Suchergebnis: Katalogdaten im Herbstsemester 2017

Nummer	Titel	Typ	ECTS	Umfang	Dozierende
Mathematik Master
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
401-3225-00L	Introduction to Lie Groups	W	8 KP	4G	A. Iozzi
Kurzbeschreibung	Topological 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.
Lernziel	The 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.
Literatur	A. 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 / Besonderes	Topology 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: https://metaphor.ethz.ch/x/2017/hs/401-3225-00L/
401-3001-61L	Algebraic Topology I	W	8 KP	4G	W. Merry
Kurzbeschreibung	This is an introductory course in algebraic topology. Topics covered include: the fundamental group, covering spaces, singular homology, cell complexes and cellular homology and the Eilenberg-Steenrod axioms. Along the way we will introduce the basics of homological algebra and category theory.
Lernziel
Skript	I will produce full lecture notes, available on my website at www.merry.io/algebraic-topology
Literatur	"Algebraic Topology" (CUP, 2002) by Hatcher is excellent and covers all the material from both Algebraic Topology I and Algebraic Topology II. You can also download it (legally!) for free from Hatcher's webpage: www.math.cornell.edu/%7ehatcher/AT/ATpage.html Another classic book is Spanier's "Algebraic Topology" (Springer, 1963). This book is very dense and somewhat old-fashioned, but again covers everything you could possibly want to know on the subject.
Voraussetzungen / Besonderes	You should know the basics of point-set topology (topological spaces, and what it means for a topological space to be compact or connected, etc). Some (very elementary) group theory and algebra will also be needed.
401-4147-67L	Algebraic Geometry II	W	10 KP	4V + 1U	R. Pink
Kurzbeschreibung	Quasicoherent sheaves, cohomology, Serre duality, Riemann-Roch theorem, algebraic curves, moduli schemes
Lernziel
Literatur	Primary reference: * Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer. * 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. * 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 / Besonderes	Algebraic Geometry I Spring 2017
401-3132-00L	Commutative Algebra	W	10 KP	4V + 1U	P. D. Nelson
Kurzbeschreibung	This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry.
Lernziel	We shall cover approximately the material from --- most of the textbook by Atiyah-MacDonald, or --- the first half of the textbook by Bosch. Topics include: * Basics about rings, ideals and modules * Localization * Primary decomposition * Integral dependence and valuations * Noetherian rings * Completions * Basic dimension theory
Literatur	Primary Reference: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) Secondary Reference: 2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013) Tertiary References: 3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)
Voraussetzungen / Besonderes	Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
401-3581-67L	Symplectic Geometry	W	8 KP	4V + 1U	A. Cannas da Silva
Kurzbeschreibung	This course is an introduction to symplectic geometry -- the geometry of manifolds equipped with a closed non-degenerate 2-form. We will discuss symplectic manifolds and transformations, the relation of symplectic to other geometries and some of the interplay with dynamics, eventually in the presence of symmetry groups. Guided homework assignments will complement the exposition.
Lernziel	Introduction to symplectic geometry

Kernfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3651-00L	Numerical Methods for Elliptic and Parabolic Partial Differential Equations (University of Zurich) 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. Der Kurs muss direkt an der UZH belegt werden. UZH Modulkürzel: MAT802 Beachten Sie die Einschreibungstermine an der UZH: https://www.uzh.ch/cmsssl/de/studies/application/mobilitaet.html	W	10 KP	4V + 1U + 1P	S. Sauter
Kurzbeschreibung	This 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.
Lernziel	Participants 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
Inhalt	A 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
Skript	Course slides will be made available to the audience.
Literatur	S. 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, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 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] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006).
Voraussetzungen / Besonderes	Practical exercises based on MATLAB
401-3621-00L	Fundamentals of Mathematical Statistics	W	10 KP	4V + 1U	S. van de Geer
Kurzbeschreibung	The course covers the basics of inferential statistics.
Lernziel
401-4889-00L	Mathematical Finance	W	11 KP	4V + 2U	J. Teichmann
Kurzbeschreibung	Advanced introduction to mathematical finance: - absence of arbitrage and martingale measures - option pricing and hedging - optimal investment problems - additional topics
Lernziel	Advanced level introduction to mathematical finance, presupposing knowledge in probability theory and stochastic processes
Inhalt	This is an advanced level introduction to mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this in both discrete- and continuous-time models. Topics include absence of arbitrage and martingale measures, option pricing and hedging, optimal investment problems, and probably others. Prerequisites are probability theory and stochastic processes (for which lecture notes are available).
Skript	Course homepage: https://metaphor.ethz.ch/x/2017/hs/401-4889-00L/ -Lecture notes -Exercise sheets -A list of relevant literature
Voraussetzungen / Besonderes	Prerequisites are probability theory and stochastic processes (for which lecture notes are available).
401-3901-00L	Mathematical Optimization	W	11 KP	4V + 2U	R. Weismantel
Kurzbeschreibung	Mathematical treatment of diverse optimization techniques.
Lernziel	Advanced optimization theory and algorithms.
Inhalt	1) 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.
Literatur	1) D. Bertsimas & R. Weismantel, "Optimization over Integers". Dynamic Ideas, 2005. 2) A. Schrijver, "Theory of Linear and Integer Programming". John Wiley, 1986. 3) D. Bertsimas & J.N. Tsitsiklis, "Introduction to Linear Optimization". Athena Scientific, 1997. 4) Y. Nesterov, "Introductory Lectures on Convex Optimization: a Basic Course". Kluwer Academic Publishers, 2003. 5) C.H. Papadimitriou, "Combinatorial Optimization". Prentice-Hall Inc., 1982.
Voraussetzungen / Besonderes	Linear algebra.

Bachelor-Kernfächer aus Bereichen der reinen Mathematik Nebst weiteren Einschränkungen gilt: 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 (www.math.ethz.ch/studiensekretariat).
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3461-00L	Functional Analysis I Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar.	E-	10 KP	4V + 1U	A. Carlotto
Kurzbeschreibung	Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces; Fourier transform and applications.
Lernziel	Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps.
Skript	Lecture Notes on "Funktionalanalysis I" by Michael Struwe
Literatur	A primary reference for the course is the textbook by H. Brezis: Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. Other useful, and recommended references are the following: Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
Voraussetzungen / Besonderes	Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces).
401-3531-00L	Differential Geometry I Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar.	E-	10 KP	4V + 1U	D. A. Salamon
Kurzbeschreibung	Submanifolds of R^n, tangent bundle, embeddings and immersions, vector fields, Lie bracket, Frobenius' Theorem. Geodesics, exponential map, completeness, Hopf-Rinow. Levi-Civita connection, parallel transport, motions without twisting, sliding, and wobbling. Isometries, Riemann curvature, Theorema Egregium. Cartan-Ambrose-Hicks, symmetric spaces, constant curvature, Hadamard's theorem.
Lernziel	Introduction to Differential Geometry. Submanifolds of Euclidean space, tangent bundle, embeddings and immersions, vector fields and flows, Lie bracket, foliations, the Theorem of Frobenius. Geodesics, exponential map, injectivity radius, completeness Hopf-Rinow Theorem, existence of minimal geodesics. Levi-Civita connection, parallel transport, Frame bundle, motions without twisting, sliding, and wobbling. Isometries, the Riemann curvature tensor, Theorema Egregium. Cartan-Ambrose-Hicks, symmetric spaces, constant curvature, nonpositive sectional curvature, Hadamard's theorem.
Literatur	Joel Robbin and Dietmar Salamon "Introduction to Differential Geometry", https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf

Bachelor-Kernfächer aus Bereichen der angewandten Mathematik .. Nebst weiteren Einschränkungen gilt: 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. Neu ist 402-0205-00L Quantenmechanik I als angewandtes Kernfach anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik (letztmals im FS 2016 angeboten) nicht angerechnet wird oder wurde (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat).
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3601-00L	Probability Theory Höchstens eines der drei Bachelor-Kernfächer 401-3461-00L Funktionalanalysis I / Functional Analysis I 401-3531-00L Differentialgeometrie I / Differential Geometry I 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist im Master-Studiengang Mathematik anrechenbar.	E-	10 KP	4V + 1U	A.‑S. Sznitman
Kurzbeschreibung	Basics of probability theory and the theory of stochastic processes in discrete time
Lernziel	This 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.
Inhalt	This 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.
Skript	available, will be sold in the course
Literatur	R. 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
402-0205-00L	Quantum Mechanics I	W	10 KP	3V + 2U	C. Anastasiou
Kurzbeschreibung	Einführung in die nicht-relativistische Einteilchen-Quantenmechanik. Diskussion grundlegender Ideen der Quantenmechanik, insbesondere Quantisierung klassischer Systeme, Wellenfunktionen und die Beschreibung von Observablen durch Operatoren auf einem Hilbertraum, und die Analyse von Symmetrien. Grundlegende Phänomene werden analysiert und durch generische Beispiele illustriert.
Lernziel	Einführung in die Einteilchen Quantenmechanik. Beherrschung grundlegender Ideen (Quantisierung, Operatorformalismus, Symmetrien, Störungstheorie) und generischer Beispiele und Anwendungen (gebunden Zustände, Tunneleffekt, Streutheorie in ein- und dreidimensionalen Problemen). Fähigkeit zur Lösung einfacher Probleme.
Inhalt	Stichworte: Schrödinger-Gleichung, Formalismus der Quantenmechanik (Zustände, Operatoren, Kommutatoren, Messprozess), Symmetrien (Translation, Rotationen), Quantenmechanik in einer Dimension, Zentralkraftprobleme, Potentialstreuung, Störungstheorie, Variations-Verfahren, Drehimpuls, Spin, Drehimpulsaddition, Relation QM und klassische Physik.
Literatur	J.J. Sakurai: Modern Quantum Mechanics Lectures on Quantum Mechanics, S. Weinberg

Wahlfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Algebra, Topologie, diskrete Mathematik, Logik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3034-00L	Axiomatische Mengenlehre	W	8 KP	3V + 1U	L. Halbeisen
Kurzbeschreibung	Es werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Zudem werden Ultrafilter untersucht und es wird das Martinaxiom eingeführt.
Lernziel
Inhalt	Es werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Insbesondere wird die Kontinuumshypothese behandelt und einige Konsequenzen besprochen. Zudem werden Ultrafilter untersucht und die Existenz gewisser Ultrafilter diskutiert. Im letzten Teil der Vorlesung wird das Martin-Axiom eingeführt, mit dessen Hilfe sich interessante Konsistenzresultate in Topologie und Masstheorie, sowie Resultate über Ultrafilter, beweisen lassen.
Skript	Ich werde mich weitgehend an mein Buch "Combinatorial Set Theory" (2nd ed., erscheint im Herbst 2017) halten.
Literatur	"Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2012) http://www.springer.com/mathematics/book/978-1-4471-2172-5
401-3118-67L	Classical Modular Forms	W	8 KP	4G	I. N. Petrow
Kurzbeschreibung
Lernziel
401-3129-67L	Differential Galois Theory	W	4 KP	2V	P. S. Jossen
Kurzbeschreibung	Algebraic theory of linear differential equations, Picard-Vessiot theory, Differential Galois groups, local theory of differential equations, the Frobenius method, Newton polygons, Connections and local systems, Riemann-Hilbert correspondence on ℙ¹.
Lernziel	We introduce differential Galois theory and mathematical concepts surrounding it. We formulate and prove an important case of the Riemann-Hilbert correspondence.
Inhalt	We study linear differential equations from an algebraic perspective, introducing differential rings, fields and differential modules (so-called Picard-Vessiot theory), and very soon the Galois group of a differential equation. We relate then the algebraic theory with the analytic theory, which leads us to the classical Riemann-Hilbert correspondence. In particular we will prove that differential equations on the complex projective line ℙ¹ with regular singularities in a finite set S correspond to representations of the fundamental group of ℙ¹∖S. If time permits, we have a look at differential equations in positive characteristic.
Literatur	M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. Vol 328, Springer 2003
401-3203-67L	Small Cancellation Theory	W	4 KP	2V	D. Gruber
Kurzbeschreibung	Small cancellation theory studies groups given by presentations in which defining relations have small common subwords. By translating group theoretic questions into geometric objects and applying concepts of negative curvature, it produces a variety of theorems on infinite groups. We will give an introduction to the theory, discuss important results, and touch on more recent developments.
Lernziel	Familiarity with the fundamental methods of small cancellation theory and its main applications; ability to apply the methods to create new examples of infinite groups with prescribed properties; basic understanding of connections with Gromov hyperbolicity.
Inhalt	We plan to cover a selection (depending on time) of the following topics: - Methods of classical small cancellation theory (e.g. van Kampen diagrams, van Kampen's lemma, Greendlinger's lemma) - Fundamental properties of small cancellation groups (e.g. Torsion Theorem, asphericity, linear/quadratic Dehn function) - Connections with algorithmic decision problems in groups (e.g. Dehn's algorithm for solving the word problem in surface groups, solvability of word and conjugacy problems in small cancellation groups) - Easy examples of small cancellation monsters (e.g. Pride's example, Rips construction) - Graphical generalization of small cancellation theory and applications (e.g. groups with expander graphs embedded in their Cayley graphs) - Connections with Gromov hyperbolicity
Literatur	V. Guirardel, Geometric small cancellation. Geometric group theory, 55-90, IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, RI, 2014. R. C. Lyndon, P. E. Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ISBN: 3-540-41158-5. A. Yu. Olshanskii, Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. ISBN: 0-7923-1394-1. R. Strebel, Appendix. Small cancellation groups. In: Sur les groupes hyperbolic d'après Mikhael Gromov (Bern, 1988), 227-273, Progr. Math. 83, Birkhäuser Boston, Boston, MA, 1990.
Voraussetzungen / Besonderes	Familiarity with very basic notions of group theory, definitions of free groups, group presentations, and graphs.
401-3177-67L	Introduction to Vertex Operator Algebras	W	4 KP	2V	C. A. Keller
Kurzbeschreibung	A first introduction to the theory of vertex operator algebras.
Lernziel	Understand the basic concepts of vertex operator algebras and their most important examples.
Inhalt	Tentative plan: 1) Formal power series, local fields 2) Vertex Algebras 3) Conformal symmetry 4) Vertex Operator Algebras 5) Correlation functions 6) VOAs from lattices 7) Connection to modular forms: Zhu's Theorem 8) Connection to Monstrous Moonshine
Literatur	Victor Kac: Vertex Algebras for Beginners James Lepowksy, Haisheng Li: Introduction to Vertex Operator Algebras and Their Representations
Voraussetzungen / Besonderes	Basic algebra and linear algebra. Some background in quantum mechanics is helpful, but not necessary.
401-3059-00L	Kombinatorik II	W	4 KP	2G	N. Hungerbühler
Kurzbeschreibung	Der Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende Kombinatorik.
Lernziel	Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
Inhalt	Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.

Auswahl: Geometrie
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3375-67L	Homogeneous Dynamics and Applications	W	8 KP	4G	M. Einsiedler, M. Akka Ginosar, Ç. Sert
Kurzbeschreibung	The aim is to reach a few of the applications of homogeneous dynamics to number theory, e.g. counting results concerning quadratic forms, but also develop the theory from scratch. The first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume.
Lernziel	The aim is to reach a few of the applications of homogeneous dynamics to number theory, e.g. counting results concerning quadratic forms, but also develop the theory from scratch. The first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume.
Inhalt	The first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume. Some of the aims of the course are: -) Pointwise ergodic theorem for a certain class of amenable groups -) Dynamics on hyperbolic surfaces, equidistribution of periodic horocycle orbits -) Applications to counting -) Some cases of Ratner theorems in Unipotent dynamics Course website: https://metaphor.ethz.ch/x/2017/hs/401-3375-67L/

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