Suchergebnis: Katalogdaten im Herbstsemester 2021
Mathematik Master  | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 WahlfächerFür das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Wahlfächer aus Bereichen der reinen Mathematik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Algebra, Zahlentheorie, Topologie, diskrete Mathematik, Logik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-3059-00L | Kombinatorik II | W | 4 KP | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Der Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3033-00L | Die Gödel'schen Sätze | W | 8 KP | 3V + 1U | L. Halbeisen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Das Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Syntax und Semantik der Prädikatenlogik Gödel'scher Vollständigkeitssatz Gödel'sche Unvollständigkeitssätze | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | L. Halbeisen und R. Krapf: Gödel's Theorems and Zermelo's Axioms: a firm foundation of mathematics, Birkhäuser-Verlag, Basel (2020) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Geometrie | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3533-70L | Topics in Riemannian Geometry | W | 6 KP | 3V | U. Lang | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | Lecture notes will be provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4207-71L | Coxeter Groups from a Geometric Viewpoint | W | 4 KP | 2V | M. Cordes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Introduction to Coxeter groups and the spaces on which they act. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Understand the basic properties of Coxeter groups. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | Brown, Kenneth S. "Buildings" Davis, Michael "The geometry and topology of Coxeter groups" | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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 | Endliche Geometrien II Findet dieses Semester nicht statt. | W | 4 KP | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Endliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Endliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Endliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | - 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Analysis | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4421-71L | Harmonic Analysis | W | 4 KP | 2V | A. Figalli | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | I plan to write some notes of the class. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | There is no official textbook. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4475-71L | Microlocal Analysis | W | 6 KP | 3G | P. Hintz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | Lecture notes will be made available on the course website. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | 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". | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kompetenzen |
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| Auswahl: Weitere Gebiete | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 KP | 4A | Betreuer/innen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 KP | 6A | Betreuer/innen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 KP | 9A | Betreuer/innen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3504-02L | Reading Course (No. 2) To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 4 KP | 9A | Betreuer/innen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-0000-00L | Communication in Mathematics Findet dieses Semester nicht statt. | W | 2 KP | 1V | W. Merry | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Knowing how to present written mathematics in a structured and clear manner. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | Full lecture notes will be made available on my website: https://www.merry.io/teaching/ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | There are no formal mathematical prerequisites. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Wahlfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Numerische Mathematik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 KP | 3V + 1U | A. Stein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | There will be English, typed lecture notes for registered participants in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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 KP | 4G | H. Ammari | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | 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 KP | 2V | E. Spence | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Postgraduate degree lecture | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4607-67L | Schramm-Loewner Evolutions | W | 4 KP | 2V | W. Werner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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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| Voraussetzungen / Besonderes | 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 KP | 2V | V. Tassion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | - Probability Theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3628-14L | Bayesian Statistics | W | 4 KP | 2V | F. Sigrist | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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 KP | 2V + 1U | L. Meier | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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 KP | 2V + 1U | M. Dettling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | A script will be available. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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