Suchergebnis: Katalogdaten im Frühjahrssemester 2021

Nummer	Titel	Typ	ECTS	Umfang	Dozierende
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
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 I Findet dieses Semester nicht statt.	W	4 KP	2G	N. Hungerbühler
Kurzbeschreibung	Endliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate.
Lernziel	Endliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne.
Inhalt	Endliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne
Literatur	- Max Jeger, Endliche Geometrien, ETH Skript 1988 - Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983 - Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press - Dembowski: Finite Geometries.
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

Seite 1 von 1