# Search result: Catalogue data in Spring Semester 2020

Mathematics Master | ||||||

Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||

Electives: Pure Mathematics | ||||||

Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3201-00L | Algebraic Groups | W | 8 credits | 4G | P. D. Nelson | |

Abstract | Introduction to the theory of linear algebraic groups. Lie algebras, the Jordan Chevalley decomposition, semisimple and reductive groups, root systems, Borel subgroups, classification of reductive groups and their representations. | |||||

Objective | ||||||

Literature | A. L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups | |||||

Prerequisites / Notice | Abstract algebra: groups, rings, fields, tensor product, etc. Some familiarity with the basics of Lie groups and their Lie algebras would be helpful, but is not absolutely necessary. We will develop what we need from algebraic geometry, without assuming prior knowledge. | |||||

401-3109-65L | Probabilistic Number Theory Does not take place this semester. | W | 8 credits | 4G | E. Kowalski | |

Abstract | 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. | |||||

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

Content | 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. | |||||

Lecture notes | The lecture notes for the class are available at Link | |||||

Prerequisites / Notice | Prerequisites: Complex analysis, measure and integral; some probability theory is useful but the main concepts needed will be recalled. Some knowledge of number theory is useful but the main results will be summarized. | |||||

401-3202-09L | The Representation Theory of the Finite Symmetric Groups NOTICE: No physical class for the next few weeks until further notice. Instead a video recording will be offered. | W | 4 credits | 2V | L. Wu | |

Abstract | This course is an Introduction to the Representation Theory of the Groups. | |||||

Objective | Our goal is to give an introduction of the Representation Theory using the examples of the Finite Symmetry Groups. | |||||

Literature | * Jean-Pierre Serre: Linear Representations of Finite Groups, Graduate Texts in Mathematics, Springer. * William Fulton and Joe Harris: Representation Theory A First Course, Graduate Texts in Mathematics, Springer. * G. D. James: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Springer. * Bruce E. Sagan: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, Springer. | |||||

Prerequisites / Notice | Some basic knowledge of the Group Theory and Linear Algebra. | |||||

401-8112-20L | Geometry of Numbers (University of Zurich)No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT548 Mind the enrolment deadlines at UZH: Link | W | 9 credits | 4V + 1U | University lecturers | |

Abstract | The Geometry of Numbers studies distribution of lattice points in the n dimensional space, for instance, existence of lattice points in various domains and existence of integral solutions of polynomial inequalities. This subject is also closely related to the Theory of Diophantine Approximation, which seeks good rational approximations for real vectors. | |||||

Objective | Learn basic techniques in the Geometry of Numbers | |||||

Literature | 1. Cassels, An introduction to Diophantine Approximation 2. Cassels, An introduction to the Geometry of Numbers 3. Schmidt, Diophantine approximation 4. Siegel, Lectures on the Geometry of Numbers | |||||

401-3058-00L | Combinatorics IDoes not take place this semester. | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | The course Combinatorics I and II is an introduction into the field of enumerative combinatorics. | |||||

Objective | Upon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them. | |||||

Content | Contents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers. | |||||

Prerequisites / Notice | Recognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008). | |||||

Selection: Geometry | ||||||

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

401-3556-20L | Topics in Symplectic Topology | W | 6 credits | 3V | P. Biran | |

Abstract | This will be an introductory course in symplectic geometry and topology. We will cover the simplest instances of symplectic rigidity phenomena, and techniques to detect and study them. The last part of the course will be devoted to more advanced techniques such as Floer theory. | |||||

Objective | Get acquainted with the basics of symplectic topology and phenomena of symplectic rigidity. | |||||

Literature | 1) Book: "Introduction to Symplectic Topology", 3'rd edition, by McDuff and Salamon. Oxford Graduate Texts in Mathematics 2) Some published articles that will be announced during the semester. | |||||

401-3056-00L | Finite Geometries I | W | 4 credits | 2G | N. Hungerbühler | |

Abstract | Finite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares. | |||||

Objective | Finite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design. | |||||

Content | Finite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design | |||||

Literature | - Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries. | |||||

401-4532-20L | Introduction to 3-Manifolds | W | 4 credits | 2V | M. Nagel | |

Abstract | This course provides an introduction to the basic notions and tools of geometric topology with a special focus on three dimensional manifolds. | |||||

Objective | In this course, we become familiar with the basic notions and tools of geometric topology, which concerns low-dimensional manifolds and their embeddings. We will focus on 3–dimensional manifolds. While this class of manifolds is very rich, it still allows for many structural results. An important goal of the lecture is to learn how to manipulate these manifolds: build them from simple pieces, cut them apart, isotope and simplify submanifolds etc. These techniques from differential topology are combined with invariants from algebraic topology, which are incredibly powerful in encoding properties of a 3–manifold. We discuss applications, which give new intuition for these invariants, and answer many questions about manifolds of dimension three or less. There are many synergies with Algebraic Topology II, which I encourage you to take in parallel. | |||||

Content | Background in differential topology Foundational results on the topology of 3–manifolds Knots and concordance | |||||

Literature | Knots and links by D. Rolfsen 3–Manifolds by J. Hempel Differential topology by T. Bröcker and K. Jänich | |||||

Prerequisites / Notice | Algebraic Topology I Differential Geometry I | |||||

401-3574-61L | Introduction to Knot Theory Does not take place this semester. | W | 6 credits | 3G | ||

Abstract | 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. | |||||

Objective | 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. | |||||

Content | 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.....) | |||||

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

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

Selection: Analysis No offering in this semester yet | ||||||

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

401-3462-99L | Reading Course: Functional Analysis II'Not yet open for registration. This supplementary reading course on functional analysis is only for students who already got credits for the course unit 401-3462-00L Functional Analysis II that was taught in the Spring Semester 2019. | W | 4 credits | 4V + 1U | M. Struwe | |

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Selection: Further Realms | ||||||

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

401-3502-20L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 2 credits | 4A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-3503-20L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 3 credits | 6A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

Objective | ||||||

401-3504-20L | Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |

Abstract | For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. | |||||

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401-4504-20L | Reading Course Link and register your reading course in myStudies. | W | 4 credits | 9A | Supervisors | |

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

Selection: Numerical Analysis | ||||||

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

401-4658-00L | Computational Methods for Quantitative Finance: PDE Methods | W | 6 credits | 3V + 1U | C. Schwab | |

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

Objective | 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. Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation. | |||||

Content | 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. | |||||

Lecture notes | There will be english, typed lecture notes as well as MATLAB software for registered participants in the course. | |||||

Literature | 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. N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013. | |||||

401-4788-16L | Mathematics of (Super-Resolution) Biomedical ImagingNOTICE: The exercise class scheduled for 5 March has been cancelled | W | 8 credits | 4G | H. Ammari | |

Abstract | The aim of this course is to review different methods used to address challenging problems in biomedical imaging. The emphasis will be on scale separation techniques, hybrid imaging, spectroscopic techniques, and nanoparticle imaging. These approaches allow one to overcome the ill-posedness character of imaging reconstruction in biomedical applications and to achieve super-resolution imaging. | |||||

Objective | Super-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other. In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold: (i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence of small anomalies; (ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies; (iii) To design efficient inversion algorithms in multi-wave modalities; (iv) to develop inversion techniques using multi-frequency measurements; (v) to develop a mathematical and numerical framework for nanoparticle imaging. In this course we shall consider both analytical and computational matters in biomedical imaging. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, inverse problems, mathematical imaging, optimal control, stochastic modelling, and analysis of physical phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in biomedical imaging, requires a deep understanding of the different scales in the physical models, an accurate mathematical modelling of the imaging techniques, and fine analysis of complex physical phenomena. An emphasis is put on mathematically analyzing acoustic-electric imaging, thermo-elastic imaging, Lorentz force based imaging, elastography, multifrequency electrical impedance tomography, and plasmonic resonant nanoparticles. | |||||

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

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

401-4605-20L | Selected Topics in Probability | W | 4 credits | 2V | A.‑S. Sznitman | |

Abstract | This course will discuss some questions of current interest in probability theory. Among examples of possible subjects are for instance topics in random media, large deviations, random walks on graphs, branching random walks, random trees, percolation, concentration of measures, large random matrices, stochastic calculus, stochastic partial differential equations. | |||||

Objective | This course will discuss some questions of current interest in probability theory. Among examples of possible subjects are for instance topics in random media, large deviations, random walks on graphs, branching random walks, random trees, percolation, concentration of measures, large random matrices, stochastic calculus, stochastic partial differential equations. | |||||

Prerequisites / Notice | Lecture Probability Theory. | |||||

401-4626-00L | Advanced Statistical Modelling: Mixed Models | W | 4 credits | 2V | M. Mächler | |

Abstract | 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. | |||||

Objective | - 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. | |||||

Content | 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. | |||||

Lecture notes | 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 Link | |||||

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

Prerequisites / Notice | - 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](Link) 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 credits | 2V | S. van de Geer | |

Abstract | 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. | |||||

Objective | ||||||

Content | 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. | |||||

401-4632-15L | Causality | W | 4 credits | 2G | C. Heinze-Deml | |

Abstract | In statistics, we are used to search for the best predictors of some random variable. In many situations, however, we are interested in predicting a system's behavior under manipulations. For such an analysis, we require knowledge about the underlying causal structure of the system. In this course, we study concepts and theory behind causal inference. | |||||

Objective | After this course, you should be able to - understand the language and concepts of causal inference - know the assumptions under which one can infer causal relations from observational and/or interventional data - describe and apply different methods for causal structure learning - given data and a causal structure, derive causal effects and predictions of interventional experiments | |||||

Prerequisites / Notice | Prerequisites: basic knowledge of probability theory and regression |

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