Suchergebnis: Katalogdaten im Frühjahrssemester 2023

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

Nummer

Titel

Typ

ECTS

Umfang

Dozierende

401-3112-23L

Number Theory II: Introduction to Modular Forms

Be aware that there is a big overlap with former course units on modular forms, in particular with 401-4118-22L taught in the Spring Semester 2022 as an elective course. Only one of the two course units may be recognised for credits. More precisely, it is also not allowed to have recognised one course unit for the Bachelor's and the other course unit for the Master's degree.

8 KP

Ö. Imamoglu

Kurzbeschreibung

This is a introductory course on automorphic forms covering its basic properties with emphasis on connections with number theory.

Lernziel

The aim of the course is to cover the classical theory of modular forms

Inhalt

Basic 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.

Literatur

J.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.

Voraussetzungen / Besonderes

Function theory, Algebra I and II

401-3002-12L

Algebraic Topology II

8 KP

S. Kalisnik Hintz

Kurzbeschreibung

This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including:
cohomology of spaces, operations in homology and cohomology, duality.

Lernziel

Literatur

1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

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

The book can be downloaded for free at:
http://www.math.cornell.edu/~hatcher/AT/ATpage.html

3) E. Spanier, "Algebraic topology", Springer-Verlag

Voraussetzungen / Besonderes

General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").

Some knowledge of differential geometry and differential topology
is useful but not absolutely necessary.

401-3146-12L

Algebraic Geometry

Findet dieses Semester nicht statt.

10 KP

4V + 1U

keine Angaben

Kurzbeschreibung

This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).

Lernziel

Learning Algebraic Geometry.

Literatur

Primary References:
* John Ottem, Geir Ellingsrud: Introduction to algebraic varieties,  https://www.uio.no/studier/emner/matnat/math/MAT4210/data/mastermat4210.pdf
* James Milne: Algebraic Geometry,
http://www.jmilne.org/math/CourseNotes/AG.pdf 

Secondary References:
* Miles Reid: Undergraduate Algebraic Geometry, Cambridge University Press.
* Ravi Vakil: Foundations of Algebraic Geometry,
http://math.stanford.edu/~vakil/216blog/
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer. 

Other textbooks:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* Igor Shafarevich: Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.

Voraussetzungen / Besonderes

Requirement: Basic knowledge of Commutative Algebra.

401-8142-21L

Algebraic Geometry II (University of Zurich)

Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MAT517

Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html

9 KP

4V + 1U

Uni-Dozierende

Kurzbeschreibung

We continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc.

Lernziel

Voraussetzungen / Besonderes

MAT507 Algebraic Geometry = 401-8141-00L Algebraic Geometry (University of Zurich) offered in the Autumn Semester 2022.
This is also a good follow-up course unit for ETH students who visited Maria Yakerson's lecture Algebraic Geometry in Spring 2022.

401-3226-00L

Symmetric Spaces

8 KP

A. Iozzi

Kurzbeschreibung

* Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples
* Symmetric spaces of non-compact type: flats and rank, roots and root spaces
* Iwasawa decomposition, Weyl group, Cartan decomposition
* Geometry at infinity

Lernziel

Learn the basics of symmetric spaces

401-3532-08L

Differential Geometry II

10 KP

4V + 1U

J. Serra

Kurzbeschreibung

This is a continuation course of Differential Geometry I. Topics covered include:
Introduction to Riemannian geometry: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, and isoperimetric inequalities.

Lernziel

Providing an introductory invitation to Riemannian geometry.

Literatur

- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,

Voraussetzungen / Besonderes

Differential Geometry I (or basics of differentiable manifolds)

401-3462-00L

Functional Analysis II

10 KP

4V + 1U

P. Hintz

Kurzbeschreibung

Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles. Basic results for hyperbolic PDE.

Lernziel

Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods to the study of elliptic boundary value problems, and to initial value problems for hyperbolic PDE.

Literatur

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001.

Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.

Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.

Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

Voraussetzungen / Besonderes

Functional Analysis I and fluency in multivariable calculus.

Kernfächer aus Bereichen der angewandten Mathematik ...

vollständiger Titel:
Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten

Nummer

Titel

Typ

ECTS

Umfang

Dozierende

401-3052-10L

Graph Theory

10 KP

4V + 1U

B. Sudakov

Kurzbeschreibung

Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem

Lernziel

The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.

Skript

Lecture will be only at the blackboard.

Literatur

West, D.: "Introduction to Graph Theory"
Diestel, R.: "Graph Theory"

Further literature links will be provided in the lecture.

Voraussetzungen / Besonderes

Students are expected to have a mathematical background and should be able to write rigorous proofs.

401-3642-00L

Brownian Motion and Stochastic Calculus

10 KP

4V + 1U

D. Possamaï

Kurzbeschreibung

This course gives an introduction to Brownian motion and stochastic calculus. It includes the construction and properties of Brownian motion, basics of Markov processes in continuous time and of Levy processes, and stochastic calculus for continuous semimartingales.

Lernziel

This course gives an introduction to Brownian motion and stochastic calculus. The following topics are planned:
- Definition and construction of Brownian motion
- Some important properties of Brownian motion
- Basics of Markov processes in continuous time
- Stochastic calculus, including stochastic integration for continuous semimartingales, Ito's formula, Girsanov's theorem, stochastic differential equations and connections with partial differential equations
- Basics of Levy processes

Skript

Lecture notes will be made available in class.

Literatur

- R.F. Bass, Stochastic Processes, Cambidge University Press (2001).
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).

Voraussetzungen / Besonderes

Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).

401-3632-00L

Computational Statistics

8 KP

3V + 1U

M. Mächler

Kurzbeschreibung

We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.

Lernziel

The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.

Inhalt

See the class website

Voraussetzungen / Besonderes

At least one semester of (basic) probability and statistics.

Programming experience is helpful but not required.

Kompetenzen

Fachspezifische Kompetenzen	Konzepte und Theorien	geprüft
	Verfahren und Technologien	geprüft
Methodenspezifische Kompetenzen	Analytische Kompetenzen	geprüft
	Medien und digitale Technologien	geprüft
	Problemlösung	geprüft
Persönliche Kompetenzen	Kreatives Denken	geprüft
	Kritisches Denken	geprüft

401-3602-00L

Applied Stochastic Processes

8 KP

3V + 1U

V. Tassion

Kurzbeschreibung

Poisson-Prozesse; Erneuerungsprozesse; Markovketten in diskreter und in stetiger Zeit; einige Beispiele und Anwendungen.

Lernziel

Stochastische Prozesse dienen zur Beschreibung der Entwicklung von Systemen, die sich in einer zufälligen Weise entwickeln. In dieser Vorlesung bezieht sich die Entwicklung auf einen skalaren Parameter, der als Zeit interpretiert wird, so dass wir die zeitliche Entwicklung des Systems studieren. Die Vorlesung präsentiert mehrere Klassen von stochastischen Prozessen, untersucht ihre Eigenschaften und ihr Verhalten und zeigt anhand von einigen Beispielen, wie diese Prozesse eingesetzt werden können. Die Hauptbetonung liegt auf der Theorie; "applied" ist also im Sinne von "applicable" zu verstehen.

Literatur

R. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997
R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1
M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1
S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)

Voraussetzungen / Besonderes

Prerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).

401-3652-00L

Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich)

Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MAT827

Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html

10 KP

4V + 1U

Uni-Dozierende

Kurzbeschreibung

This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB.

Lernziel

The goal of this course is familiarity with the fundamental ideas and mathematical
consideration underlying modern numerical methods for conservation laws and wave equations.

Inhalt

* Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering.

* Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes.

* Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes.

* Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods.

* Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes.

* Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.

Skript

Lecture slides will be made available to participants. However, additional material might be covered in the course.

Literatur

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online.

R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online.

E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.

Voraussetzungen / Besonderes

Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite.

Programming exercises in MATLAB

Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"

Wahlfächer

Wahlfächer aus Bereichen der reinen Mathematik

Auswahl: Algebra, Zahlentheorie, Topologie, diskrete Mathematik, Logik

Nummer

Titel

Typ

ECTS

Umfang

Dozierende

401-3058-00L

Kombinatorik I

4 KP

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.

401-3035-00L

Forcing: Einführung in Unabhängigkeitsbeweise

8 KP

3V + 1U

L. Halbeisen

Kurzbeschreibung

Mit Hilfe der Forcing-Technik werden verschiedene Unabhaengigkeitsbeweise gefuehrt. Insbesondere wird gezeigt, dass die Kontinuumshypothese von den Axiomen der Mengenlehre unabhaengig ist.

Lernziel

Die Forcing-Technik kennenlernen und verschiedene Unabhaengigkeitsbeweise fuehren koennen.

Inhalt

Mit 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.

Skript

Ich werde mich weitgehend an mein Buch "Combinatorial Set Theory" halten, aus dem einige Kapitel aus Part III & IV behandelt werden.

Literatur

"Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2017)

http://www.springer.com/de/book/9783319602301

Voraussetzungen / Besonderes

Voraussetzung ist die Vorlesung "Axiomatische Mengenlehre" (Herbstsemester 2017) bzw. die entsprechenden Kapitel aus meinem Buch.

Auswahl: Geometrie

Nummer

Titel

Typ

ECTS

Umfang

Dozierende

401-3056-00L

Endliche Geometrien I

Findet dieses Semester nicht statt.

4 KP

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.

401-4146-22L

Derived Algebraic Geometry

4 KP

A. Bojko

Kurzbeschreibung

The main goal is to introduce this subject to a wider audience in a more intuitive way. The course should ideally end with applications of derived algebraic geometry to constructing virtual fundamental classes in enumerative geometry using perverse sheaves.

Lernziel

A keen listener should understand by the end of the course why derived algebraic geometry is useful and have an idea of where to begin in applying it to problems in enumerative geometry.

Literatur

B. Toën, Derived Algebraic Geometry, arXiv:1401.1044, 2014.
J. Lurie. Higher topos theory, Annals of Mathematics Studies. Princeton
University Press, Princeton, NJ, 2009.
J. Lurie, On Infinity Topoi, arXiv:math/0306109, 2003.
J. Lurie, Derived Algebraic Geometry, Ph.D. thesis, Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
B. Toën and G. Vezzosi. Homotopical algebraic geometry I: Topos theory”, Advances in mathematics, 2005.
B. Toën and G. Vezzosi, From HAG to DAG: Derived Moduli Stacks:Ax-iomatic, Enriched and Motivic Homotopy Theory, 2004.
B. Toën and M. Vaquié, Moduli of objects in dg-categories, Annales scien-tifiques de l’Ecole normale supérieure, 2007.
C. Brav, V. Bussi, and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, Journal of the American Mathematical Society, 2019.
D. Joyce , P. Safronov, A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes, In Annales de la Faculté des sciences de Toulouse: Mathématiques, 2019.
D. Borisov, and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geometry & Topology, 2017.
K. Tasuki, Virtual classes via vanishing cycles, arXiv:2109.06468, 2021.

Voraussetzungen / Besonderes

One should have some understanding of algebraic geometry (in particular intersection theory), algebraic topology and category theory.

Familiarity with some enumerative geometry using virtual fundamental classes would be helpful for understanding the goal of the course.

Kompetenzen

Fachspezifische Kompetenzen

Konzepte und Theorien

geprüft

401-3574-61L

Introduction to Knot Theory

Findet dieses Semester nicht statt.

6 KP

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

401-4494-23L

Variational Problems and PDEs

4 KP

A. Figalli

Kurzbeschreibung

In this class, we will study some classical variational problems to motivate the introduction and study of a series of important tools in analysis, such as:
1) the direct method of the calculus of variations
2) the use of geometric measure theory to deal with non-smooth objects
3) the regularity theory for elliptic PDEs with measurable and smooth coefficients

Lernziel

Learn important tools in the calculus of variations, elliptic PDEs, and geometric measure theory.

Voraussetzungen / Besonderes

Functional Analysis I

401-4834-23L

Nonlinear Wave Equations with Applications to General Relativity

4 KP

C. Kehle

Kurzbeschreibung

Introduction to linear and nonlinear wave equations, aspects of Lorentzian geometry and the Einstein equations in general relativity.

Lernziel

In the first part of the course, the basics of linear wave equations on Minkowski space are covered. We then go beyond to nonlinear equations as well as to curved backgrounds. To this end, the relevant background from Lorentzian geometry is introduced. The ultimate goal is to rigorously study dynamics of the Einstein equations of general relativity.

Voraussetzungen / Besonderes

It may be helpful if the participants have some familiarity with the basics of differential manifolds and basic functional analysis.

Kompetenzen

Fachspezifische Kompetenzen

Konzepte und Theorien

geprüft

401-3374-23L

Dynamical Systems and Ergodic Theory (University of Zurich)

Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MAT733

Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/deadlines.html

9 KP

4V + 2U

Uni-Dozierende

Kurzbeschreibung

Lernziel

By the end of the unit the student: will have developed a good background in the area of dynamical systems; will be familiar with the basic concepts, results, and techniques relevant to the area; will have detailed knowledge of a number of fundamental examples that help clarify and motivate the main concepts in the theory; will understand the proofs of the fundamental theorems in the area; will have mastered the application of dynamical systems techniques for solving a range of standard problems; will have a firm foundation for further study in the area.

Inhalt

Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it in average. We will give a strong emphasis on presenting many fundamental examples of dynamical systems. Driven by the examples, we will first introduce some of the phenomena and main concepts which one is interested in studying.
We will then formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications for example to number theory, information theory and Internet search engines. Topics which will be covered include:

-Basic examples of dynamical systems (e.g. rotations and doubling map; baker’s map, CAT map and hyperbolic toral automorphisms; the Gauss map and continued fractions);
-Elements of topological dynamics (minimality; topological conjugacy; topological mixing; topological entropy);
-Elements of symbolic dynamics (shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding);
-Introduction to ergodic theory: invariant measures; Poincare' recurrence; ergodicity; mixing; the Birkhoff Ergodic Theorem and applications; Markov measures; the ergodic theorem for Markov chains and applications to Internet Search; measure theoretic entropy;
-Selected topics (time permitting): Shannon-McMillan-Breiman theorem; Lyapunov exponents and multiplicative ergodic theorem; continuous time dynamical systems and some mathematical billiards.

Skript

Lecture notes for several of the topics covered will be provided.

Literatur

Textbooks which can be used as additional reference for some of the topics include:

-B. Hasselblatt and A. Katok, Dynamics: A first course. (Cambdirge University Press, 2003) – Chapters 7,8,10 and 15
-M. Brin and G. Stuck, Introduction to Dynamical Systems. (Cambridge University Press, 2002) – Chapters 1-4
-Omri Sarig, Lectures Notes on Ergodic Theory (Available Online), Topics from Chapter 1 and 2

Voraussetzungen / Besonderes

Prior Knowledge
Basic knowledge of measure theory and integration.

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