Suchergebnis: Katalogdaten im Herbstsemester 2017
Mathematik Bachelor | ||||||
Basisjahr | ||||||
» Obligatorische Fächer des Basisjahres | ||||||
» Ergänzende Fächer | ||||||
» GESS Wissenschaft im Kontext | ||||||
Obligatorische Fächer des Basisjahres | ||||||
Basisprüfungsblock 1 | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
---|---|---|---|---|---|---|
401-1151-00L | Lineare Algebra I | O | 7 KP | 4V + 2U | M. Akveld | |
Kurzbeschreibung | Einführung in die Theorie der Vektorräume für Studierende der Mathematik und der Physik: Grundlagen, Vektorräume, lineare Abbildungen, Lösungen linearer Gleichungen und Matrizen, Determinanten, Endomorphismen, Eigenwerte und Eigenvektoren. | |||||
Lernziel | - Beherrschung der Grundkonzepte der Linearen Algebra - Einführung ins mathematische Arbeiten | |||||
Inhalt | - Grundlagen - Vektorräume und lineare Abbildungen - Lineare Gleichungssysteme und Matrizen - Determinanten - Endomorphismen und Eigenwerte | |||||
Literatur | - H. Schichl und R. Steinbauer: Einführung in das mathematische Arbeiten. Springer-Verlag 2012. Siehe: Link - G. Fischer: Lineare Algebra. Springer-Verlag 2014. Siehe: Link - K. Jänich: Lineare Algebra. Springer-Verlag 2004. Siehe: Link - S. H. Friedberg, A. J. Insel und L. E. Spence: Linear Algebra. Pearson 2003. Link - R. Pink: Lineare Algebra I und II. Skript. Siehe: Link | |||||
402-1701-00L | Physik I | O | 7 KP | 4V + 2U | A. Wallraff | |
Kurzbeschreibung | Diese Vorlesung stellt eine erste Einführung in die Physik dar und behandelt Themen der klassischen Mechanik. | |||||
Lernziel | Aneignung von Kenntnissen der physikalischen Grundlagen in der klassischen Mechanik. Fertigkeiten im Lösen von physikalischen Fragen anhand von Übungsaufgaben. | |||||
252-0847-00L | Informatik | O | 5 KP | 2V + 2U | B. Gärtner | |
Kurzbeschreibung | Die Vorlesung gibt eine Einführung in das Programmieren anhand der Sprache C++. Wir behandeln fundamentale Typen, Kontrollanweisungen, Funktionen, Felder und Klassen. Die Konzepte werden dabei jeweils durch Algorithmen und Anwendungen motiviert und illustriert. | |||||
Lernziel | Das Ziel der Vorlesung ist eine algorithmisch orientierte Einführung ins Programmieren. | |||||
Inhalt | Die Vorlesung gibt eine Einführung in das Programmieren anhand der Sprache C++. Wir behandeln fundamentale Typen, Kontrollanweisungen, Funktionen, Felder und Klassen. Die Konzepte werden dabei jeweils durch Algorithmen und Anwendungen motiviert und illustriert. | |||||
Skript | Ein Skript in englischer Sprache sowie Handouts in deutscher Sprache werden semesterbegleitend elektronisch herausgegeben. | |||||
Literatur | Andrew Koenig and Barbara E. Moo: Accelerated C++, Addison-Wesley, 2000. Stanley B. Lippman: C++ Primer, 3. Auflage, Addison-Wesley, 1998. Bjarne Stroustrup: The C++ Programming Language, 3. Auflage, Addison-Wesley, 1997. Doina Logofatu: Algorithmen und Problemlösungen mit C++, Vieweg, 2006. Walter Savitch: Problem Solving with C++, Eighth Edition, Pearson, 2012 | |||||
Basisprüfungsblock 2 | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-1261-07L | Analysis I | O | 10 KP | 6V + 3U | M. Einsiedler | |
Kurzbeschreibung | Einführung in die Differential- und Integralrechnung in einer reellen Veränderlichen: Grundbegriffe des mathematischen Denkens, Zahlen, Folgen und Reihen, topologische Grundbegriffe, stetige Funktionen, differenzierbare Funktionen, gewöhnliche Differentialgleichungen, Riemannsche Integration. | |||||
Lernziel | Mathematisch exakter Umgang mit Grundbegriffen der Differential-und Integralrechnung. | |||||
Literatur | H. Amann, J. Escher: Analysis I Link J. Appell: Analysis in Beispielen und Gegenbeispielen Link R. Courant: Vorlesungen über Differential- und Integralrechnung Link O. Forster: Analysis 1 Link H. Heuser: Lehrbuch der Analysis Link K. Königsberger: Analysis 1 Link W. Walter: Analysis 1 Link V. Zorich: Mathematical Analysis I (englisch) Link A. Beutelspacher: "Das ist o.B.d.A. trivial" Link H. Schichl, R. Steinbauer: Einführung in das mathematische Arbeiten Link | |||||
Obligatorische Fächer | ||||||
Prüfungsblock I Im Prüfungsblock I muss entweder die Lerneinheit 402-2883-00L Physik III oder die Lerneinheit 402-2203-01L Allgemeine Mechanik gewählt und zur Prüfung angemeldet werden. (Die andere der beiden Lerneinheiten kann im ETH Bachelor-Studiengang Mathematik belegt, aber weder in myStudies zur Prüfung angemeldet noch für den Studiengang angerechnet werden.) | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-2303-00L | Funktionentheorie | O | 6 KP | 3V + 2U | R. Pandharipande | |
Kurzbeschreibung | Komplexe Funktionen einer komplexen Veränderlichen, Cauchy-Riemann Gleichungen, Cauchyscher Integralsatz, Singularitäten, Residuensatz, Umlaufzahl, analytische Fortsetzung, spezielle Funktionen, konforme Abbildungen. Riemannscher Abbildungssatz. | |||||
Lernziel | Fähigkeit zum Umgang mit analytischen Funktion; insbesondre Anwendungen des Residuensatzes | |||||
Literatur | Th. Gamelin: Complex Analysis. Springer 2001 E. Titchmarsh: The Theory of Functions. Oxford University Press D. Salamon: "Funktionentheorie". Birkhauser, 2011. (In German) L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. K.Jaenich: Funktionentheorie. Springer Verlag R.Remmert: Funktionentheorie I. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publications | |||||
401-2333-00L | Methoden der mathematischen Physik I | O | 6 KP | 3V + 2U | H. Knörrer | |
Kurzbeschreibung | Fourierreihen. Lineare partielle Differentialgleichungen der mathematischen Physik. Fouriertransformation. Spezielle Funktionen und Eigenfunktionenentwicklungen. Distributionen. Ausgewählte Probleme aus der Quantenmechanik. | |||||
Lernziel | ||||||
Voraussetzungen / Besonderes | Die Einschreibung in die Übungsgruppen erfolgt online. Melden Sie sich im Laufe der ersten Semesterwoche unter echo.ethz.ch mit Ihrem ETH Account an. Der Übungsbetrieb beginnt in der zweiten Semesterwoche. | |||||
402-2883-00L | Physics III | W | 7 KP | 4V + 2U | J. Home | |
Kurzbeschreibung | Einführung in das Gebiet der Quanten- und Atomphysik und in die Grundlagen der Optik und statistischen Physik. | |||||
Lernziel | Grundlegende Kenntnisse in Quanten- und Atomphysik und zudem in Optik und statistischer Physik werden erarbeitet. Die Fähigkeit zur eigenständigen Lösung einfacher Problemstellungen aus den behandelten Themengebieten wird erreicht. Besonderer Wert wird auf das Verständnis experimenteller Methoden zur Beobachtung der behandelten physikalischen Phänomene gelegt. | |||||
Inhalt | Einführung in die Quantenphysik: Atome, Photonen, Photoelektrischer Effekt, Rutherford Streuung, Compton Streuung, de-Broglie Materiewellen. Quantenmechanik: Wellenfunktionen, Operatoren, Schrödinger-Gleichung, Potentialtopf, harmonischer Oszillator, Wasserstoffatom, Spin. Atomphysik: Zeeman-Effekt, Spin-Bahn Kopplung, Mehrelektronenatome, Röntgenspektren, Auswahlregeln, Absorption und Emission von Strahlung, LASER. Optik: Fermatsches Prinzip, Linsen, Abbildungssysteme, Beugung und Brechung, Interferenz, geometrische und Wellenoptik, Interferometer, Spektrometer. Statistische Physik: Wahrscheinlichkeitsverteilungen, Boltzmann-Verteilung, statistische Ensembles, Gleichverteilungssatz, Schwarzkörperstrahlung, Plancksches Strahlungsgesetz. | |||||
Skript | Im Rahmen der Veranstaltung wird ein Skript in elektronischer Form zur Verfügung gestellt. | |||||
Literatur | Quantenmechanik/Atomphysik/Moleküle: "Atom- und Quantenphysik", H. Haken and H. C. Wolf, ISBN 978-3540026211 Optik: "Optik", E. Hecht, ISBN 978-3486588613 Statistische Mechanik: "Statistical Physics", F. Mandl ISBN 0-471-91532-7 | |||||
402-2203-01L | Allgemeine Mechanik | W | 7 KP | 4V + 2U | N. Beisert | |
Kurzbeschreibung | Begriffliche und methodische Einführung in die theoretische Physik: Newtonsche Mechanik, Zentralkraftproblem, Schwingungen, Lagrangesche Mechanik, Symmetrien und Erhaltungssätze, Kreisel, relativistische Raum-Zeit-Struktur, Teilchen im elektromagnetischen Feld, Hamiltonsche Mechanik, kanonische Transformationen, integrable Systeme, Hamilton-Jacobi-Gleichung. | |||||
Lernziel | ||||||
252-0851-00L | Algorithmen und Komplexität | O | 4 KP | 2V + 1U | A. Steger | |
Kurzbeschreibung | Einführung: RAM-Maschine, Datenstrukturen; Algorithmen: Sortieren, Medianbest., Matrixmultiplikation, kürzeste Pfade, min. spann. Bäume; Paradigmen: Divide&Conquer, dynam. Programmierung, Greedy; Datenstrukturen: Suchbäume, Wörterbücher, Priority Queues; Komplexitätstheorie: Klassen P und NP, NP-vollständig, Satz von Cook, Beispiele für Reduktionen. | |||||
Lernziel | Nach dieser Vorlesung kennen die Studierenden einige Algorithmen und übliche Werkzeuge. Sie kennen die Grundlagen der Komplexitätstheorie und können diese verwenden um Probleme zu klassifizieren. | |||||
Inhalt | Die Vorlesung behandelt den Entwurf und die Analyse von Algorithmen und Datenstrukturen. Die zentralen Themengebiete sind: Sortieralgorithmen, Effiziente Datenstrukturen, Algorithmen für Graphen und Netzwerke, Paradigmen des Algorithmenentwurfs, Klassen P und NP, NP-Vollständigkeit, Approximationsalgorithmen. | |||||
Skript | Ja. Wird zu Beginn des Semesters verteilt. | |||||
Prüfungsblock II | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-2003-00L | Algebra I | O | 7 KP | 4V + 2U | E. Kowalski | |
Kurzbeschreibung | Einführung in die grundlegenden Begriffe und Resultate der Gruppentheorie, der Ringtheorie und der Körpertheorie. | |||||
Lernziel | Einführung in grundlegende Begriffe und Resultate aus der Theorie der Gruppen, der Ringe und der Körper. | |||||
Inhalt | Gruppentheorie: grundlegende Begriffe und Beispiele von Gruppen; Untergruppen, Quotientengruppen und Homomorphismen, Sylow Theoreme, Gruppenwirkungen und Anwendungen Ringtheorie: grundlegende Begriffe und Beispiele von Ringen; Ringhomomorphismen, Ideale und Quotientenringe, Anwendungen Körpertheorie: grundlegende Begriffe und Beispiele von Körpern; endliche Körper, Anwendungen | |||||
Literatur | J. Rotman, "Advanced modern algebra, 3rd edition, part 1" Link J.F. Humphreys: A Course in Group Theory (Oxford University Press) G. Smith and O. Tabachnikova: Topics in Group Theory (Springer-Verlag) M. Artin: Algebra (Birkhaeuser Verlag) R. Lidl and H. Niederreiter: Introduction to Finite Fields and their Applications (Cambridge University Press) B.L. van der Waerden: Algebra I & II (Springer Verlag) | |||||
Kernfächer | ||||||
Kernfächer aus Bereichen der reinen Mathematik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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. | W | 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", Link | |||||
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. | W | 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). | |||||
401-3001-61L | 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 Link | |||||
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: Link 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. | |||||
401-3132-00L | 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 reinen Mathematik (Mathematik Master) | ||||||
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: Link | 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 | |||||
401-3601-00L | 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. | W | 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 | |||||
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 | ||||||
401-3901-00L | 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. | |||||
252-0057-00L | Theoretische Informatik Hinweis: Studierende, die das Fach 252-0065-00L Theoretische Informatik (8 KP) bereits abgeschlossen haben, können die LE 252-0057-00L Theoretische Informatik (7 KP) nicht anrechnen lassen. | W | 7 KP | 4V + 2U | J. Hromkovic | |
Kurzbeschreibung | Konzepte zur Beantwortung grundlegender Fragen wie: a) Was ist völlig automatisiert machbar (algorithmisch lösbar) b) Wie kann man die Schwierigkeit von Aufgaben (Problemen) messen? c) Was ist Zufall und wie kann er nützlich sein? d) Was ist Nichtdeterminisus und welche Rolle spielt er in der Informatik? e) Wie kann man unendliche Objekte durch Automaten und Grammatiken endlich darstellen? | |||||
Lernziel | Vermittlung der grundlegenden Konzepte der Informatik in ihrer geschichtlichen Entwicklung | |||||
Inhalt | Die Veranstaltung ist eine Einführung in die Theoretische Informatik, die die grundlegenden Konzepte und Methoden der Informatik in ihrem geschichtlichen Zusammenhang vorstellt. Wir präsentieren Informatik als eine interdisziplinäre Wissenschaft, die auf einer Seite die Grenzen zwischen Möglichem und Unmöglichem und die quantitativen Gesetze der Informationsverarbeitung erforscht und auf der anderen Seite Systeme entwirft, analysiert, verifiziert und implementiert. Die Hauptthemen der Vorlesung sind: - Alphabete, Wörter, Sprachen, Messung der Informationsgehalte von Wörtern, Darstellung von algorithmischen Aufgaben - endliche Automaten, reguläre und kontextfreie Grammatiken - Turingmaschinen und Berechenbarkeit - Komplexitätstheorie und NP-Vollständigkeit - Algorithmenentwurf für schwere Probleme | |||||
Skript | Die Vorlesung ist detailliert durch das Lehrbuch "Theoretische Informatik" bedeckt. | |||||
Literatur | Basisliteratur: 1. J. Hromkovic: Theoretische Informatik. 5. Auflage, Springer Vieweg 2014. 2. J. Hromkovic: Theoretical Computer Science. Springer 2004. Weiterführende Literatur: 3. M. Sipser: Introduction to the Theory of Computation, PWS Publ. Comp.1997 4. J.E. Hopcroft, R. Motwani, J.D. Ullman: Einführung in die Automatentheorie, Formale Sprachen und Komplexitätstheorie. Pearson 2002. 5. I. Wegener: Theoretische Informatik. Teubner Weitere Übungen und Beispiele: 6. A. Asteroth, Ch. Baier: Theoretische Informatik | |||||
Voraussetzungen / Besonderes | Während des Semesters werden zwei freiwillige Probeklausuren gestellt. | |||||
252-0209-00L | Algorithms, Probability, and Computing | W | 8 KP | 4V + 2U + 1A | E. Welzl, M. Ghaffari, A. Steger, D. Steurer, P. Widmayer | |
Kurzbeschreibung | Advanced design and analysis methods for algorithms and data structures: Random(ized) Search Trees, Point Location, Minimum Cut, Linear Programming, Randomized Algebraic Algorithms (matchings), Probabilistically Checkable Proofs (introduction). | |||||
Lernziel | Studying and understanding of fundamental advanced concepts in algorithms, data structures and complexity theory. | |||||
Skript | Will be handed out. | |||||
Literatur | Introduction to Algorithms by T. H. Cormen, C. E. Leiserson, R. L. Rivest; Randomized Algorithms by R. Motwani und P. Raghavan; Computational Geometry - Algorithms and Applications by M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf. | |||||
» Kernfächer aus Bereichen der angewandten Mathematik ... (Mathematik Master) | ||||||
Kernfächer aus weiteren anwendungsorientierten Gebieten 402-0205-00L Quantenmechanik I ist als angewandtes Kernfach anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik (letztmals im FS 2016 angeboten) nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link). | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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 | ||||||
Auswahl: Algebra, Topologie, diskrete Mathematik, Logik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
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. | |||||
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) Link | |||||
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. | |||||
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: Link | |||||
401-3301-67L | Introduction to Hyperbolic Geometry | W | 4 KP | 2V | Q. Chen | |
Kurzbeschreibung | Hyperbolic geometry and ideal tetrahedra, decomposition of the Figure-8 knot, The Rigidity Theorem (Compact Case), hyperbolic structures on knot complements, the Space of Hyperbolic Manifolds and the Volume Function | |||||
Lernziel | ||||||
Literatur | "Low dimensional geometry: from euclidean surfaces to hyperbolic knots" by Bonahon "Hyperbolic Knot Theory" by Purcell "Lectures on Hyperbolic Geometry" by Benedetti and Petronio "The Geometry and Topology of Three-Manifolds" by Thurston | |||||
401-3057-00L | Endliche Geometrien II Findet dieses Semester nicht statt. | W | 4 KP | 2G | 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. | |||||
Auswahl: Analysis noch kein Angebot | ||||||
Auswahl: Numerische Mathematik Kein Angebot in diesem Semester. | ||||||
Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3627-00L | High-Dimensional Statistics Findet dieses Semester nicht statt. | W | 4 KP | 2V | P. L. Bühlmann | |
Kurzbeschreibung | "High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed. | |||||
Lernziel | Knowledge of methods and basic theory for high-dimensional statistical inference | |||||
Inhalt | Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling | |||||
Literatur | Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. ISBN 978-3-642-20191-2. | |||||
Voraussetzungen / Besonderes | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||
401-4623-00L | Time Series Analysis Findet dieses Semester nicht statt. | W | 6 KP | 3G | keine Angaben | |
Kurzbeschreibung | Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R. | |||||
Lernziel | Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R. | |||||
Inhalt | This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations. Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations, spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R. | |||||
Skript | Not available | |||||
Literatur | A list of references will be distributed during the course. | |||||
Voraussetzungen / Besonderes | Basic knowledge in probability and statistics | |||||
401-0625-01L | Applied Analysis of Variance and Experimental Design | W | 5 KP | 2V + 1U | L. Meier | |
Kurzbeschreibung | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||
Lernziel | Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R. | |||||
Inhalt | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||
Literatur | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||
Voraussetzungen / Besonderes | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held. | |||||
401-0649-00L | Applied Statistical Regression | W | 5 KP | 2V + 1U | M. Dettling | |
Kurzbeschreibung | This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis. | |||||
Lernziel | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||
Inhalt | The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data. | |||||
Skript | A script will be available. | |||||
Literatur | Faraway (2005): Linear Models with R Faraway (2006): Extending the Linear Model with R Draper & Smith (1998): Applied Regression Analysis Fox (2008): Applied Regression Analysis and GLMs Montgomery et al. (2006): Introduction to Linear Regression Analysis | |||||
Voraussetzungen / Besonderes | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held. In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Regression" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit. | |||||
401-3628-14L | Bayesian Statistics | W | 4 KP | 2V | F. Sigrist | |
Kurzbeschreibung | Introduction to the Bayesian approach to statistics: Decision theory, prior distributions, hierarchical Bayes models, Bayesian tests and model selection, empirical Bayes, computational methods, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods. | |||||
Lernziel | Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis. | |||||
Inhalt | Topics that we will discuss are: Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, Jeffreys priors), tests and model selection (Bayes factors, hyper-g priors in regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods) | |||||
Skript | A script will be available in English. | |||||
Literatur | Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007. A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013). Additional references will be given in the course. | |||||
Voraussetzungen / Besonderes | Familiarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||
401-4637-67L | On Hypothesis Testing | W | 4 KP | 2V | F. Balabdaoui | |
Kurzbeschreibung | This course is a review of the main results in decision theory. | |||||
Lernziel | The goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class). | |||||
Literatur | - Statistical Inference (Casella & Berger) - Testing Statistical Hypotheses (Lehmann and Romano) | |||||
Auswahl: Finanz- und Versicherungsmathematik Im Bachelor-Studiengang Mathematik ist auch 401-3913-01L Mathematical Foundations for Finance als Wahlfach anrechenbar, aber nur unter der Bedingung, dass 401-3888-00L Introduction to Mathematical Finance nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link). | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3922-00L | Life Insurance Mathematics | W | 4 KP | 2V | M. Koller | |
Kurzbeschreibung | The classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated. | |||||
Lernziel | ||||||
401-3925-00L | Non-Life Insurance: Mathematics and Statistics | W | 8 KP | 4V + 1U | M. V. Wüthrich | |
Kurzbeschreibung | The lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial sciences. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models, credibility theory, claims reserving and solvency. | |||||
Lernziel | The student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations. | |||||
Inhalt | The following topics are treated: Collective Risk Modeling Individual Claim Size Modeling Approximations for Compound Distributions Ruin Theory in Discrete Time Premium Calculation Principles Tariffication and Generalized Linear Models Bayesian Models and Credibility Theory Claims Reserving Solvency Considerations | |||||
Skript | M. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics Link | |||||
Voraussetzungen / Besonderes | The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under Link. Prerequisites: knowledge of probability theory, statistics and applied stochastic processes. | |||||
401-3927-00L | Mathematical Modelling in Life Insurance | W | 4 KP | 2V | T. J. Peter | |
Kurzbeschreibung | The course covers various mathematical models that are used in life insurance. | |||||
Lernziel | The course's objective is to present various mathematical models that are used in life insurance for valuation or risk management purposes. | |||||
Inhalt | Following main topics are covered: 1. Guarantees & options in life insurance 2. Financial modeling 3. Valuation of life insurance contracts: Unit linked and participating contracts 4. Mortality modeling | |||||
Voraussetzungen / Besonderes | The exams ONLY take place during the official ETH examination period. The course counts towards the diploma of "Aktuar SAV". Basic knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful. | |||||
401-3928-00L | Reinsurance Analytics | W | 4 KP | 2V | P. Antal, P. Arbenz | |
Kurzbeschreibung | History of reinsurance and catastrophic events. Forms of reinsurance. Modelling of reinsurance losses through frequency severity models. Rating/Pricing of reinsurance contracts. Modelling of natural catastrophes. Reinsurance markets and companies. Risk profile and solvency implications of reinsurance. Solvency 2 modelling. Alternatives to reinsurance such as Cat Bonds. | |||||
Lernziel | Understand the following aspects: History of reinsurance. Role of reinsurance in society and history of catastrophic events. Forms of reinsurance (proportional and nonproportional). Covered types of business (property, casualty, specialties, life, health). Modelling of reinsurance losses through frequency severity models (typical distributions and parameters). Rating/Pricing of reinsurance contracts (experience and exposure). Modelling of natural catastrophes (methodological approaches and techniques). Natural catastrophes in Switzerland (importance, insurance, reinsurance). Reinsurance markets and companies. Risk profile implications of reinsurance (Catastrophe risk, reserving risk, Credit risk, basis risk, etc). Solvency implications of reinsurance (primary insurance and reinsurance side). Solvency 2 modelling (standard models, internal models, FINMA StandRe). Alternatives to reinsurance (insurance linked securities, subordinate debt). Trigger types of cat bonds (indemnity, modeled loss, industry loss, parametric) | |||||
Inhalt | History of reinsurance. Role of reinsurance in society and history of catastrophic events. Forms of reinsurance (proportional and nonproportional). Covered types of business (property, casualty, specialties, life, health). Modelling of reinsurance losses through frequency severity models (typical distributions and parameters). Rating/Pricing of reinsurance contracts (experience and exposure). Modelling of natural catastrophes (methodological approaches and techniques). Natural catastrophes in Switzerland (importance, insurance, reinsurance). Reinsurance markets and companies. Risk profile implications of reinsurance (Catastrophe risk, reserving risk, Credit risk, basis risk, etc). Solvency implications of reinsurance (primary insurance and reinsurance side). Solvency 2 modelling (standard models, internal models, FINMA StandRe). Alternatives to reinsurance (insurance linked securities, subordinate debt). Trigger types of cat bonds (indemnity, modeled loss, industry loss, parametric) | |||||
Skript | Slides, lecture notes, and references to literature will be made available. | |||||
401-4935-67L | Mean Field Games | W | 4 KP | 2V | M. Burzoni | |
Kurzbeschreibung | The analysis of differential games with a large number of players finds applications in various research fields, from physics to economics and finance. The aim of Mean Field Games theory is to provide a suitable approximation of such problems with a higher tractability. | |||||
Lernziel | This course aims to give a broad understanding of the basic ideas of Mean Field Games, the main mathematical tools and the possible applications. | |||||
Inhalt | We first present and analyze toy models of Mean Field Games in order to familiarize with the subject and to understand what kind of problems can be solved with this theory. We recall some basic principles of optimal control theory and stochastic differential equations. We explore two different approaches to Mean Field Games. From an analytic point of view it consists of a coupled system of PDEs. From a probabilistic point of view it amounts to a particular type of stochastic differential equations. We will concentrate, in particular, in the probabilistic approach. | |||||
Literatur | 1) Notes on Mean Field Games. P. Cardaliaguet 2) Mean Field Games. J.M. Lasry, P.L. Lions 3) Probabilistic theory of Mean Field Games and applications. R. Carmona, F. Delarue | |||||
Voraussetzungen / Besonderes | Basic courses in analysis including basic knowledge of ordinary/partial differential equations. Basic knowledge of stochastic analysis including Brownian Motion and stochastic differential equations. | |||||
Auswahl: Mathematische Physik, Theoretische Physik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3833-65L | Chaotically Singular Spacetimes Findet dieses Semester nicht statt. | W | 6 KP | 3V | H. Knörrer | |
Kurzbeschreibung | One might have, more provacatively, entitled the course: How does time end (in, Einstein's general relativity)? In a word, badly. Not in a whimper, nor in a crunch, but in something much more exotic. | |||||
Lernziel | ||||||
Inhalt | One might have, more provacatively, entitled the course: How does time end (in, Einstein's general relativity)? In a word, badly. Not in a whimper, nor in a crunch, but in something much more exotic. More, technically, what does a generic singular point, restricting time, in solutions to the Einstein gravitational field equations look like? Special cosmological solutions, such as Freedman's, do have singularities. In 1963, Lifshitz and Khalatnikov 'constructed a class' of singular solutions and concluded that '... the presence of a singularity in time is NOT a necessary property of cosmological models of the general theory of relativity, and that the general case of an arbitrary distribution of matter and gravitational field does not lead to the appearance of a singularity.' In 1965 Penrose and Hawking formulated and proved 'incompleteness' theorems that convinced even Lifshitz and Khalatnikov that singularities in time ARE a necessary property of cosmological models of the general theory of relativity. Penrose and Hawking proved, that under very general, physically reasonable conditions, a spacetime (that is, a solution to the Einstein equations) has a light ray (null geodesic) that suddenly ends ('incompleteness') sufficiently far in the past. They adroitly sidestep the problem of defining what a singularity acturally is, by saying it is the 'place' where their light rays end. The proofs of incompleteness theorems are not hard. That's good. Unfortunately, they are by their very nature completely non constructive and provide no quantitative information at all about what a 'singularity' really looks like. In 1970, Belinskii, Khalatnikov and Lifshitz revisited the work of 1963 and found that Khalatnikov and Lifshitz had missed something and that '... we shall show that there exists a general solution which exhibits a physical singularity with respect to time.' In 1982 they revised the 1970 proposal. Their work culminates in a series of fascinating, but very, very heuristic, statements about the possible existence of a class of singular solutions to the field equations. These heuristic statements are referred to as the 'BKL Conjectures'. Next semester, we will rigorously formulate and prove the 'BKL Conjectures' for homogeneous spacetimes. That is, we will construct a set of initial data with positive measure which evolve into homogeneous, chaotically singular spacetimes that exhibit all of the BKL phenomenology. Most importantly, there are chaotic oscillations, growing in magnitude, whose distribution is governed by the continued fraction expansion of a parameter appearing in the initial data. The lectures will be completely self contained. One doesn't need to know anything about general relativity; the Einstein field equations will be introduced from scratch. We will classify real, three dimensional Lie algebras, introduce tensor analysis and discuss the geometry of homogeneneous spacetimes. We will also derive the basic properties of continued fractions and the Gauss map $\displaystyle x \mapsto \frac 1x - \Bigl\lfloor \frac 1x \Bigr\rfloor$ from $(0,1) \smallsetminus \mathbb Q$ to itself. | |||||
Skript | There will be lecture notes. | |||||
Voraussetzungen / Besonderes | First year analysis and linear algebra are the only prerequisites. | |||||
402-0830-00L | General Relativity | W | 10 KP | 4V + 2U | G. M. Graf | |
Kurzbeschreibung | Manifold, Riemannian metric, connection, curvature; Special Relativity; Lorentzian metric; Equivalence principle; Tidal force and spacetime curvature; Energy-momentum tensor, field equations, Newtonian limit; Post-Newtonian approximation; Schwarzschild solution; Mercury's perihelion precession, light deflection. | |||||
Lernziel | Basic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts. | |||||
Literatur | Suggested textbooks: C. Misner, K, Thorne and J. Wheeler: Gravitation S. Carroll - Spacetime and Geometry: An Introduction to General Relativity R. Wald - General Relativity S. Weinberg - Gravitation and Cosmology N. Straumann - General Relativity with applications to Astrophysics | |||||
Auswahl: Mathematische Optimierung, Diskrete Mathematik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3055-64L | Algebraic Methods in Combinatorics | W | 6 KP | 2V + 1U | B. Sudakov | |
Kurzbeschreibung | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. | |||||
Lernziel | ||||||
Inhalt | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools. One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to): Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at Link | |||||
Auswahl: Theoretische Informatik | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
252-1407-00L | Algorithmic Game Theory | W | 7 KP | 3V + 2U + 1A | P. Penna | |
Kurzbeschreibung | Game theory provides a formal model to study the behavior and interaction of self-interested users and programs in large-scale distributed computer systems without central control. The course discusses algorithmic aspects of game theory. | |||||
Lernziel | Learning the basic concepts of game theory and mechanism design, acquiring the computational paradigm of self-interested agents, and using these concepts in the computational and algorithmic setting. | |||||
Inhalt | The Internet is a typical example of a large-scale distributed computer system without central control, with users that are typically only interested in their own good. For instance, they are interested in getting high bandwidth for themselves, but don't care about others, and the same is true for computational load or download rates. Game theory provides a particularly well-suited model for the behavior and interaction of such selfish users and programs. Classic game theory dates back to the 1930s and typically does not consider algorithmic aspects at all. Only a few years back, algorithms and game theory have been considered together, in an attempt to reconcile selfish behavior of independent agents with the common good. This course discusses algorithmic aspects of game-theoretic models, with a focus on recent algorithmic and mathematical developments. Rather than giving an overview of such developments, the course aims to study selected important topics in depth. Outline: - Introduction to classic game-theoretic concepts. - Existence of stable solutions (equilibria), algorithms for computing equilibria, computational complexity. - Speed of convergence of natural game playing dynamics such as best-response dynamics or regret minimization. - Techniques for bounding the quality-loss due to selfish behavior versus optimal outcomes under central control (a.k.a. the 'Price of Anarchy'). - Design and analysis of mechanisms that induce truthful behavior or near-optimal outcomes at equilibrium. - Selected current research topics, such as Google's Sponsored Search Auction, the U.S. FCC Spectrum Auction, Kidney Exchange. | |||||
Skript | Lecture notes will be usually posted on the website shortly after each lecture. | |||||
Literatur | "Algorithmic Game Theory", edited by N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, Cambridge University Press, 2008; "Game Theory and Strategy", Philip D. Straffin, The Mathematical Association of America, 5th printing, 2004 Several copies of both books are available in the Computer Science library. | |||||
Voraussetzungen / Besonderes | Audience: Although this is a Computer Science course, we encourage the participation from all students who are interested in this topic. Requirements: You should enjoy precise mathematical reasoning. You need to have passed a course on algorithms and complexity. No knowledge of game theory is required. | |||||
252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 8 KP | 3V + 2U + 2A | A. Steger, E. Welzl | |
Kurzbeschreibung | Las Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks | |||||
Lernziel | After this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas. | |||||
Inhalt | Randomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas. | |||||
Skript | Yes. | |||||
Literatur | - Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995) - Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005) | |||||
252-1425-00L | Geometry: Combinatorics and Algorithms | W | 6 KP | 2V + 2U + 1A | E. Welzl, L. F. Barba Flores, M. Hoffmann, A. Pilz | |
Kurzbeschreibung | Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) | |||||
Lernziel | The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. | |||||
Inhalt | Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations. | |||||
Skript | yes | |||||
Literatur | Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. | |||||
Voraussetzungen / Besonderes | Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area. | |||||
263-4500-00L | Advanced Algorithms | W | 6 KP | 2V + 2U + 1A | M. Ghaffari | |
Kurzbeschreibung | This is an advanced course on the design and analysis of algorithms, covering a range of topics and techniques not studied in typical introductory courses on algorithms. | |||||
Lernziel | This course is intended to familiarize students with (some of) the main tools and techniques developed over the last 15-20 years in algorithm design, which are by now among the key ingredients used in developing efficient algorithms. | |||||
Inhalt | the lectures will cover a range of topics, including the following: graph sparsifications while preserving cuts or distances, various approximation algorithms techniques and concepts, metric embeddings and probabilistic tree embeddings, online algorithms, multiplicative weight updates, streaming algorithms, sketching algorithms, and a bried glance at MapReduce algorithms. | |||||
Voraussetzungen / Besonderes | This course is designed for masters and doctoral students and it especially targets those interested in theoretical computer science, but it should also be accessible to last-year bachelor students. Sufficient comfort with both (A) Algorithm Design & Analysis and (B) Probability & Concentrations. E.g., having passed the course Algorithms, Probability, and Computing (APC) is highly recommended, though not required formally. If you are not sure whether you're ready for this class or not, please consulte the instructor. | |||||
Auswahl: Weitere Gebiete | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3502-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 2 KP | 4A | Professor/innen | |
Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||
Lernziel | ||||||
401-3503-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 3 KP | 6A | Professor/innen | |
Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||
Lernziel | ||||||
401-3504-66L | Reading Course DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT. Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben: 1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten; 2) in welchem Semester; 3) für welchen Studiengang; 4) Ihr Name und Vorname; 5) Ihre Studierenden-Nummer; 6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses. | W | 4 KP | 9A | Professor/innen | |
Kurzbeschreibung | In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet. | |||||
Lernziel | ||||||
Kern- und Wahlfächer (Mathematik Master) | ||||||
» Kernfächer (Mathematik Master) | ||||||
» Wahlfächer (Mathematik Master) | ||||||
Seminare Bitte Seminare frühzeitig im myStudies belegen, damit wir einen allfälligen Bedarf an weiteren Seminaren rechtzeitig erkennen. Bei einigen Seminaren werden Wartelisten geführt. Belegen Sie trotzdem höchstens zwei Mathematik-Seminare. In diesem Fall bekunden Sie für das Seminar, das Sie zuerst belegen, eine höhere Präferenz. | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-3680-67L | Persistent Homology and Topological Data Analysis Maximale Teilnehmerzahl: 8 | W | 4 KP | 2S | P. S. Jossen | |
Kurzbeschreibung | We study the fundamental tools of topological data analysis: Persistent homology, persistence modules and barcodes. Our goal is to read and understand parts of the paper "Principal Component Analysis of Persistent Homology..." by Vanessa Robins and Kate Turner (ArXiV 1507.01454v1). | |||||
Lernziel | To get familiar with the basic concepts of topological data analysis and see some applications thereof. | |||||
Literatur | Herbert Edelsbrunner and John L. Harer: Computational Topology, An Introduction. AMS 2010 | |||||
Voraussetzungen / Besonderes | Participants are supposed to be familiar with singular homology. | |||||
401-3180-67L | Algebraic K-Theory Maximale Teilnehmerzahl: 26 | W | 4 KP | 2S | C. Busch | |
Kurzbeschreibung | The "algebraic K-theory" describes a branch of algebra which centers about two functors which assign to each associative ring R an abelian group. | |||||
Lernziel | We will introduce the functors K0 and K1 and consider the further development of K-theory. | |||||
Literatur | John Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press and University of Tokyo Press (1971). | |||||
Voraussetzungen / Besonderes | Basic knowledge of Algebra as taught in a course Algebra I + II. Every week two students will give a talk and deliver a summary containing the main results of their subject. The weekly attendance of the seminar is mandatory. | |||||
401-3370-67L | Seminar on Homogeneous Dynamics and Applications Maximale Teilnehmerzahl: 12 | W | 4 KP | 2S | M. Einsiedler, M. Akka Ginosar, Ç. Sert | |
Kurzbeschreibung | This seminar is offered to students taking the course Homogeneous Dynamics and Applications. It will give some more details and fill in some of the background of the material in the course. Exercises will also be an integral part of the seminar. | |||||
Lernziel | ||||||
Inhalt | Seminar website: Link | |||||
Voraussetzungen / Besonderes | The seminar is restricted to 12 students, registration will be finalised in the first week of the semester. | |||||
401-3650-67L | Numerical Analysis Seminar: Tensor Numerics and Deep Neural Networks Maximale Teilnehmerzahl: 10 | W | 4 KP | 2S | C. Schwab | |
Kurzbeschreibung | The seminar addresses recently discovered _mathematical_ connections between Deep Learning and Tensor-formatted numerical analysis, with particular attention to the numerical solution of partial differential equations, with random input data. | |||||
Lernziel | The aim of the seminar is to review recent [2015-] research work and results, together with recently published software such as the TT-Toolbox, and Google's TENSORFLOW. The focus is on the mathematical analysis and interpretation of current learning approaches and related mathematical and technical fields, e.g. high-dimensional approximation, tensor structured numerical methods for the numerical solution of highdimensional PDEs, with applications in computational UQ. For theory, we refer to the references in the survey Link Numerical experiments will be done with TENSORFLOW and with the TT toolbox at Link | |||||
Skript | The seminar will study a set of 10 orginial papers from 2015 to today. | |||||
Literatur | Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok, Philipp Petersen Optimal Approximation with Sparsely Connected Deep Neural Networks arXiv:1705.01714 N. Cohen, O. Sharir, Y. Levine, R. Tamari, D. Yakira and A. Shashua (May 2017): Analysis and design of convolutional networks via hierarchical tensor decompositions, arXiv:1705.02302v3. N. Cohen and A. Shashua (March 2016), Convolutional rectifier networks as generalized tensor decompositions, Technical report, arXiv:1603.00162. Proceedings of The 33rd International Conference on Machine Learning, pp. 955-963, 2016. N. Cohen, O. Sharir and A. Shashua (Sept. 2015), On the expressive power of deep learning: A tensor analysis, Technical report, arXiv:1509.05009. Journal-ref: 29th Annual Conference on Learning Theory, pp. 698-728, 2016. | |||||
Voraussetzungen / Besonderes | Completed BSc MATH exam. | |||||
401-3620-67L | Student Seminar in Statistics: Computer Age Statistical Inference Maximale Teilnehmerzahl: 24 Hauptsächlich für Studierende im Studiengang Mathematik Bachelor oder Master, welche zusätzlich zum Einführungskurs 401-2604-00L Wahrscheinlichkeit und Statistik / Probability and Statistics mindestens ein Kern- oder Wahlfach in Statistik besucht haben. | W | 4 KP | 2S | M. H. Maathuis, P. L. Bühlmann, N. Meinshausen, S. van de Geer | |
Kurzbeschreibung | We study selected chapters from the book "Computer Age Statistical Inference: Algorithms, Evidence and Data Science" by Bradley Efron and Trevor Hastie. | |||||
Lernziel | During this seminar, we will study roughly one chapter per week from the book "Computer Age Statistical Inference: Algorithms, Evidence and Data Science" by Bradley Efron and Trevor Hastie. You will obtain a good overview of the field of modern statistics. Moreover, you will practice your self-studying and presentation skills. | |||||
Inhalt | In the words of Efron and Hastie: "The twenty-first century has seen a breathtaking expansion of statistical methodology, both in scope and in influence. “Big data,” “data science,” and “machine learning” have become familiar terms in the news, as statistical methods are brought to bear upon the enormous data sets of modern science and commerce. How did we get here? And where are we going? This book takes us on a journey through the revolution in data analysis following the introduction of electronic computation in the 1950s. Beginning with classical inferential theories – Bayesian, frequentist, Fisherian – individual chapters take up a series of influential topics: survival analysis, logistic regression, empirical Bayes, the jackknife and bootstrap, random forests, neural networks, Markov chain Monte Carlo, inference after model selection, and dozens more. The book integrates methodology and algorithms with statistical inference, and ends with speculation on the future direction of statistics and data science." | |||||
Literatur | Bradley Efron and Trevor Hastie (2016). Computer Age Statistical Inference: Algorithms, Evidence and Data Science. Cambridge University Press, New York. ISBN: 9781107149892. | |||||
Voraussetzungen / Besonderes | We require at least one course in statistics in addition to the 4th semester course Introduction to Probability and Statistics, as well as some experience with the statistical software R. Topics will be assigned during the first meeting. | |||||
401-3920-67L | Optimal Stopping Maximale Teilnehmerzahl: 26 | W | 4 KP | 2S | P. Cheridito | |
Kurzbeschreibung | In this seminar different methods to solve optimal stopping problems are studied and various applications are discussed. | |||||
Lernziel | The goal is to learn different methods that can be used to solve optimal stopping problems in discrete and continuous time. | |||||
Inhalt | Methods of optimal stopping theory in both discrete and continuous time using both martingale and Markovian approaches. Various concrete problems from the theory of probability, mathematical statistics and mathematical finance that can be reformulated as problems of optimal stopping of stochastic processes. | |||||
Literatur | Optimal Stopping and Free-Boundary Problems. G. Peskir and A. Shiryaev. 2006 Springer. | |||||
Voraussetzungen / Besonderes | Probability theory, stochastic processes, martingales, Brownian motion, stochastic calculus | |||||
» Seminare (Mathematik Master) | ||||||
Ergänzende Fächer | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-1511-00L | Geometrie | W | 3 KP | 2V + 1U | T. Ilmanen | |
Kurzbeschreibung | Wir betrachten die Geometrie und Topologie 2 und 3-dimensionaler Raeume (Mannigfaltigkeiten) aus einem intuitiven Standpunkt. | |||||
Lernziel | -Wie ist es in einem nicht-euklidischen Raum (z.B. in einer Flaeche) zu leben? -Orientierung, Genus, Kruemmung -Klassifikation der geschlossenen orientierbaren Flaechen -Elliptische, euklidische, und hyperbolische Geometrie -3-Mannigfaltigkeiten aus dem Thurstonschen Standpunkt | |||||
Literatur | Jeffrey R. Weeks. The Shape of Space. Edwin A. Abbott. Flatland. 1884. | |||||
402-0351-00L | Astronomie | W | 2 KP | 2V | S. P. Quanz | |
Kurzbeschreibung | Ein Überblick über die wichtigsten Gebiete der heutigen Astronomie: Planeten, Sonne, Sterne, Milchstrasse, Galaxien und Kosmologie. | |||||
Lernziel | Einführung in die Astronomie mit einem Überblick über die wichtigsten Gebiete der heutigen Astronomie. Diese Vorlesung dient auch als Grundlage für die Astrophysikvorlesungen der höheren Semester. | |||||
Inhalt | Planeten, Sonne, Sterne, Milchstrasse, Galaxien und Kosmologie. | |||||
Skript | Kopien der Präsentationen werde zur Verfügung gestellt. | |||||
Literatur | Astronomie. Harry Nussbaumer, Hans Martin Schmid vdf Vorlesungsskripte (8. Auflage) Der Neue Kosmos. A. Unsöld, B. Baschek, Springer | |||||
Bachelor-Arbeit | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-2000-00L | Scientific Works in Mathematics Zielpublikum: Bachelor-Studierende im dritten Jahr; Master-Studierende, welche noch keine entsprechende Ausbildung vorweisen können. Obligatorisch für alle Bachelor- und Master-Studierenden mit Immatrikulation ab dem HS 2014. Weisung Link | O | 0 KP | E. Kowalski | ||
Kurzbeschreibung | Introduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.) | |||||
Lernziel | Learn the basic standards of scientific works in mathematics. | |||||
Inhalt | - Types of mathematical works - Publication standards in pure and applied mathematics - Data handling - Ethical issues - Citation guidelines | |||||
Skript | Moodle of the Mathematics Library: Link | |||||
Voraussetzungen / Besonderes | This course is completed by the optional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library. For more details see: Link | |||||
401-3990-10L | Bachelor-Arbeit Voraussetzung: erfolgreicher Abschluss der Lerneinheit 401-2000-00L Scientific Works in Mathematics Weitere Angaben unter Link | O | 8 KP | 11D | Betreuer/innen | |
Kurzbeschreibung | Die Bachelor-Arbeit dient der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Sie soll die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern. | |||||
Lernziel | ||||||
GESS Wissenschaft im Kontext | ||||||
Wissenschaft im Kontext | ||||||
» siehe Studiengang Wissenschaft im Kontext: Typ A: Förderung allgemeiner Reflexionsfähigkeiten | ||||||
» Empfehlungen aus dem Bereich Wissenschaft im Kontext (Typ B) für das D-MATH. | ||||||
Sprachkurse | ||||||
» siehe Studiengang Wissenschaft im Kontext: Sprachkurse ETH/UZH | ||||||
Zusätzliche Veranstaltungen | ||||||
Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |
401-5000-00L | Zurich Colloquium in Mathematics | E- | 0 KP | A. Iozzi, S. Mishra, R. Pandharipande, Uni-Dozierende | ||
Kurzbeschreibung | The lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians. | |||||
Lernziel | ||||||
401-5990-00L | Zurich Graduate Colloquium | E- | 0 KP | 1K | A. Iozzi, Uni-Dozierende | |
Kurzbeschreibung | The Graduate Colloquium is an informal seminar aimed at graduate students and postdocs whose purpose is to provide a forum for communicating one's interests and thoughts in mathematics. | |||||
Lernziel | ||||||
401-5960-00L | Kolloquium über Mathematik, Informatik und Unterricht Fachdidaktik für Mathematik- und Informatiklehrpersonen. | E- | 0 KP | N. Hungerbühler, M. Akveld, J. Hromkovic, H. Klemenz | ||
Kurzbeschreibung | Didaktikkolloquium | |||||
Lernziel | ||||||
402-0101-00L | The Zurich Physics Colloquium | E- | 0 KP | 1K | R. Renner, G. Aeppli, C. Anastasiou, N. Beisert, G. Blatter, S. Cantalupo, C. Degen, G. Dissertori, K. Ensslin, T. Esslinger, J. Faist, T. K. Gehrmann, G. M. Graf, R. Grange, J. Home, S. Huber, A. Imamoglu, P. Jetzer, S. Johnson, U. Keller, K. S. Kirch, S. Lilly, L. M. Mayer, J. Mesot, B. Moore, D. Pescia, A. Refregier, A. Rubbia, T. C. Schulthess, M. Sigrist, A. Vaterlaus, R. Wallny, A. Wallraff, W. Wegscheider, A. Zheludev, O. Zilberberg | |
Kurzbeschreibung | Research colloquium | |||||
Lernziel | ||||||
402-0800-00L | The Zurich Theoretical Physics Colloquium | E- | 0 KP | 1K | O. Zilberberg, C. Anastasiou, N. Beisert, G. Blatter, T. K. Gehrmann, G. M. Graf, S. Huber, P. Jetzer, L. M. Mayer, B. Moore, T. C. Schulthess, M. Sigrist, Uni-Dozierende | |
Kurzbeschreibung | Research colloquium | |||||
Lernziel | The Zurich Theoretical Physics Colloquium is jointly organized by the University of Zurich and ETH Zurich. Its mission is to bring both students and faculty with diverse interests in theoretical physics together. Leading experts explain the basic questions in their field of research and communicate the fascination for their work. | |||||
251-0100-00L | Kolloquium für Informatik | E- | 0 KP | 2K | Dozent/innen | |
Kurzbeschreibung | Eingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm. | |||||
Lernziel | Das Kolloquium des Departements Informatik bietet die Gelegenheit, international renommierte Wissenschaftler zu aktuellen Themen der Informatik zu hören. Die Veranstaltungsreihe ist öffentlich und Besucher sind sehr willkommen. Studierenden des Departements wird besonders empfohlen, am Kolloquium teilzunehmen. Die Vorträge umfassen auch Antritts- und Abschiedsvorlesungen der Professorinnen und Professoren des Departements. | |||||
Inhalt | Eingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm. |