# Search result: Catalogue data in Autumn Semester 2021

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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401-3059-00L | Combinatorics II | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3033-00L | Gödel's Theorems | W | 8 credits | 3V + 1U | L. Halbeisen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Die Vorlesung besteht aus drei Teilen: Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe. Teil II behandelt den Gödel'schen Vollständigkeitssatz Teil III behandelt die Gödel'schen Unvollständigkeitssätze | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | L. Halbeisen and R. Krapf: Gödel's Theorems and Zermelo's Axioms: a firm foundation of mathematics, Birkhäuser-Verlag, Basel (2020) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

401-3533-70L | Topics in Riemannian Geometry | W | 6 credits | 3V | U. Lang | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Selected topics from Riemannian geometry in the large: triangle and volume comparison theorems, Milnor's results on growth of the fundamental group, Gromov-Hausdorff convergence, Cheeger's diffeomorphism finiteness theorem, the Besson-Courtois-Gallot barycenter method and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Lecture notes | Lecture notes will be provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4207-71L | Coxeter Groups from a Geometric Viewpoint | W | 4 credits | 2V | M. Cordes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Introduction to Coxeter groups and the spaces on which they act. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Understand the basic properties of Coxeter groups. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Brown, Kenneth S. "Buildings" Davis, Michael "The geometry and topology of Coxeter groups" | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Students must have taken a first course in algebraic topology or be familiar with fundamental groups and covering spaces. They should also be familiar with groups and group actions. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3057-00L | Finite Geometries IIDoes 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

401-4421-71L | Harmonic Analysis | W | 4 credits | 2V | A. Figalli | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The goal of this class is to give an introduction to harmonic analysis, covering a series of classical important results such as: 1) interpolation theorems 2) convergence properties of Fourier series 3) Calderón-Zygmund operators 4) Littlewood-Paley decomposition 5) Hardy and BMO spaces | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | I plan to write some notes of the class. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | There is no official textbook. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4475-71L | Microlocal Analysis | W | 6 credits | 3G | P. Hintz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Microlocal analysis is the analysis of partial differential equations in phase space. The first half of the course introduces basic notions such as pseudodifferential operators, wave front sets of distributions, and elliptic parametrices. The second half develops modern tools for the study of nonelliptic equations, with applications to wave equations arising in general relativity. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Students will be able to analyze linear partial differential operators (with smooth coefficients) and their solutions in phase space, i.e. in the cotangent bundle. For various classes of operators including, but not limited to, elliptic and hyperbolic operators, they will be able to prove existence and uniqueness (possibly up to finite-dimensional obstructions) of solutions, and study the precise regularity properties of solutions. The first goal is to construct and apply parametrices (approximate inverses) or approximate solutions of PDEs using suitable calculi of pseudodifferential operators (ps.d.o.s). This requires defining ps.d.o.s and the associated symbol calculus on Euclidean space, proving the coordinate invariance of ps.d.o.s, and defining a ps.d.o. calculus on manifolds (including mapping properties on Sobolev spaces). The second goal is to analyze distributions and operations on them (such as: products, restrictions to submanifolds) using information about their wave front sets or other microlocal regularity information. Students will in particular be able to compute the wave front set of distributions. The third goal is to infer microlocal properties (in the sense of wave front sets) of solutions of general linear PDEs, with a focus on elliptic, hyperbolic and certain degenerate hyperbolic PDE. For hyperbolic operators, this includes proving the Duistermaat-Hörmander theorem on the propagation of singularities. For certain degenerate hyperbolic operators, students will apply positive commutator methods to prove results on the propagation of microlocal regularity at critical or invariant sets for the Hamiltonian vector field of the principal symbol of the partial differential operator under study. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Tempered distributions, Sobolev spaces, Schwartz kernel theorem. Symbols, asymptotic summation. Pseudodifferential operators on Euclidean space: composition, principal symbols and the symbol calculus, elliptic parametrix construction, boundedness on Sobolev spaces. Pseudodifferential operators on manifolds, elliptic operators on compact manifolds and Fredholm theory, basic symplectic geometry. Microlocalization: wave front set, characteristic set; pairings, products, restrictions of distributions. Hyperbolic evolution equations: existence and uniqueness of solutions, Egorov's theorem. Propagation of singularities: the Duistermaat-Hörmander theorem, microlocal estimates at radial sets. Applications to general relativity: asymptotic behavior of waves on de Sitter space. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture notes will be made available on the course website. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Lars Hörmander, "The Analysis of Linear Partial Differential Operators", Volumes I and III. Alain Grigis and Johannes Sjöstrand, "Microlocal Analysis for differential operators: an introduction". | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Students are expected to have a good understanding of functional analysis. Familiarity with distribution theory, the Fourier transform, and analysis on manifolds is useful but not strictly necessary; the relevant notions will be recalled in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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Selection: Further Realms | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3502-71L | 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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401-3503-71L | 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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401-3504-71L | 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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401-0000-00L | Communication in MathematicsDoes not take place this semester. | W | 2 credits | 1V | W. Merry | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Don'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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Knowing how to present written mathematics in a structured and clear manner. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Topics 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 notes | Full lecture notes will be made available on my website: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | There are no formal mathematical prerequisites. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

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

401-4657-00L | Numerical Analysis of Stochastic Ordinary Differential Equations Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods" | W | 6 credits | 3V + 1U | A. Stein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Course 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Generation 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 notes | There will be English, typed lecture notes for registered participants in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | P. 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 / Notice | Prerequisites: Mandatory: Probability and measure theory, basic numerical analysis and basics of MATLAB/Python programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday September 22, 2021. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4785-00L | Mathematical and Computational Methods in Photonics | W | 8 credits | 4G | H. Ammari | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-5003-71L | At the Interface Between Semiclassical Analysis and Numerical Analysis of Wave-Scattering Problems | W | 4 credits | 2V | E. Spence | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Postgraduate degree lecture | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Content | Semiclassical analysis (SCA) is a branch of microlocal analysis concerned with rigorously analysing PDEs with large (or small) parameters. On the other hand, numerical analysis (NA) seeks to design numerical methods that are accurate, efficient, and robust, with theorems guaranteeing these properties. In the context of high-frequency wave scattering, both SCA and NA share the same goal – that of understanding the behaviour of the scattered wave – but these two fields have operated largely in isolation, mainly because the tools and techniques of the two fields are somewhat disjoint. This by-and-large self-contained course focuses on the Helmholtz equation, which is arguably the simplest possible model of wave propagation. Our first goal will be to show how even relatively-simple tools from semiclassical analysis can be used to prove fundamental results about the numerical analysis of finite-element method applied to the high-frequency Helmholtz equation. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The course will aim at being accessible both to students coming from a numerical-analysis/applied-maths background and to students coming from an analysis background. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

401-4607-67L | Schramm-Loewner Evolutions | W | 4 credits | 2V | W. Werner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This advanced course will be an introduction to SLE (Schramm-Loewner Evolutions), which are a class of conformally invariant random curves in the plane. We will discuss their construction and some of their main properties. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Prerequisites / Notice | Knowledge of Brownian motion and stochastic calculus and basic knowledge of complex analysis (Riemann's mapping theorem). Familiarity of lattice models such as percolation or the Ising model can be useful but not necessary. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3822-17L | Ising Model | W | 4 credits | 2V | V. Tassion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Prerequisites / Notice | - Probability Theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3628-14L | Bayesian Statistics | W | 4 credits | 2V | F. Sigrist | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Topics 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 notes | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Christian 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 / Notice | Familiarity 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-0625-01L | Applied Analysis of Variance and Experimental Design | W | 5 credits | 2V + 1U | L. Meier | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Principles 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Participants 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Principles 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-0649-00L | Applied Statistical Regression | W | 5 credits | 2V + 1U | M. Dettling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The 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 notes | A script will be available. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Faraway (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 / Notice | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3627-00L | High-Dimensional Statistics | W | 4 credits | 2V | P. 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Lasso 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Peter 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 / Notice | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4623-00L | Time Series AnalysisDoes not take place this semester. | W | 6 credits | 3G | F. Balabdaoui | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | This 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | The main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3612-00L | Stochastic SimulationDoes not take place this semester. | W | 5 credits | 3G | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | P. 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 / Notice | Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3925-00L | Non-Life Insurance: Mathematics and Statistics | W | 8 credits | 4V + 1U | M. V. Wüthrich | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The 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 notes | M.V. Wüthrich, Non-Life Insurance: Mathematics & Statistics Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | M.V. Wüthrich, M. Merz. Statistical Foundations of Actuarial Learning and its Applications Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3922-00L | Life Insurance Mathematics | W | 4 credits | 2V | M. Koller | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3928-00L | Reinsurance Analytics | W | 4 credits | 2V | P. Antal, P. Arbenz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

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

Lecture notes | Slides and lecture notes will be made available. An excerpt of last year's lecture notes is available here: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Basic knowledge in statistics, probability theory, and actuarial techniques | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3927-00L | Mathematical Modelling in Life Insurance | W | 4 credits | 2V | T. J. Peter | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | In 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Following 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 notes | Lectures notes and slides will be provided | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

402-0843-00L | Quantum Field Theory ISpecial Students UZH must book the module PHY551 directly at UZH. | W | 10 credits | 4V + 2U | G. M. Graf | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Will be provided as the course progresses | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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402-0861-00L | Statistical Physics | W | 10 credits | 4V + 2U | M. Sigrist | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This lecture covers the concepts of classical and quantum statistical physics. Several techniques such as second quantization formalism for fermions, bosons, photons and phonons as well as mean field theory and self-consistent field approximation. These are used to discuss phase transitions, critical phenomena and superfluidity. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | This lecture gives an introduction in 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Kinetic approach to statistical physics: H-theorem, detailed balance and equilibirium conditions. Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems. Quantum statistical physics: density matrix, ensembles, Fermi gas, Bose gas (Bose-Einstein condensation), photons and phonons. Identical quantum particles: many body wave functions, second quantization formalism, equation of motion, correlation functions, selected applications, e.g. Bose-Einstein condensate and coherent state, phonons in elastic media and melting. One-dimensional interacting systems. Phase transitions: mean field approach to Ising model, Gaussian transformation, Ginzburg-Landau theory (Ginzburg criterion), self-consistent field approach, critical phenomena, Peierls' arguments on long-range order. Superfluidity: Quantum liquid Helium: Bogolyubov theory and collective excitations, Gross-Pitaevskii equations, Berezinskii-Kosterlitz-Thouless transition. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture notes available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | No specific book is used for the course. Relevant literature will be given in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0830-00L | General Relativity Special Students UZH must book the module PHY511 directly at UZH. | W | 10 credits | 4V + 2U | C. Anastasiou | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Basic 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Introduction 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Suggested 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0897-00L | Introduction to String Theory | W | 6 credits | 2V + 1U | J. Brödel | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. It is related to numerous interesting topics and questions in quantum field theory. In this course, an introduction to the basics of string theory is provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Within this course, a basic understanding and overview of the concepts and notions employed in string theory shall be given. More advanced topics will be touched upon towards the end of the course briefly in order to foster further research. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | - mechanics of point particles and extended objects - string modes and their quantisation; higher dimensions, supersymmetry - D-branes, T-duality - supergravity as a low-energy effective theory, strings on curved backgrounds - two-dimensional field theories (classical/quantum, conformal/non-conformal) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | D. Lust, S. Theisen, Lectures on String Theory, Lecture Notes in Physics, Springer (1989). M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory I, CUP (1987). B. Zwiebach, A First Course in String Theory, CUP (2004). J. Polchinski, String Theory I & II, CUP (1998). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Recommended: Quantum Field Theory I (in parallel) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Selection: Mathematical Optimization, Discrete Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3055-64L | Algebraic Methods in Combinatorics | W | 6 credits | 2V + 1U | B. Sudakov | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools. One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to): Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

Auswahl: Theoretical Computer Science, Discrete Mathematics | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

263-4500-00L | Advanced Algorithms Takes place for the last time. | W | 9 credits | 3V + 2U + 3A | M. Ghaffari, G. Zuzic | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | This course familiarizes the students with some of the main tools and techniques in modern subareas of algorithm design. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | The 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 notes | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | This 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

252-1425-00L | Geometry: Combinatorics and Algorithms | W | 8 credits | 3V + 2U + 2A | B. Gärtner, E. Welzl, M. Hoffmann, M. Wettstein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Geometric 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?) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | The 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Planar 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 notes | yes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literature | Mark 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 / Notice | Prerequisites: 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 10 credits | 3V + 2U + 4A | A. Steger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Las 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | After 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Randomized 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 notes | Yes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

401-4944-20L | Mathematics of Data Science | W | 8 credits | 4G | A. Bandeira | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Mostly self-contained, but fast-paced, introductory masters level course on various theoretical aspects of algorithms that aim to extract information from data. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Introduction to various mathematical aspects of Data Science. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | These topics lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature a couple of Mathematical Open Problem(s) related to Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The main mathematical tools used will be Probability, Linear Algebra (and real analysis), and a working knowledge of these subjects is required. In addition to these prerequisites, this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs. We encourage students who are interested in mathematical data science to take both this course and ``227-0434-10L Mathematics of Information'' taught by Prof. H. Bölcskei. The two courses are designed to be complementary. A. Bandeira and H. Bölcskei | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

227-0423-00L | Neural Network Theory | W | 4 credits | 2V + 1U | H. Bölcskei | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | The class focuses on fundamental mathematical aspects of neural networks with an emphasis on deep networks: Universal approximation theorems, capacity of separating surfaces, generalization, fundamental limits of deep neural network learning, VC dimension. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | After 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 neural networks. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | 1. 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. Vapnik-Chervonenkis (VC) dimension 7. VC dimension of neural networks 8. Generalization error in neural network learning | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Detailed lecture notes are available on the course web page Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | This course is aimed at students with a strong mathematical background in general, and in linear algebra, analysis, and probability theory in particular. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3502-71L | Reading Course Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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401-3503-71L | Reading Course Link and register your reading course in myStudies. | W | 3 credits | 6A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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401-3504-71L | Reading Course Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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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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401-0000-00L | Communication in MathematicsDoes not take place this semester. | W | 2 credits | 1V | W. Merry | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Abstract | Don'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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Objective | Knowing how to present written mathematics in a structured and clear manner. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Content | Topics 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 notes | Full lecture notes will be made available on my website: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | There are no formal mathematical prerequisites. |