# Suchergebnis: Katalogdaten im Herbstsemester 2021

Doktorat Departement Mathematik Mehr Informationen unter: Link Die Liste der Lehrveranstaltungen (samt der zugehörigen Anzahl Kreditpunkte) für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht. Link ACHTUNG: Kreditpunkte fürs Doktoratsstudium sind nicht mit ECTS-Kreditpunkten zu verwechseln! | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Graduate School / Graduiertenkolleg Offizielle Website der Zurich Graduate School in Mathematics: Link | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-5005-71L | Randomization and Dimensionality in Risk Modeling | W | 0 KP | 2V | H. Albrecher | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Nachdiplom lecture | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Over the years, randomization has proven to be a powerful tool in the modeling of risks on several levels: for computational purposes, in uncovering connections between different models, but also in the consideration and generation of physical and/or synthetic scenarios in risk management. A second, and in part connected, theme is the parsimonious and structure-preserving refinement of stochastic models via matrix-valued parameters, and related questions concerning the appropriate and effective dimension of models for a given purpose. This lecture will deal with various recent advances in these fields, and also illustrate concrete applications in insurance and finance, including the optimal design of reinsurance treaties and the probabilistic analysis of the profitability of blockchain mining. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3225-00L | Introduction to Lie Groups | W | 8 KP | 4G | A. Iozzi | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser) A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73) F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer) H. Samelson: "Notes on Lie algebras" (Springer, '90) S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78) A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester. Course webpage: Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3533-70L | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3001-61L | Algebraic Topology I | W | 8 KP | 4G | W. Merry | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | 1) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 2) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3059-00L | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

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

401-4607-67L | 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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Voraussetzungen / Besonderes | - Probability Theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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

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

Inhalt | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Link | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3621-00L | Fundamentals of Mathematical Statistics | W | 10 KP | 4V + 1U | S. van de Geer | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The course covers the basics of inferential statistics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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401-3622-00L | Statistical Modelling | W | 8 KP | 4G | C. Heinze-Deml | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | In der Regression wird die Abhängigkeit einer zufälligen Response-Variablen von anderen Variablen untersucht. Wir betrachten die Theorie der linearen Regression mit einer oder mehreren Ko-Variablen, hoch-dimensionale lineare Modelle, nicht-lineare Modelle und verallgemeinerte lineare Modelle, Robuste Methoden, Modellwahl und nicht-parametrische Modelle. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Einführung in Theorie und Praxis eines umfassenden und vielbenutzten Teilgebiets der Statistik, unter Berücksichtigung neuerer Entwicklungen. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | In der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Hoch-dimensionale Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression. Durchrechnung und Diskussion von Anwendungsbeispielen. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | This is the course unit with former course title "Regression". Credits cannot be recognised for both courses 401-3622-00L Statistical Modelling and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4623-00L | Time Series AnalysisFindet dieses Semester nicht statt. | W | 6 KP | 3G | F. Balabdaoui | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

Voraussetzungen / Besonderes | Basic knowledge in probability and statistics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3627-00L | High-Dimensional Statistics | W | 4 KP | 2V | P. L. Bühlmann | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | "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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

Inhalt | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

401-3612-00L | Stochastic SimulationFindet dieses Semester nicht statt. | W | 5 KP | 3G | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | 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). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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

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