Suchergebnis: Katalogdaten im Frühjahrssemester 2020

Mathematik Master Information
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
NummerTitelTypECTSUmfangDozierende
401-3201-00LAlgebraic Groups Information W8 KP4GP. D. Nelson
KurzbeschreibungIntroduction 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.
Lernziel
LiteraturA. L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups
Voraussetzungen / BesonderesAbstract 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.
401-3109-65LProbabilistic Number Theory Information
Findet dieses Semester nicht statt.
W8 KP4GE. Kowalski
KurzbeschreibungThe 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.
LernzielThe goal of the course is to present some results of probabilistic number theory in a unified manner.
InhaltThe 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.
SkriptThe lecture notes for the class are available at

Link
Voraussetzungen / BesonderesPrerequisites: 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.
401-3202-09LThe Representation Theory of the Finite Symmetric Groups Information
NOTICE: No physical class for the next few weeks until further notice. Instead a video recording will be offered.
W4 KP2VL. Wu
KurzbeschreibungThis course is an Introduction to the Representation Theory of the Groups.
LernzielOur goal is to give an introduction of the Representation Theory using the examples of the Finite Symmetry Groups.
Literatur* 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.
Voraussetzungen / BesonderesSome basic knowledge of the Group Theory and Linear Algebra.
401-8112-20LGeometry of Numbers (University of Zurich)
Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MAT548

Beachten Sie die Einschreibungstermine an der UZH: Link
W9 KP4V + 1UUni-Dozierende
KurzbeschreibungThe 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.
LernzielLearn basic techniques in the Geometry of Numbers
Literatur1. 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
401-3058-00LKombinatorik I
Findet dieses Semester nicht statt.
W4 KP2GN. Hungerbühler
KurzbeschreibungDer Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik.
LernzielDie Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
InhaltInhalt 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 / BesonderesWer 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
NummerTitelTypECTSUmfangDozierende
401-3556-20LTopics in Symplectic TopologyW6 KP3VP. Biran
KurzbeschreibungThis 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.
LernzielGet acquainted with the basics of symplectic topology and phenomena
of symplectic rigidity.
Literatur1) 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.
401-3056-00LEndliche Geometrien IW4 KP2GN. Hungerbühler
KurzbeschreibungEndliche 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.
LernzielEndliche 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.
InhaltEndliche 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.
401-4532-20LIntroduction to 3-ManifoldsW4 KP2VM. Nagel
KurzbeschreibungThis course provides an introduction to the basic notions and tools of geometric topology with a special focus on three dimensional manifolds.
LernzielIn 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.
InhaltBackground in differential topology
Foundational results on the topology of 3–manifolds
Knots and concordance
LiteraturKnots and links by D. Rolfsen
3–Manifolds by J. Hempel
Differential topology by T. Bröcker and K. Jänich
Voraussetzungen / BesonderesAlgebraic Topology I
Differential Geometry I
401-3574-61LIntroduction to Knot Theory Information
Findet dieses Semester nicht statt.
W6 KP3G
KurzbeschreibungIntroduction 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.
LernzielThe 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.
InhaltDefinition 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.....)
LiteraturAn extensive bibliography will be handed out in the course.
Voraussetzungen / BesonderesPrerequisites are some elementary knowledge of algebra and topology.
Auswahl: Analysis
(noch) kein Angebot in diesem Semester
NummerTitelTypECTSUmfangDozierende
401-3462-99LReading 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.
W4 KP4V + 1UM. Struwe
Kurzbeschreibung
Lernziel
Auswahl: Weitere Gebiete
NummerTitelTypECTSUmfangDozierende
401-3502-20LReading Course Belegung eingeschränkt - Details anzeigen
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W2 KP4ABetreuer/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-20LReading Course Belegung eingeschränkt - Details anzeigen
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W3 KP6ABetreuer/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-20LReading Course Belegung eingeschränkt - Details anzeigen
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 KP9ABetreuer/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-4504-20LReading Course Belegung eingeschränkt - Details anzeigen
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 KP9ABetreuer/innen
KurzbeschreibungIn 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
NummerTitelTypECTSUmfangDozierende
401-4658-00LComputational Methods for Quantitative Finance: PDE Methods Information Belegung eingeschränkt - Details anzeigen W6 KP3V + 1UC. Schwab
KurzbeschreibungIntroduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming
and knowledge of numerical mathematics at ETH BSc level.
LernzielIntroduce 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.
Inhalt1. 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.
SkriptThere will be english, typed lecture notes as well as MATLAB software for registered participants in the course.
LiteraturR. 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.
401-4788-16LMathematics of (Super-Resolution) Biomedical Imaging
NOTICE: The exercise class scheduled for 5 March has been cancelled
W8 KP4GH. Ammari
KurzbeschreibungThe 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.
LernzielSuper-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.
Auswahl: Wahrscheinlichkeitstheorie, Statistik
NummerTitelTypECTSUmfangDozierende
401-4605-20LSelected Topics in Probability Information W4 KP2VA.‑S. Sznitman
KurzbeschreibungThis 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.
LernzielThis 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.
Voraussetzungen / BesonderesVorlesung Probability Theory.
401-4626-00LAdvanced Statistical Modelling: Mixed ModelsW4 KP2VM. Mächler
KurzbeschreibungMixed 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.
InhaltThe 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.
SkriptWe 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
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](Link) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00LEmpirical Process Theory and ApplicationsW4 KP2VS. van de Geer
KurzbeschreibungEmpirical 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
InhaltIn 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.
401-4632-15LCausality Information W4 KP2GC. Heinze-Deml
KurzbeschreibungIn 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.
LernzielAfter 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
Voraussetzungen / BesonderesPrerequisites: basic knowledge of probability theory and regression
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