Das Herbstsemester 2020 findet in einer gemischten Form aus Online- und Präsenzunterricht statt.
Bitte lesen Sie die publizierten Informationen zu den einzelnen Lehrveranstaltungen genau.

Suchergebnis: Katalogdaten im Frühjahrssemester 2017

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
NummerTitelTypECTSUmfangDozierende
401-3146-12LAlgebraic Geometry Information W10 KP4V + 1UR. Pink
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, http://math.stanford.edu/~vakil/216blog/
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html

"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, http://www.jmilne.org/math/CourseNotes/AG.pdf

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 IIW8 KP4GP. S. Jossen
KurzbeschreibungThis is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations, characteristic classes, spectral sequences etc.
Lernziel
Literatur1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

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

See also:
http://www.math.cornell.edu/~hatcher/#anchor1772800

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

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

4) R. Bott & L. Tu, "Differential forms in algebraic topology",
Graduate Texts in Mathematics, 82. Springer-Verlag, 1982.

5) J. Milnor & J. Stasheff, "Characteristic classes",
Annals of Mathematics Studies, No. 76.
Princeton University Press, 1974.
Voraussetzungen / BesonderesGeneral topology, linear algebra.
Basic knowledge of singular homolgoy and cohomology 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.
401-3226-01LRepresentation Theory of Lie GroupsW8 KP4GE. Kowalski
KurzbeschreibungThis course will contain two parts:
* Introduction to unitary representations of Lie groups
* Introduction to the study of discrete subgroups of Lie groups and some applications.
LernzielThe goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups.
Inhalt* Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula
* Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C)
* Example: Property (T) for SL(n,R)
* Discrete subgroups of Lie groups: examples and some applications
LiteraturBekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press.
Voraussetzungen / BesonderesDifferential geometry, Functional analysis, Introduction to Lie Groups (or equivalent).
Notice that this course has a large overlap with 401-3226-01L Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups taught in FS 2016. Therefore it is not possible to acquire credits for both courses.
401-3372-00LDynamical Systems IIW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic dynamics.
LernzielMastery of the basic methods and principal themes of some aspects of hyperbolic dynamical systems.
InhaltTopics covered include:

- Circle homeomorphisms and rotation numbers.
- 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.
SkriptI will provide full lecture notes, available here:

http://www.merry.io/dynamical-systems/
LiteraturThe most useful textbook is

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

Another (more advanced) useful book is

- Introduction to the Modern Theory of Dynamical Systems, Katok and Hasselblatt, CUP, 1995.
Voraussetzungen / BesonderesIt will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:

http://www.merry.io/dynamical-systems/

However we will only really use material covered in the first 12 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 12 lectures.

In addition, it would be useful to have some familiarity with basic differential geometry.
401-3532-08LDifferential Geometry IIW10 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, curvature and topology, spaces of riemannian manifolds.
LernzielThe aim of this course is to give an introduction to Riemannian Geometry in combination with some elements of modern metric geometry.
InhaltRiemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form of submanifolds, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of riemannian manifolds.
LiteraturRiemannian Geometry:
- 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
Metric Geometry:
- M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999
- D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001
Voraussetzungen / BesonderesPrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, tangent and tensor bundles, and differential forms.
401-3462-00LFunctional Analysis IIW10 KP4V + 1UM. Struwe
KurzbeschreibungSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity theory, Schauder estimates
LernzielThe lecture course will focus on weak solutions of elliptic boundary value problems in Sobolev spaces and discuss their regularity properties, possibly followed by a proof of the Calderon-Zygmund
inequality and some basic results on parabolic regularity, with
applications to geometry, if time allows.
Kernfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel:
Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
NummerTitelTypECTSUmfangDozierende
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.
401-3652-00LNumerical Methods for Hyperbolic Partial Differential Equations Information W10 KP4V + 1UU. S. Fjordholm
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"
401-3642-00LBrownian Motion and Stochastic CalculusW10 KP4V + 1UM. Larsson
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- 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 Statistics Information W10 KP3V + 2UM. Mächler, P. L. Bühlmann
Kurzbeschreibung"Computational Statistics" deals with modern methods of data analysis (aka "data science") for prediction and inference. An overview of existing methodology is provided and also by the exercises, the student is taught to choose among possible models and about their algorithms and to validate them using graphical methods and simulation based approaches.
LernzielGetting to know modern methods of data analysis for prediction and inference.
Learn to choose among possible models and about their algorithms.
Validate them using graphical methods and simulation based approaches.
InhaltDas Schliessen von beobachteten Daten auf komplexe Modelle ist ein zentrales Thema der rechnerorientierten Statistik. Die Modelle sind oft unendlich-dimensional und die statistischen Verfahren deshalb Computer-intensiv.
Als Grundlage wird die klassische multiple Regression eingeführt. Danach werden einige nichtparametrische Verfahren für die Regression und die Klassifikation vorgestellt: Kernschätzer, glättende Splines, Regressions-/Klassifikationsbäume, additive Modelle, Projection Pursuit und (kurz) Neuronale Netze, wobei einige davon gut interpretierbar und andere für genaue Prognosen geeignet sind. Insbesondere werden auch die Problematik des Fluchs der Dimension und die stochastische Regularisierung diskutiert. Hochdimensionale Modelle werden mit LASSO u.ä. Verfahren regularisiert. Nebst dem Anpassen eines (komplexen) Modells werden auch die Evaluation, Güte und Unsicherheit von Verfahren und Modellen anhand von Resampling, Bootstrap und Kreuz-Validierung behandelt.

In den Übungen wird mit dem Statistik-Paket R (https://www.R-project.org) gearbeitet. Es werden dabei auch praxis-bezogene Probleme bearbeitet. Aktive Teilnahme an den Übungen wird sehr empfohlen. Detailinformation sind auf https://stat.ethz.ch/lectures/ (-> "Computational Statistics").
Skriptlecture notes are available online; see
http://stat.ethz.ch/lectures/ (-> "Computational Statistics").
Literatur(see the link above, and the lecture notes)
Voraussetzungen / BesonderesBasic "applied" mathematical calculus (incl. simple two-dimensional) and linear algebra (including Eigenvalue decomposition) similar to two semester "Analysis" in an ETH (math or) engineer's bachelor.

At least one semester of (basic) probability and statistics, as e.g., taught in an ETH engineer's or math bachelor.

Programming experience in either a compiler-based computer language (such as C++) or a high-level language such as python, R, julia, or matlab. The language used in the exercises and the final exam will be R (https://www.r-project.org) exclusively. If you don't know it already, some extra effort will be required for the exercises.
401-3602-00LApplied Stochastic ProcessesW8 KP3V + 1UA.‑S. Sznitman
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: 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 / BesonderesPrerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3622-00LRegression
Findet dieses Semester nicht statt.
W8 KP4Gkeine Angaben
KurzbeschreibungIn der Regression wird die Abhängigkeit einer zufälligen Response-Variablen von anderen Variablen untersucht. Wir betrachten die Theorie der linearen Regression mit einer oder mehreren Co-Variablen, nicht-lineare Modelle und verallgemeinerte lineare Modelle, Robuste Methoden, Modellwahl und nicht-parametrische Modelle. Verschiedene numerische Beispiele werden die Theorie illustrieren.
LernzielEinführung in Theorie und Praxis eines umfassenden und vielbenutzten Teilgebiets der angewandten Statistik, unter Berücksichtigung neuerer Entwicklungen.
InhaltIn der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression, Numerik, Ridge Regression. Durchrechnung und Diskussion von Anwendungsbeispielen.
SkriptVorlesungsskript
Voraussetzungen / BesonderesCredits cannot be recognised for both courses 401-3622-00L Regression and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden).
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
401-4142-17LAlgebraic CurvesW6 KP3GR. Pandharipande
KurzbeschreibungI will discuss the classical theory of algebraic curves. The topics will include:
divisors, Riemann-Roch, linear systems, differentials, Clifford's theorem,
curves on surfaces, singularities, curves in projective space, elliptic curves,
hyperelliptic curves, families of curves, moduli, and enumerative geometry.
There will be many examples and calculations.
Lernziel
InhaltLecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-4142-17L/
LiteraturForster, "Lectures on Riemann Surfaces"

Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves"

Mumford, "Curves and their Jacobians"
Voraussetzungen / BesonderesFor background, a semester course in algebraic geometry should be
sufficient (perhaps even if taken concurrently). You should know the definitions
of algebraic varieties and algebraic morphisms and their basic properties.
401-3106-17LClass Field TheoryW6 KP2V + 1UJ. Fresán
KurzbeschreibungClass Field Theory aims at describing the Galois group of the maximal abelian extension of global and local fields.
Lernziel
Literatur[1] D. Harari, Cohomologie galoisienne et théorie du corps de classes, EDP Sciences, CNRS Éditions, Paris, 2017.
[2] K. Kato, N. Kurokawa, T. Saito, Number theory 2. Introduction to class field theory, Translations of Mathematical Monographs 240, AMS, 2011.
[3] J. S. Milne, Class Field Theory (available at http://www.jmilne.org/math/CourseNotes/cft.html)
[4] J-P. Serre, Local fields, Grad. Texts Math. 67. Springer-Verlag, 1979.
401-3033-00LDie Gödel'schen SätzeW8 KP3V + 1UL. Halbeisen
KurzbeschreibungDie Vorlesung besteht aus drei Teilen:
Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe.
Teil II behandelt den Gödel'schen Vollständigkeitssatz
Teil III behandelt die Gödel'schen Unvollständigkeitssätze
LernzielDas Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln.
InhaltSyntax und Semantik der Prädikatenlogik
Gödel'scher Vollständigkeitssatz
Gödel'sche Unvollständigkeitssätze
LiteraturErgänzende Literatur wird in der Vorlesung angegeben.
401-3058-00LKombinatorik IW4 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.
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.
401-3112-17LIntroduction to Number TheoryW4 KP2VC. Busch
KurzbeschreibungThis course gives an introduction to number theory. The focus will be on algebraic number theory.
Lernziel
InhaltThe following subjects will be covered:
- Euclidean algorithm, greatest common divisor, ...
- Congruences, Chinese Remainder Theorem
- Quadratic residues, Legendre symbol, law of quadratic reciprocity
- Quadratic number fields, integers and primes
- Units of quadratic number fields, Pell's equation, Dirichlet unit theorem
- Continued fractions and quadratic irrationalities, Theorem of Euler Lagrange, relation to units.
Literatur- A. Fröhlich, M.J. Taylor, Algebraic number theory, Cambridge studies in advanced mathematics 27, Cambridge University Press, 1991
- S. Lang, Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics, 110, Springer, 1994
- J. Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften 322, Springer 1999
- R. Remmert, P. Ullrich, Elementare Zahlentheorie, Grundstudium Mathematik, Basel Birkhäuser, 2008
- P. Samuel, Algebraic Theory of Numbers, Kershaw Publishing Company LTD, 1972 (Original edition in French at Hermann)
- J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer 1973 (Original edition in French at Presses Universitaires de France)
Voraussetzungen / BesonderesBasic knowledge of Algebra as taught in a course Algebra I + II.
Auswahl: Geometrie
NummerTitelTypECTSUmfangDozierende
401-4206-17LGroup Actions on TreesW4 KP2VN. Lazarovich
KurzbeschreibungAs a main theme, we will explain how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. After introducing the general theory, we will cover various topics in this general theme.
LernzielIntroduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups.
InhaltDepending on time we will cover some of the following topics.
- Free groups and their subgroups.
- The general theory of actions on trees, i.e, Bass-Serre theory.
- Trees as 1-dimensional buildings.
- Stallings' theorem.
- Grushko's and Dunwoody's accessibility results.
- Actions on R-trees and the Rips machine.
LiteraturJ.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9

C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Voraussetzungen / BesonderesFamiliarity with the basics of fundamental group (and covering theory).
401-4148-17LReading Course: Introduction to the Moduli of Maps and Gromow-Witten InvariantsW2 KP4AG. Bérczi
KurzbeschreibungEnumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology.
LernzielThe aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position.
InhaltTopics covered:
1) Brief survey on moduli spaces: fine and coarse moduli.
2) Stable n-pointed curves
3) Stable maps
4) Enumerative geometry via stable maps
5) Gromov-Witten invariants
6) Quantum cohomology and quantum product
7) Kontsevich's formula
LiteraturThe main reference for the course is:
J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves
www.math.utah.edu/%7eyplee/teaching/gw/Koch.pdf

Background material:
-Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
-Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag
-Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag
Voraussetzungen / BesonderesSome minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves).
Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples.
401-3056-00LEndliche Geometrien I
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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