotman, "Advanced Modern Algebra" third edition, part 1,
Graduate Studies in Mathematics,Volume 165
American Mathematical Society
406-2303-AALComplex 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.
E-6 credits13RA. Bandeira
AbstractComplex 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.
Objective
LiteratureL. 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
406-2284-AALMeasure and Integration
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.
E-6 credits13RF. Da Lio
AbstractIntroduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language.
ObjectiveBasic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral.
Literature1. Lecture notes by Professor Michael Struwe (Link)
2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions"
3. Walter Rudin "Real and complex analysis"
4. R. Bartle The elements of Integration and Lebesgue Measure
5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. Link
406-2554-AALTopology
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.
E-6 credits13RA. Carlotto
AbstractTopological spaces, continuous maps, connectedness, compactness, metric spaces, quotient spaces, homotopy, fundamental group and covering spaces, van Kampen Theorem.
Objective
LiteratureJames Munkres: Topology
Prerequisites / NoticeThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2604-AALProbability and Statistics
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.
E-7 credits15RM. Schweizer
AbstractIntroduction to probability and statistics with many examples, based on chapters from the books "Probability and Random Processes" by G. Grimmett and D. Stirzaker and "Mathematical Statistics and Data Analysis" by J. Rice.
ObjectiveThe goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed.
ContentProbability:
Chapters 1-5 (Probabilities and events, Discrete and continuous random variables, Generating functions) and Sections 7.1-7.5 (Convergence of random variables) from the book "Probability and Random Processes". Most of this material is also covered in Chap. 1-5 of "Mathematical Statistics and Data Analysis", on a slightly easier level.

Statistics:
Sections 8.1 - 8.5 (Estimation of parameters), 9.1 - 9.4 (Testing Hypotheses), 11.1 - 11.3 (Comparing two samples) from "Mathematical Statistics and Data Analysis".
LiteratureGeoffrey Grimmett and David Stirzaker, Probability and Random Processes.
3rd Edition. Oxford University Press, 2001.

John A. Rice, Mathematical Statistics and Data Analysis, 3rd edition.
Duxbury Press, 2006.