# Search result: Catalogue data in Spring Semester 2023

Mathematics Master | ||||||||||||||||||||||||

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 | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||
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401-3058-00L | Combinatorics I | W | 4 credits | 2G | N. Hungerbühler | |||||||||||||||||||

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

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

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

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

401-3035-00L | Forcing: An Introduction to Independence Proofs | W | 8 credits | 3V + 1U | L. Halbeisen | |||||||||||||||||||

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

Objective | Die Forcing-Technik kennenlernen und verschiedene Unabhaengigkeitsbeweise fuehren koennen. | |||||||||||||||||||||||

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

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

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

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

Selection: Geometry | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-3056-00L | Finite Geometries I Does not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |||||||||||||||||||

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

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

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

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | |||||||||||||||||||||||

401-4146-22L | Derived Algebraic Geometry | W | 4 credits | 2V | A. Bojko | |||||||||||||||||||

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

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

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

Prerequisites / Notice | One should have some understanding of algebraic geometry (in particular intersection theory), algebraic topology and category theory. Familiarity with some enumerative geometry using virtual fundamental classes would be helpful for understanding the goal of the course. | |||||||||||||||||||||||

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401-3574-61L | Introduction to Knot Theory Does not take place this semester. | W | 6 credits | 3G | ||||||||||||||||||||

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

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

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

Literature | An extensive bibliography will be handed out in the course. | |||||||||||||||||||||||

Prerequisites / Notice | Prerequisites are some elementary knowledge of algebra and topology. | |||||||||||||||||||||||

Selection: Analysis | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-4494-23L | Variational Problems and PDEs | W | 4 credits | 2V | A. Figalli | |||||||||||||||||||

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

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

Prerequisites / Notice | Functional Analysis I | |||||||||||||||||||||||

401-4834-23L | Nonlinear Wave Equations with Applications to General Relativity | W | 4 credits | 2V | C. Kehle | |||||||||||||||||||

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

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

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

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401-3374-23L | Dynamical Systems and Ergodic Theory (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT733 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 2U | University lecturers | |||||||||||||||||||

Abstract | Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. | |||||||||||||||||||||||

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

Content | Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it in average. We will give a strong emphasis on presenting many fundamental examples of dynamical systems. Driven by the examples, we will first introduce some of the phenomena and main concepts which one is interested in studying. We will then formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications for example to number theory, information theory and Internet search engines. Topics which will be covered include: -Basic examples of dynamical systems (e.g. rotations and doubling map; baker’s map, CAT map and hyperbolic toral automorphisms; the Gauss map and continued fractions); -Elements of topological dynamics (minimality; topological conjugacy; topological mixing; topological entropy); -Elements of symbolic dynamics (shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding); -Introduction to ergodic theory: invariant measures; Poincare' recurrence; ergodicity; mixing; the Birkhoff Ergodic Theorem and applications; Markov measures; the ergodic theorem for Markov chains and applications to Internet Search; measure theoretic entropy; -Selected topics (time permitting): Shannon-McMillan-Breiman theorem; Lyapunov exponents and multiplicative ergodic theorem; continuous time dynamical systems and some mathematical billiards. | |||||||||||||||||||||||

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

Literature | Textbooks which can be used as additional reference for some of the topics include: -B. Hasselblatt and A. Katok, Dynamics: A first course. (Cambdirge University Press, 2003) – Chapters 7,8,10 and 15 -M. Brin and G. Stuck, Introduction to Dynamical Systems. (Cambridge University Press, 2002) – Chapters 1-4 -Omri Sarig, Lectures Notes on Ergodic Theory (Available Online), Topics from Chapter 1 and 2 | |||||||||||||||||||||||

Prerequisites / Notice | Prior Knowledge Basic knowledge of measure theory and integration. | |||||||||||||||||||||||

Selection: Further Realms and Some UZH Courses | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-3502-22L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |||||||||||||||||||

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

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401-3503-22L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 3 credits | 6A | Supervisors | |||||||||||||||||||

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

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401-3504-22L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |||||||||||||||||||

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

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401-3504-02L | Reading Course (No. 2) Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |||||||||||||||||||

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401-3504-03L | Reading Course (No. 3) Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |||||||||||||||||||

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401-8144-23L | Singular Foliations (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT555 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 2U | University lecturers | |||||||||||||||||||

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

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

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

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

Electives: Applied Mathematics and Further Application-Oriented Fields ¬ | ||||||||||||||||||||||||

Selection: Numerical Analysis | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-4658-00L | Numerical Methods for Finance | W | 6 credits | 3V + 1U | C. Schwab, A. Stein | |||||||||||||||||||

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

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

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

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

Literature | Main reference (course text): N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013. Supplementary texts: R. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004. Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005. D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008. J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000. | |||||||||||||||||||||||

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

401-4656-21L | Deep Learning in Scientific Computing Aimed at students in a Master's Programme in Mathematics, Engineering and Physics. | W | 6 credits | 2V + 1U | S. Mishra, B. Moseley | |||||||||||||||||||

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

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

Content | A selection of the following topics will be presented in the lectures. 1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces. 2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets. 3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors. 4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error. 5. Uncertainty Quantification for PDEs with supervised learning algorithms. 6. Deep Neural Networks as Reduced order models and prediction of solution fields. 7. Active Learning algorithms for PDE constrained optimization. 8. Recurrent Neural Networks and prediction of time series for dynamical systems. 9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs. 10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation. All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others. | |||||||||||||||||||||||

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

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

Prerequisites / Notice | The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications. Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful. | |||||||||||||||||||||||

401-4652-23L | Inverse Problems | W | 4 credits | 2G | R. Alaifari | |||||||||||||||||||

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

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

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

Lecture notes | The lecture notes will be made available during the semester. | |||||||||||||||||||||||

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

Prerequisites / Notice | Analysis, linear algebra, numerical analysis, ideal but not necessary: functional analysis | |||||||||||||||||||||||

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Selection: Probability Theory, Statistics | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-4604-23L | First Passage Percolation and Large Deviations | W | 4 credits | 2V | B. Dembin | |||||||||||||||||||

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

Objective | The model of first passage percolation (FPP) was introduced in 1965 by Hammersley and Welsh to study the spread of a fluid through a random medium. The model is defined on the lattice (Z^d,E^d) by assigning independently to each edge a positive random variable representing the time to cross the edge. This induces a random metric on the lattice, where the time between two vertices corresponds to the time of the shortest path. In this course, our goal is to study the asymptotic properties of this random metric, as well as the time-minimizing paths (geodesics). In particular, we will study time and spatial fluctuations of geodesics. The objectives of this class are two-fold. First, discover an active field of research, have an overview of recent results and major open questions. Second, get familiar with various tools and concepts of probability that are not specific to this model. In particular, this class will contain an introduction to large deviations theory. | |||||||||||||||||||||||

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

Prerequisites / Notice | Basic knowledge in probability | |||||||||||||||||||||||

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401-4626-00L | Advanced Statistical Modelling: Mixed ModelsDoes not take place this semester. | W | 4 credits | 2V | M. Mächler | |||||||||||||||||||

Abstract | Mixed Models = (*| generalized| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms. In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences. | |||||||||||||||||||||||

Objective | - Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models. - Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis: "fit -> interpret & diagnose -> modify the fit -> interpret & ...." - Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software. | |||||||||||||||||||||||

Content | The lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online. Inference (significance of factors, confidence intervals) will focus on the more realistic *un*balanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference. | |||||||||||||||||||||||

Lecture notes | We will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples. These lecture notes and all R scripts are made available from Link | |||||||||||||||||||||||

Literature | (see web page and lecture notes) | |||||||||||||||||||||||

Prerequisites / Notice | - We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..). Typically this means at least two classes of (math based) statistics, say 1. Intro to probability and statistics 2. (Applied) regression including Matrix-Vector notation Y = X b + E - Basic (1 semester) "Matrix calculus" / linear algebra is also assumed. - If familiarity with [R](Link) is not given, it should be acquired during the course (by the student on own initiative). | |||||||||||||||||||||||

Selection: Financial and Insurance Mathematics | ||||||||||||||||||||||||

Number | Title | Type | ECTS | Hours | Lecturers | |||||||||||||||||||

401-3629-00L | Quantitative Risk Management | W | 4 credits | 2V + 1U | P. Cheridito | |||||||||||||||||||

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

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

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

Lecture notes | Course material is available on Link | |||||||||||||||||||||||

Literature | Quantitative Risk Management: Concepts, Techniques and Tools AJ McNeil, R Frey and P Embrechts Princeton University Press, Princeton, 2015 (Revised Edition) Link | |||||||||||||||||||||||

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

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