# Suchergebnis: Katalogdaten im Frühjahrssemester 2021

Mathematik Master | ||||||

Wahlfächer Für 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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401-4116-12L | Lectures on Drinfeld Modules | W | 6 KP | 3V | R. Pink | |

Kurzbeschreibung | Drinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc. | |||||

Lernziel | ||||||

Inhalt | A central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations. A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them. | |||||

Literatur | Drinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592. Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91. Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996. Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004. Further literature will be indicated during the course | |||||

401-3109-65L | Probabilistic Number Theory | W | 8 KP | 4G | E. Kowalski | |

Kurzbeschreibung | The course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums. | |||||

Lernziel | The goal of the course is to present some results of probabilistic number theory in a unified manner. | |||||

Inhalt | The main concepts will be presented in parallel with the proof of a few main theorems: (1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions; (2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line; (3) the Chebychev bias for primes in arithmetic progressions; (4) functional limit theorems for the paths of partial sums of families of exponential sums. | |||||

Skript | The lecture notes for the class are available at https://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf | |||||

Voraussetzungen / Besonderes | Prerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled). Some knowledge of number theory is useful but the main results will also be summarized. | |||||

401-3362-21L | Spectral Theory of Eisenstein Series | W | 4 KP | 2V | P. D. Nelson | |

Kurzbeschreibung | We plan to discuss the basic theory of Eisenstein series and the spectral decomposition of the space of automorphic forms, with focus on the groups GL(2) and GL(n). | |||||

Lernziel | ||||||

Voraussetzungen / Besonderes | Some familiarity with basics on Lie groups and functional analysis would be helpful, and some prior exposure to modular forms or homogeneous spaces may provide useful motivation. | |||||

401-3058-00L | Kombinatorik I | 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. | |||||

Voraussetzungen / Besonderes | Wer 401-3052-00L Kombinatorik (letztmals im FS 2008 gelesen) für den Bachelor- oder Master-Studiengang Mathematik anrechnen lässt, darf 401-3058-00L Kombinatorik I nur noch fürs Mathematik Lehrdiplom oder fürs Didaktik-Zertifikat Mathematik anrechnen lassen. | |||||

Auswahl: Geometrie | ||||||

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

401-4118-21L | Spectral Theory of Hyperbolic Surfaces | W | 4 KP | 2V | C. Burrin | |

Kurzbeschreibung | The Laplacian plays a prominent role in many parts of mathematics. On a flat surface like the torus, understanding its spectrum is the topic of Fourier analysis, whose 19th century development allowed to solve the heat and wave equations. On the sphere, one studies spherical harmonics. In this course, we will study the spectrum of hyperbolic surfaces and its Maass forms (eigenfunctions). | |||||

Lernziel | We will start from scratch, with an overview of hyperbolic geometry and harmonic analysis on the hyperbolic plane. The objectives are to prove the spectral theorem and Selberg's trace formula, and explore applications in geometry and number theory. | |||||

Inhalt | Tentative syllabus: Hyperbolic geometry (the hyperbolic plane and Fuchsian groups) Construction of arithmetic hyperbolic surfaces Harmonic analysis on the hyperbolic plane The spectral theorem Selberg's trace formula Applications in geometry (isoperimetric inequalities, geodesic length spectrum) and number theory (links to the Riemann zeta function and Riemann hypothesis) Possible further topics (if time permits): Eisenstein series Explicit constructions of Maass forms (after Maass) A special case of the Jacquet-Langlands correspondence (after the exposition of Bergeron, see references) | |||||

Literatur | Nicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer Universitext 2011. Armand Borel, Automorphic forms on SL(2,R), Cambridge University Press 1997. Peter Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser 1992. Henryk Iwaniec, Spectral methods of automorphic forms. Graduate studies in mathematics, AMS 2002. | |||||

Voraussetzungen / Besonderes | Knowledge of the material covered in the first two years of bachelor studies is assumed. Prior knowledge of differential geometry, functional analysis, or Riemann surfaces is not required. | |||||

401-4206-17L | Groups Acting on Trees | W | 6 KP | 3G | B. Brück | |

Kurzbeschreibung | As a main theme, we will see how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. | |||||

Lernziel | Learn basics of Bass-Serre theory; get to know concepts from geometric group theory. | |||||

Inhalt | As a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces. This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field. Topics that will be covered in the lecture include: - Trees and their automorphisms - Different characterisations of free groups - Amalgamated products and HNN extensions - Graphs of groups - Kurosh's theorem on subgroups of free (amalgamated) products | |||||

Literatur | J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9 O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8 C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101 | |||||

Voraussetzungen / Besonderes | Basic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary. In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient. | |||||

401-3056-00L | Endliche Geometrien IFindet 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. | |||||

401-3574-61L | Introduction to Knot Theory Findet dieses Semester nicht statt. | W | 6 KP | 3G | ||

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

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

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

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

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

Auswahl: Analysis | ||||||

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

401-4422-21L | An Introduction to the Calculus of Variations | W | 4 KP | 2V | A. Figalli | |

Kurzbeschreibung | Calculus of variations is a fundamental tool in mathematical analysis, used to investigate the existence, uniqueness, and properties of minimizers to variational problems. Classic examples include, for instance, the existence of the shortest curve between two points, the equilibrium shape of an elastic membrane, and so on. | |||||

Lernziel | ||||||

Inhalt | In the course, we will study both 1-dimensional and multi-dimensional problems. | |||||

Voraussetzungen / Besonderes | Basic knowledge of Sobolev spaces is important, so some extra additional readings would be required for those unfamiliar with the topic. | |||||

401-3378-19L | Entropy in Dynamics | W | 8 KP | 4G | M. Einsiedler | |

Kurzbeschreibung | Definition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group. | |||||

Lernziel | The course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group. | |||||

Skript | Entropy book under construction, available online under https://tbward0.wixsite.com/books/entropy | |||||

Voraussetzungen / Besonderes | No prior knowledge of dynamical systems will be assumed but measure theory will be assumed and very important. Doctoral students are welcome to attend the course for 2KP. | |||||

Auswahl: Weitere Gebiete | ||||||

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

401-3502-21L | 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 | ||||||

401-3503-21L | 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 | ||||||

401-3504-21L | 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 | ||||||

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 | |

401-4658-00L | Computational Methods for Quantitative Finance: PDE Methods | W | 6 KP | 3V + 1U | C. Marcati, A. Stein | |

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

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

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

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

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

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

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

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

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

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

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

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

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

401-4652-21L | Nonlocal Inverse Problems | W | 4 KP | 2V | J. Railo | |

Kurzbeschreibung | This course is an introduction to the Calderón problem and nonlocal inverse problems for the fractional Schrödinger equation. These are examples of nonlinear inverse problems. The classical Calderón problem models electrical impedance tomography (EIT) and fractional operators appear, for example, in some mathematical models in finance. | |||||

Lernziel | Students become familiar with the Calderón problem and some nonlocal phenomena related to the fractional Laplacian. Advanced students should be able to read research articles on the fractional Calderón problems after the course. | |||||

Inhalt | In the beginning of the course, we will introduce some basic theory for the classical Calderón problem. The focus of the course will be in the study of nonlocal inverse problems for the fractional Schrödinger equation with lower order perturbations. We discuss necessary preliminaries on Sobolev spaces, Fourier analysis, functional analysis and theory of PDEs. Our scope will be in the uniqueness properties. Classical Calderón problem (about 1/3): Conductivity and Schrödinger equations, Dirichlet-to-Neumann maps, Cauchy data, and related boundary value inverse problems. The methods include, for example, complex geometric optics (CGO) solutions. Fractional Calderón problem (about 2/3): Nonlocal unique continuation principles (UCP), Runge approximation properties, and uniqueness for the fractional Calderón problem. The methods include, for example, Caffarelli-Silvestre extensions, the fractional Poincaré inequality and Riesz transforms. | |||||

Skript | Lecture notes and exercises | |||||

Literatur | 1. M. Salo: Calderón problem. Lecture notes, University of Helsinki (2008). (Available at http://users.jyu.fi/~salomi/index.html.) 2. T. Ghosh, M. Salo, G. Uhlmann: The Calderón problem for the fractional Schrödinger equation. Analysis & PDE 13 (2020), no. 2, 455-475. 3. A. Rüland, M. Salo: The fractional Calderón problem: low regularity and stability. Nonlinear Analysis 193 (2020), special issue "Nonlocal and Fractional Phenomena", 111529. 4. Other literature will be specified in the course. | |||||

Voraussetzungen / Besonderes | Functional Analysis I & II or similar knowledge. Any additional knowledge of Fourier analysis, Sobolev spaces, distributions and PDEs will be an asset. | |||||

401-3426-21L | Time-Frequency Analysis | W | 4 KP | 2G | R. Alaifari | |

Kurzbeschreibung | This course gives a basic introduction to time-frequency analysis from the viewpoint of applied harmonic analysis. | |||||

Lernziel | By the end of the course students should be familiar with the concept of the short-time Fourier transform, the Bargmann transform, quadratic time-frequency representations (ambiguity function and Wigner distribution), Gabor frames and modulation spaces. The connection and comparison to time-scale representations will also be subject of this course. | |||||

Inhalt | Time-frequency analysis lies at the heart of many applications in signal processing and aims at capturing time and frequency information simultaneously (as opposed to the classical Fourier transform). This course gives a basic introduction that starts with studying the short-time Fourier transform and the special role of the Gauss window. We will visit quadratic representations and then focus on discrete time-frequency representations, where Gabor frames will be introduced. Later, we aim at a more quantitative analysis of time-frequency information through modulation spaces. At the end, we touch on wavelets (time-scale representation) as a counterpart to the short-time Fourier transform. | |||||

Literatur | Gröchenig, K. (2001). Foundations of time-frequency analysis. Springer Science & Business Media. | |||||

Voraussetzungen / Besonderes | Functional analysis, Fourier analysis, complex analysis, operator theory | |||||

Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||

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

401-4611-21L | Rough Path Theory | W | 4 KP | 2V | A. Allan, J. Teichmann | |

Kurzbeschreibung | The aim of this course is to provide an introduction to the theory of rough paths, with a particular focus on their integration theory and associated rough differential equations, and how the theory relates to and enhances the field of stochastic calculus. | |||||

Lernziel | Our first motivation will be to understand the limitations of classical notions of integration to handle paths of very low regularity, and to see how the rough integral succeeds where other notions fail. We will construct rough integrals and establish solutions of differential equations driven by rough paths, as well as the continuity of these objects with respect to the paths involved, and their consistency with stochastic integration and SDEs. Various applications and extensions of the theory will then be discussed. | |||||

Skript | Lecture notes will be provided by the lecturer. | |||||

Literatur | P. K. Friz and M. Hairer, A course on rough paths with an introduction to regularity structures, Springer (2014). P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths, Cambridge University Press (2010). | |||||

Voraussetzungen / Besonderes | The aim will be to make the course as self-contained as possible, but some knowledge of stochastic analysis is highly recommended. The course “Brownian Motion and Stochastic Calculus” would be ideal, but not strictly required. | |||||

401-4626-00L | Advanced Statistical Modelling: Mixed ModelsFindet dieses Semester nicht statt. | W | 4 KP | 2V | M. Mächler | |

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

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

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

Skript | We will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples. These lecture notes and all R scripts are made available from https://github.com/mmaechler/MEMo | |||||

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

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

401-4627-00L | Empirical Process Theory and Applications | W | 4 KP | 2V | S. van de Geer | |

Kurzbeschreibung | Empirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc. | |||||

Lernziel | ||||||

Inhalt | In this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection. |

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