Suchergebnis: Katalogdaten im Herbstsemester 2016
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, Topologie, diskrete Mathematik, Logik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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401-3117-66L | Introduction to the Circle Method | W | 6 KP | 2V + 1U | E. Kowalski | |
Kurzbeschreibung | The circle method, invented by Hardy and Ramanujan and developped by Hardy and Littlewood and Kloosterman, is one of the most versatile methods currently available to determine the asymptotic behavior of the number of integral solutions to polynomial equations, when the number of solutions is sufficiently large. | |||||
Lernziel | ||||||
Inhalt | The circle method, invented by Hardy and Ramanujan and developped by Hardy and Littlewood and Kloosterman, is one of the most versatile methods currently available to determine the asymptotic behavior of the number of integral solutions to polynomial equations, when the number of solutions is sufficiently large. The lecture will present an introduction to this method. In particular, it will present the solution of Waring's Problem concerning the representability of integers as sums of a bounded numbers of (fixed) powers of integers. | |||||
Literatur | H. Davenport, "Analytic methods for Diophantine equations and Diophatine inequalities", Cambridge H. Iwaniec and E. Kowalski, "Analytic number theory", chapter 20; AMS R. Vaughan, "The Hardy-Littlewood method", Cambridge | |||||
401-4209-66L | Group and Representation Theory: Beyond an Introduction | W | 8 KP | 3V + 1U | T. H. Willwacher | |
Kurzbeschreibung | The goal of the course is to study several classical and important (and beautiful!) topics in group and representation theory, that are otherwise often overlooked in a standard curriculum. In particular, we plan to study reflection and Coxeter groups, classical invariant theory, and the theory of real semi simple Lie algebras and their representations. | |||||
Lernziel | Despite the title, the course will begin by a recollection of basic concepts of group and representation theory, in particular that of finite groups and Lie groups. Hence the course should be accessible also for students who only had a brief exposure to representation theory, as for example in the MMP course. | |||||
401-3059-00L | Kombinatorik II Findet dieses Semester nicht statt. | W | 4 KP | 2G | N. Hungerbühler | |
Kurzbeschreibung | Der Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende Kombinatorik. | |||||
Lernziel | Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden. | |||||
Inhalt | Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele. | |||||
401-4145-66L | Reading Course: Abelian Varieties over Finite Fields | W | 2 KP | 4A | J. Fresán, P. S. Jossen | |
Kurzbeschreibung | ||||||
Lernziel | ||||||
Auswahl: Geometrie | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-4531-66L | Topics in Rigidity Theory | W | 6 KP | 3G | M. Burger | |
Kurzbeschreibung | The aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups. | |||||
Lernziel | Understand the basic techniques of rigidity theory. | |||||
Inhalt | This course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are: - Describe all its proper quotients. - Classify all its finite dimensional linear representations. - More generally, can this group act by diffeomorphisms on "small" manifolds like the circle? - Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure? In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction. | |||||
Literatur | - R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984. - D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv - Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage. - M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online. | |||||
Voraussetzungen / Besonderes | For this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques. | |||||
401-3309-66L | Riemann Surfaces (Part 2) | W | 4 KP | 2V | A. Buryak | |
Kurzbeschreibung | The program will be the following: * Proof of the Serre duality; * Riemann-Hurwitz formula; * Functions and differential forms on a compact Riemann surface with prescribed principal parts; * Weierstrass points on a compact Riemann surface; * The Jacobian and the Picard group of a compact Riemann surface; * Holomorphic vector bundles; * Non-compact Riemann surfaces. | |||||
Lernziel | ||||||
Literatur | O. Forster. Lectures on Riemann Surfaces. | |||||
Voraussetzungen / Besonderes | This is a continuation of 401-3308-16L Riemann Surfaces that was taught in the spring semester (FS 2016), see Link for the lecture notes. The students are also assumed to be familiar with what would generally be covered in one semester courses on general topology and on algebra. | |||||
401-3057-00L | Endliche Geometrien II | 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. | |||||
Auswahl: Analysis | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3536-11L | Geometric Aspects of Hamiltonian Dynamics | W | 6 KP | 3V | P. Biran | |
Kurzbeschreibung | The course will concentrate on the geometry of the group of Hamiltonian diffeomorphisms introduced by Hofer in the early 1990's and its relations to various topics in symplectic geometry such as capacities, Lagrangian submanifolds, holomorphic curves, as well as recent algebraic structures on the group of Hamiltonian diffeomorphisms such as quasi-morphisms. | |||||
Lernziel | ||||||
Literatur | Books: * L. Polterovich: "The geometry of the group of symplectic diffeomorphisms" * H. Hofer & E. Zehnder: "Symplectic invariants and Hamiltonian dynamics" | |||||
Voraussetzungen / Besonderes | Prerequisites. Good knowledge of undergraduate mathematics (analysis, complex functions, topology, and differential geometry). Some knowledge of elementary algebraic topology would be useful. | |||||
401-4767-66L | Partial Differential Equations (Hyperbolic PDEs) | W | 7 KP | 4V | D. Christodoulou | |
Kurzbeschreibung | The course begins with characteristics, the definition of hyperbolicity, causal structure and the domain of dependence theorem. The course then focuses on nonlinear systems of equations in two independent variables, in particular the Euler equations of compressible fluids with plane symmetry and the Einstein equations of general relativity with spherical symmetry. | |||||
Lernziel | The objective is to introduce students in mathematics and physics to an area of mathematical analysis involving differential geometry which is of fundamental importance for the development of classical macroscopic continuum physics. | |||||
Inhalt | The course shall begin with the basic structure associated to hyperbolic partial differential equations, characteristic hypersurfaces and bicharacteristics, causal structure, and the domain of dependence theorem. The course shall then focus on nonlinear systems of equations in two independent variables. The first topic shall be the Euler equations of compressible fluids under plane symmetry where we shall study the formation of shocks, and second topic shall be the Einstein equations of general relativity under spherical symmetry where we shall study the formation of black holes and spacetime singularities. | |||||
Voraussetzungen / Besonderes | Basic real analysis and differential geometry. | |||||
401-4831-66L | Mathematical Themes in General Relativity I | W | 4 KP | 2V | A. Carlotto | |
Kurzbeschreibung | First part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems. | |||||
Lernziel | Acquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis. | |||||
Inhalt | Lorentzian geometry; geometric review of special relativity; the Einstein equations and their basic classes of special solutions; the Einstein equations as an initial-value problem; causality theory and hyperbolicity; singularities and trapped domains; Penrose diagrams; asymptotically flat spaces: ADM invariants, positive mass theorems, Penrose inequalities, geometric properties. | |||||
Skript | Lecture notes written by the instructor will be provided to all enrolled students. | |||||
Voraussetzungen / Besonderes | The content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable. | |||||
401-4497-66L | Free Boundary Problems | W | 4 KP | 2V | A. Figalli | |
Kurzbeschreibung | ||||||
Lernziel | ||||||
401-4463-62L | Fourier Analysis in Function Space Theory | W | 6 KP | 3V | T. Rivière | |
Kurzbeschreibung | In the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. | |||||
Lernziel | ||||||
Inhalt | During the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces. In the second part of the course we will study fundamental properties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occasion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings. In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators. This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course. If time permits we shall present the notion of Paraproduct, Paracompositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE. | |||||
Literatur | 1) Elias M. Stein, "Singular Integrals and Differentiability Properties of Functions" (PMS-30) Princeton University Press. 2) Javier Duoandikoetxea, "Fourier Analysis" AMS. 3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer. 4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer. | |||||
Voraussetzungen / Besonderes | Notions from ETH courses in Measure Theory, Functional Analysis I and II (Fundamental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions) | |||||
401-4475-66L | Partial Differential Equations and Semigroups of Bounded Linear Operators | W | 4 KP | 2G | A. Jentzen | |
Kurzbeschreibung | In this course we study the concept of a semigroup of bounded linear operators and we use this concept to investigate existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs) of the evolutionary type. | |||||
Lernziel | The aim of this course is to teach the students a decent knowledge (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. | |||||
Inhalt | The course includes content (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. Key example PDEs that are treated in this course are heat and wave equations. | |||||
Skript | Lecture Notes are available in the lecture homepage (please follow the link in the Learning materials section). | |||||
Literatur | 1. Amnon Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). 2. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations. Springer-Verlag, New York (2000). | |||||
Voraussetzungen / Besonderes | Mandatory prerequisites: Functional analysis Start of lectures: Friday, September 23, 2016 For more details, please follow the link in the Learning materials section. | |||||
401-3303-00L | Ausgewählte Themen der Funktionentheorie | W | 6 KP | 3V | H. Knörrer | |
Kurzbeschreibung | Hypergeometrische Funktionen, Randwerte holomorpher Funktionen, Nevanlinna Theorie und andere spezielle Themen | |||||
Lernziel | Fortgeschrittene Methoden der Funktionentheorie | |||||
Literatur | R. Remmert: Funktionentheorie II. Springer Verlag E.Titchmarsh: The Theory of Functions. Oxford University Press C.Caratheodory: Funktionentheorie. Birkhaeuser E.Hille: Analytic Function Theory. AMS Chelsea Publishing A.Gogolin:Komplexe Integration. Springer | |||||
Voraussetzungen / Besonderes | Funktionentheorie | |||||
Auswahl: Weitere Gebiete | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3502-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 2 KP | 4A | Professor/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 | ||||||
401-3503-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 3 KP | 6A | Professor/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 | ||||||
401-3504-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 4 KP | 9A | Professor/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 |
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