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 applicationoriented fields.  
Electives: Pure Mathematics  
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic  
Number  Title  Type  ECTS  Hours  Lecturers  

401305900L  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.  
401303300L  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, BirkhäuserVerlag, Basel (2020)  
Selection: Geometry  
Number  Title  Type  ECTS  Hours  Lecturers  
401353370L  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, GromovHausdorff convergence, Cheeger's diffeomorphism finiteness theorem, the BessonCourtoisGallot barycenter method and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces.  
Objective  
Lecture notes  Lecture notes will be provided.  
401420771L  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.  
401305700L  Finite Geometries II 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 thirtysix officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and PappusPascal, 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  
401442171L  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) CalderónZygmund operators 4) LittlewoodPaley decomposition 5) Hardy and BMO spaces  
Objective  
Lecture notes  I plan to write some notes of the class.  
Literature  There is no official textbook.  
401447571L  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 finitedimensional 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 DuistermaatHörmander 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 DuistermaatHörmander 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 
 
Selection: Further Realms  
Number  Title  Type  ECTS  Hours  Lecturers  
401350271L  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.  
Objective  
401350371L  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.  
Objective  
401350471L  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.  
Objective  
401350402L  Reading Course (No. 2) 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.  
Objective  
401000000L  Communication in Mathematics Does 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 ApplicationOriented Fields ¬  
Selection: Numerical Analysis  
Number  Title  Type  ECTS  Hours  Lecturers  
401465700L  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. SpringerVerlag, New York, 2004. P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. SpringerVerlag, 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.  
401478500L  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, superfocusing & superresolution 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. Lightbased technologies can be used effectively for the very early detection of diseases, with noninvasive imaging techniques or pointofcare 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 pacemakers to synthetic bones, and from endoscopes to the microcameras used in invivo 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, superresolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multimathematics 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 operatorvalued 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, superresolution, and metamaterials.  
401500371L  At the Interface Between Semiclassical Analysis and Numerical Analysis of WaveScattering Problems  W  4 credits  2V  E. Spence  
Abstract  Postgraduate degree lecture  
Objective  
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 highfrequency 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 byandlarge selfcontained 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 relativelysimple tools from semiclassical analysis can be used to prove fundamental results about the numerical analysis of finiteelement method applied to the highfrequency Helmholtz equation.  
Prerequisites / Notice  The course will aim at being accessible both to students coming from a numericalanalysis/appliedmaths background and to students coming from an analysis background.  
Selection: Probability Theory, Statistics  
Number  Title  Type  ECTS  Hours  Lecturers  
401460767L  SchrammLoewner Evolutions  W  4 credits  2V  W. Werner  
Abstract  This advanced course will be an introduction to SLE (SchrammLoewner Evolutions), which are a class of conformally invariant random curves in the plane. We will discuss their construction and some of their main properties.  
Objective  
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.  
401382217L  Ising Model  W  4 credits  2V  V. Tassion  
Abstract  
Objective  
Prerequisites / Notice   Probability Theory.  
401362814L  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, hyperg 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.  
401062501L  Applied Analysis of Variance and Experimental Design  W  5 credits  2V + 1U  L. Meier  
Abstract  Principles of experimental design, oneway analysis of variance, contrasts and multiple comparisons, multifactor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, splitplot designs, incomplete block designs, twoseries 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, oneway analysis of variance, contrasts and multiple comparisons, multifactor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, splitplot designs, incomplete block designs, twoseries 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, opensource statistical software R, for which an introduction will be held.  
401064900L  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, opensource statistical software package R, for which an introduction will be held. In the Mathematics Bachelor and Master programmes, the two course units 401064900L "Applied Statistical Regression" and 401362200L "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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