Suchergebnis: Katalogdaten im Herbstsemester 2015

Mathematik Master Information
Kernfächer
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.
Kernfächer aus Bereichen der reinen Mathematik
(auch Bachelor-)Kernfächer (Link) sind unter gewissen Bedingungen anrechenbar.
NummerTitelTypECTSUmfangDozierende
401-3225-00LIntroduction to Lie GroupsW8 KP4GM. Einsiedler
KurzbeschreibungTopological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.
LernzielThe goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.
LiteraturA. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)
A.Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
F.Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
H. Samelson: "Notes on Lie algebras" (Springer, '90)
S.Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
A.Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)
Voraussetzungen / BesonderesTopology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.

Course webpage: Link
401-3118-09LModular FormsW8 KP3V + 1UÖ. Imamoglu
KurzbeschreibungThis is a introductory course on automorphic forms covering its basic properties with emphasis on connections with number theory.
LernzielThe aim of the course is to cover the classical theory of modular forms.
InhaltBasic definitions and properties of SL(2,Z), its subgroups and modular forms for SL(2,Z). Eisenstein and Poincare series. L-functions of modular forms. Hecke operators. Theta functions. Possibly Maass forms. Possibly automorphic forms for more general groups.
LiteraturJ.P. Serre, A Course in Arithmetic;
N. Koblitz, Introduction to Elliptic Curves and Modular Forms;
D. Zagier, The 1-2-3 of Modular Forms;
H. Iwaniec, Topics in Classical Automorphic Forms.
Kernfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
(auch Bachelor-)Kernfächer (Link) sind unter gewissen Bedingungen anrechenbar.
NummerTitelTypECTSUmfangDozierende
401-3651-00LNumerical Methods for Elliptic and Parabolic Partial Differential Equations
Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.
W10 KP4V + 1UC. Schwab
KurzbeschreibungThis course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
LernzielParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
InhaltA selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
SkriptCourse slides will be made available to the audience.
LiteraturS. C. Brenner and L. Ridgway Scott: The mathematical theory of Finite Element Methods. New York, Berlin [etc]: Springer-Verl, cop.1994.

A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer, 2004.

V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).

Additional Literature:
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]

R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013
Voraussetzungen / BesonderesPractical exercises based on MATLAB
401-3621-00LFundamentals of Mathematical StatisticsW10 KP4V + 1US. van de Geer
KurzbeschreibungThe course covers the basics of inferential statistics.
Lernziel
401-3901-00LMathematical Optimization Information W11 KP4V + 2UR. Weismantel
KurzbeschreibungMathematical treatment of diverse optimization techniques.
LernzielAdvanced optimization theory and algorithms.
Inhalt1. Linear optimization: The geometry of linear programming, the simplex method for solving linear programming problems, Farkas' Lemma and infeasibility certificates, duality theory of linear programming.

2. Nonlinear optimization: Lagrange relaxation techniques, Newton method and gradient schemes for convex optimization.

3. Integer optimization: Ties between linear and integer optimization, total unimodularity, complexity theory, cutting plane theory.

4. Combinatorial optimization: Network flow problems, structural results and algorithms for matroids, matchings and, more generally, independence systems.
(auch Bachelor-)Kernfächer aus Bereichen der reinen Mathematik
Die Anrechnung von 401-3531-00L Differentialgeometrie I / Differential Geometry I im Master-Studiengang ist nur dann zulässig, wenn 401-3532-00L Differentialgeometrie II / Differential Geometry II nicht für den Bachelor-Studiengang angerechnet wurde.
Ebenso für:
401-3461-00L Funktionalanalysis I / Functional Analysis I - 401-3462-00L Funktionalanalysis II / Functional Analysis II
401-3001-61L Algebraische Topologie I / Algebraic Topology I - 401-3002-12L Algebraische Topologie II / Algebraic Topology II
401-3132-00L Kommutative Algebra / Commutative Algebra - 401-3146-12L Algebraische Geometrie / Algebraic Geometry
401-3371-00L Dynamische Systeme I / Dynamical Systems I - 401-3372-00L Dynamische Systeme II / Dynamical Systems II
Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link).
NummerTitelTypECTSUmfangDozierende
401-3531-00LDifferential Geometry IW10 KP4V + 1UM. Burger
KurzbeschreibungThis course is an introduction to differential and riemannian geometry.
LernzielThe aim is to lead students from a reasonable knowledge of advanced calculus, basic knowledge of general topology and solid knowledge of linear algebra to fundamental knowledge of differentiable manifolds and their basic tools. Riemannian geometry, some basic Lie theory, and de Rham cohomology will be developed as applications.
LiteraturW.Boothby "An introduction to differentiable manifolds and Riemannian geometry"

J.M.Lee "Introduction to smooth manifolds"

M.P. Do Carmo "Riemannian Geometry"
401-3461-00LFunctional Analysis IW10 KP4V + 1UD. A. Salamon
KurzbeschreibungBaire category; Banach and Hilbert spaces, bounded linear operators; Three Fundamental Principles: Uniform Boundedness, Open Mapping/Closed Graph, Hahn-Banach; Convexity; Dual Spaces: weak and weak* topologies, Banach-Alaoglu, reflexive spaces; Ergodic Theorem; compact operators and Fredholm theory, Closed Image Theorem; Spectral theory, self-adjoint operators.
Lernziel
SkriptLecture Notes on "Functional Analysis" by D.A. Salamon
401-3001-61LAlgebraic Topology IW8 KP4GP. Biran
KurzbeschreibungThis is an introductory course in algebraic topology. The course will cover the following main topics: introduction to homotopy theory, homology and cohomology of spaces.
Lernziel
Literatur1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
Link

See also:
Link

3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / BesonderesGeneral topology, linear algebra.

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3132-00LCommutative AlgebraW10 KP4V + 1UP. D. Nelson
KurzbeschreibungThis course is meant to provide an introduction to commutative algebra that equips the student to start studying the basics of algebraic geometry.
LernzielAbout the course: We shall closely follow the text "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald. Wherever possible, there will be extra focus on exercises that lead towards the basics of Algebraic Geometry. Topics include
* Basics about rings, ideals and modules
* Localisation
* Primary decomposition
* Integral dependence and valuations
* Noetherian rings
* Completions
* Basic dimension theory
LiteraturReferences:
1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)
2. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
Voraussetzungen / BesonderesPrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
401-3371-00LDynamical Systems IW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is a Part I of a broad introduction to dynamical systems. Topic covered include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics. In Part II (FS 2016), we will cover low-dimensional dynamics, complex dynamics, measure-theoretic entropy and Hamiltonian dynamics.
LernzielMastery of the basic methods and principal themes of dynamical systems.
InhaltThe course introduces the principal themes of modern dynamical systems. Topics covered include:

1. Topological dynamics
(transitivity, attractors, chaos, structural stability)

2. Symbolic dynamics
(Perron-Frobenius theorem, zeta functions)

3. Ergodic theory
(Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)

4. Hyperbolic dynamics
(Grobman-Hartman theorem, Shadowing lemma, Closing lemma and applications)
LiteraturThe most relevant textbook for this course is

Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.

Another excellent book (which will be relevant also for Dynamical Systems II) is

Lectures on Dynamical Systems, Zehnder, EMS 2010.

A more advanced textbook which covers everything in both Dynamical Systems I and II (and much more!) is

Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP, 1995.
Voraussetzungen / BesonderesThe material of the basic courses of the first two years of the program at ETH is assumed. Some basic differential geometry and functional analysis would be useful but not essential.
(auch Bachelor-)Kernfächer aus Bereichen der angewandten Mathematik
Die Anrechnung von 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory im Master-Studiengang ist nur dann zulässig, wenn weder 401-3642-00L Brownian Motion and Stochastic Calculus noch 401-3602-00L Applied Stochastic Processes für den Bachelor-Studiengang angerechnet wurde.
Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link).
NummerTitelTypECTSUmfangDozierende
401-3601-00LProbability TheoryW10 KP4V + 1UA.‑S. Sznitman
KurzbeschreibungBasics of probability theory and the theory of stochastic processes in discrete time
LernzielThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
InhaltThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Skriptavailable, will be sold in the course
LiteraturR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
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
NummerTitelTypECTSUmfangDozierende
401-3035-00LForcing: Einführung in UnabhängigkeitsbeweiseW8 KP3V + 1UL. Halbeisen
KurzbeschreibungMit Hilfe der Forcing-Technik werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese von den Axiomen der Mengenlehre unabhaengig ist.
LernzielDie Forcing-Technik kennenlernen und verschiedene Unabhaengigkeitsbeweise fuehren koennen.
InhaltMit Hilfe der sogenannten Forcing-Technik, welche anfangs der 1960er Jahre von Paul Cohen entwickelt wurde, werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese CH von den Axiomen der Mengenlehre ZFC unabhaengig ist. Weiter wird in Modellen von ZFC, in denen CH nicht gilt, die Groesse verschiedener Kardinalzahlcharakteristiken untersucht. Zum Schluss der Vorlesung wird ein Modell von ZFC konstruiert, in dem es (bis auf Isomorphie) genau n Ramsey-Ultrafilter gibt, wobei n fuer irgend eine nicht-negative ganze Zahl steht.
SkriptIch werde mich weitgehend an mein Buch "Combinatorial Set Theory" halten, aus dem einige Kapitel aus Part II & III behandelt werden.
Literatur"Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2012)

Link
Voraussetzungen / BesonderesVoraussetzung ist die Vorlesung "Axiomatische Mengenlehre" (Fruehlingssemester 2015) bzw. die entsprechenden Kapitel aus meinem Buch.
401-3109-65LProbabilistic Number TheoryW6 KP2V + 1UE. Kowalski
KurzbeschreibungThe course presents some aspects of probabilistic number theory, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.
Lernziel
InhaltThe goal of the course is to present some results of probabilistic
number theory in a unified manner. The main concepts will be presented
in parallel with the proof of three main theorems: (1) the Erdös-Kac
theorem and its variants concerning the number of prime divisors of
integers in various sequences; (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) functional limit
theorems for the paths of partial sums of families of exponential sums
such as Kloosterman sums.
LiteraturH. Iwaniec and E. Kowalski: "Analytic number theory", and additional
lecture notes will be prepared.
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.
401-3149-65LElliptic CurvesW4 KP2VE. Viada
KurzbeschreibungWe will study elliptic curves from different point of view: as varieties, as equations, as quotients. We will then study some properties of algebraic points on an elliptic curve. We will finally describe special subset of the rational points of curves that are embedded in products of elliptic curves.
LernzielThe aim of this course is to get used to geometric objects and algebraic tools, such as elliptic curves, curves, heights and degree.
InhaltWe will first study the properties of elliptic curves. We prove that an elliptic curve can be described as the quotient of the complex numbers by a lattice and equivalently as the zero set of an equation of degree 3. We will introduce the notion of height and degree of algebraic points and describe subsets of algebraic points of bounded height and degree.
Finally, We will study some properties of algebraic points on curves embedded in products of elliptic curves.
LiteraturJ. Silverman "The arithmetic of Elliptic Curves"
J. Silverman " Advanced Topics in the arithmetic of Elliptic Curves"
E. Bombieri & W. Gubler " Heights in Diophantine Geometry"
Voraussetzungen / BesonderesAlgebra and Linear Algebra, Topology, Geometry, some basic Algebraic Geometry.
401-3059-00LKombinatorik IIW4 KP2GN. Hungerbühler
KurzbeschreibungDer Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende 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.
401-3202-09LRepresentation Theory of Finite Groups, and in Particular Symmetric Groups Information W4 KP2VA. Buryak
KurzbeschreibungThe first part of the course will be devoted to the general theory of linear representations of finite groups. In the second part we will discuss in details the representation theory of the symmetric groups and some of its applications.
Lernziel
LiteraturJ.-P. Serre. Linear Representations of Finite Groups.
G. D. James. The Representation Theory of the Symmetric Groups.
D. M. Goldschmidt. Groups Characters, Symmetric Functions, and the Hecke Algebra.
Voraussetzungen / BesonderesIt will be assumed that the listeners know the material from a basic linear algebra course and also basic facts about groups and rings.
401-4149-65LReading Course: Geometric Invariant TheoryW2 KP4AJ. Fresán, P. S. Jossen
KurzbeschreibungGeometric Invariant Theory (GIT) is concerned with the problem of defining quotients of algebraic varieties by group actions, a crucial step in the construction of moduli spaces. Although some of the ideas go back to Hilbert, it was developed in its present form by Mumford in the 60s.
LernzielThe goal of this reading course is to give an introduction to GIT, with emphasis on examples rather than the most general statements.

After a couple of introductory sessions, participants will contribute with talks.
InhaltWe will cover topics as:
-existence of affine and projective quotients
-the Hilbert-Mumford criterion
-construction of the moduli space of elliptic curves
-toric varieties as GIT quotients
-semistable vector bundles on curves
LiteraturD. Mumford and K. Suominen. "Introduction to the theory of moduli". Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math.), pp. 171-222. Wolters-Noordhoff, Groningen, 1972.

J. Le Potier. Lectures on vector bundles. Cambridge Studies in Advanced Mathematics, 54. Cambridge University Press, Cambridge, 1997.
Voraussetzungen / BesonderesBasic knowledge of algebraic geometry will be assumed.
Auswahl: Geometrie
NummerTitelTypECTSUmfangDozierende
401-3523-65LEquidecomposability of PolytopesW4 KP2VL. Parapatits
KurzbeschreibungA polygon in the plane can be decomposed into finitely many (convex) pieces and reassembled to form another polygon if and only if they have the same area. Hilbert's third problem asks if the analogous is also true for two polyhedra in space. Whether or not it is possible to define volume without the use of approximation arguments depends on the answer to this question.
LernzielThe course will cover classical results on equidecomposabilty including the Dehn-Sydler theorem, i.e. the solution to Hilbert's third problem. We will then describe the connection between equidecomposability and valuation theory. Finally, we will discuss some recent classification results of valuations that are invariant under certain groups of motions.
Voraussetzungen / BesonderesOffice hours: Thursday 11:00 - 12:00
401-4573-65LSurfaces and 3-ManifoldsW4 KP2VA. Sisto
KurzbeschreibungThis course is an introduction and invitation to the theory of manifolds of dimension 2 and 3, with focus on the connections between the two dimensions.
LernzielThe goal is to give an overview of hyperbolic surfaces, Mapping Class Groups, construction of 3-manifolds, and the geometrisation theorem. The starting point will be the statement of the geometrisation theorem in dimension 2 and the goal the statement of the geometrisation theorem in dimension 3.
The choice of topics to discuss, especially in the second part of the course, can vary depending on the interests of the audience.
Voraussetzungen / BesonderesThe prerequisite is essentially just knowing the definition of manifold. It could help but it's not strictly necessary to know basic covering theory and Riemannian geometry.
The exam will consist in presenting a result from the course whose proof has been skipped during the course.
401-3057-00LEndliche Geometrien II
Findet dieses Semester nicht statt.
W4 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.
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