Suchergebnis: Katalogdaten im Frühjahrssemester 2020

Mathematik Master Information
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
401-3146-12LAlgebraic Geometry Information W10 KP4V + 1UD. Johnson
KurzbeschreibungThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
LernzielLearning Algebraic Geometry.
LiteraturPrimary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.

Secondary reference:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013).

Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry, Link
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link

"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry, Link

Further readings:
* Günter Harder: Algebraic Geometry 1 & 2
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Voraussetzungen / BesonderesRequirement: Some knowledge of Commutative Algebra.
401-3002-12LAlgebraic Topology II Information W8 KP4GA. Sisto
KurzbeschreibungThis 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.
Literatur1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

The book can be downloaded for free at:

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

3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / BesonderesGeneral 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.
401-3226-00LSymmetric Spaces Information W8 KP4GM. Burger
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
* Hints of the geometry at infinity of SL(n,R)/SO(n).
LernzielLearn the basics of symmetric spaces
401-3372-00LDynamical Systems IIW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.
LernzielMastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.
InhaltTopics covered include:

- Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem.
- Hyperbolic sets, Anosov diffeomorphisms.
- The (Un)stable Manifold Theorem.
- Shadowing Lemmas and stability.
- The Lambda Lemma.
- Transverse homoclinic points, horseshoes, and chaos.
- Complex dynamics of rational maps on the Riemann sphere
- Julia sets and Fatou sets.
- Fractals and the Mandelbrot set.
SkriptI will provide full lecture notes, available here:

LiteraturThe most useful textbook is

- Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
Voraussetzungen / BesonderesIt will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:


However we will only really use material covered in the first 10 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 10 lectures.

In addition, it would be useful to have some familiarity with basic differential geometry and complex analysis.
401-3532-08LDifferential Geometry II Information W10 KP4V + 1UU. Lang
KurzbeschreibungIntroduction to Riemannian geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of Riemannian manifolds.
LernzielLearn the basics of Riemannian geometry and some elements of modern metric geometry.
Literatur- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983
Voraussetzungen / BesonderesPrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms.
401-3462-00LFunctional Analysis II Information W10 KP4V + 1UM. Struwe
KurzbeschreibungSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity
LernzielAcquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates
SkriptFunktionalanalysis II, Michael Struwe
LiteraturFunktionalanalysis II, Michael Struwe

Functional Analysis, Spectral Theory and Applications.
Manfred Einsiedler and Thomas Ward, GTM Springer 2017
Voraussetzungen / BesonderesFunctional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.
Kernfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel:
Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
401-3052-10LGraph Theory Information W10 KP4V + 1UB. Sudakov
KurzbeschreibungBasics, 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
LernzielThe 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.
SkriptLecture will be only at the blackboard.
LiteraturWest, D.: "Introduction to Graph Theory"
Diestel, R.: "Graph Theory"

Further literature links will be provided in the lecture.
Voraussetzungen / BesonderesStudents are expected to have a mathematical background and should be able to write rigorous proofs.
401-3642-00LBrownian Motion and Stochastic Calculus Information W10 KP4V + 1UW. Werner
KurzbeschreibungThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
LernzielThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
SkriptLecture notes will be distributed in class.
Literatur- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- 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).
- D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
Voraussetzungen / BesonderesFamiliarity 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).
401-3632-00LComputational StatisticsW8 KP3V + 1UM. H. Maathuis
KurzbeschreibungWe 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.
LernzielThe 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.
Voraussetzungen / BesonderesAt least one semester of (basic) probability and statistics.

Programming experience is helpful but not required.
401-3602-00LApplied Stochastic Processes Information
Findet dieses Semester nicht statt.
W8 KP3V + 1Ukeine Angaben
KurzbeschreibungPoisson-Prozesse; Erneuerungsprozesse; Markovketten in diskreter und in stetiger Zeit; einige Beispiele und Anwendungen.
LernzielStochastische 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.
LiteraturR. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: Link
R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: Link
M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: Link
S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)
Voraussetzungen / BesonderesPrerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3652-00LNumerical 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: Link
W10 KP4V + 2UUni-Dozierende
KurzbeschreibungThis 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.
LernzielThe 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.
SkriptLecture slides will be made available to participants. However, additional material might be covered in the course.
LiteraturH. 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 / BesonderesHaving 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"
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
401-3201-00LAlgebraic 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.
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

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
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.
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.
  •  Seite  1  von  7 Nächste Seite Letzte Seite     Alle