Suchergebnis: Katalogdaten im Herbstsemester 2016

Mathematik Master Information
Kernfächer
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Kernfächer aus Bereichen der reinen Mathematik
NummerTitelTypECTSUmfangDozierende
401-3225-00LIntroduction to Lie GroupsW8 KP4GP. D. Nelson
KurzbeschreibungTopological 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.
LernzielThe 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.
LiteraturA. 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 / BesonderesTopology 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: Link
Kernfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel: Kernfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
NummerTitelTypECTSUmfangDozierende
401-3651-00LNumerical Methods for Elliptic and Parabolic Partial Differential Equations Information
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.
W10 KP4V + 1UC. Schwab
KurzbeschreibungThis 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.
LernzielParticipants 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
InhaltA 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
SkriptCourse slides will be made available to the audience.
LiteraturS. 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 / BesonderesPractical exercises based on MATLAB
401-3621-00LFundamentals of Mathematical StatisticsW10 KP4V + 1UF. Balabdaoui
KurzbeschreibungThe course covers the basics of inferential statistics.
Lernziel
401-4889-00LMathematical FinanceW11 KP4V + 2UM. Schweizer
KurzbeschreibungAdvanced introduction to mathematical finance:
- absence of arbitrage and martingale measures
- option pricing and hedging
- optimal investment problems
- additional topics
LernzielAdvanced level introduction to mathematical finance, presupposing knowledge in probability theory and stochastic processes
InhaltThis 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).
SkriptNone available
LiteraturDetails will be announced in the course.
Voraussetzungen / BesonderesPrerequisites are probability theory and stochastic processes (for which lecture notes are available).
401-3901-00LMathematical Optimization Information W11 KP4V + 2UR. Weismantel
KurzbeschreibungMathematical treatment of diverse optimization techniques.
LernzielAdvanced optimization theory and algorithms.
Inhalt1. 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.
(auch 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 (Link).
NummerTitelTypECTSUmfangDozierende
401-3461-00LFunktionalanalysis I
Das Bachelor-Kernfach 401-3461-00L Funktionalanalysis I / Functional Analysis I ist für Studierende mit einem ETH Zürich Bachelor-Abschluss in Mathematik für den Master-Studiengang Mathematik anrechenbar, falls sie im vorangegangenen Bachelor-Studium weder 401-3461-00L Funktionalanalysis I / Functional Analysis I noch 401-3462-00L Funktionalanalysis II / Functional Analysis II für den Bachelor-Abschluss anrechnen liessen.
Ausserdem ist höchstens eines der drei Fächer
401-3461-00L Funktionalanalysis I / Functional Analysis I
401-3531-00L Differentialgeometrie I / Differential Geometry I
401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory
im Master-Studiengang Mathematik anrechenbar.
W10 KP4V + 1UM. Struwe
KurzbeschreibungBaire-Kategorie; Banach- and Hilberträume, stetige lineare Abbildungen; Prinzipien: Gleichmässige Beschränktheit, Sätze von der offenen Abbildung/vom abgeschlossenen Graphen; Hahn-Banach; Dualraum; Konvexität; schwache/schwach*-Topologie; Banach-Alaoglu; reflexive Räume; Operatoren mit abgeschlossenem Bild; kompakte Operatoren; Fredholmtheorie; Spektraltheorie selbst-adjungierter Operatoren.
Lernziel
SkriptSkript zur "Funktionalanalysis I" von Michael Struwe
401-3531-00LDifferentialgeometrie I
Das Bachelor-Kernfach 401-3531-00L Differentialgeometrie I / Differential Geometry I ist für Studierende mit einem ETH Zürich Bachelor-Abschluss in Mathematik für den Master-Studiengang Mathematik anrechenbar, falls sie im vorangegangenen Bachelor-Studium weder 401-3531-00L Differentialgeometrie I / Differential Geometry I noch 401-3532-00L Differentialgeometrie II / Differential Geometry II für den Bachelor-Abschluss anrechnen liessen.
Ausserdem ist höchstens eines der drei Fächer
401-3461-00L Funktionalanalysis I / Functional Analysis I
401-3531-00L Differentialgeometrie I / Differential Geometry I
401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory
im Master-Studiengang Mathematik anrechenbar.
W10 KP4V + 1UU. Lang
KurzbeschreibungKurven im R^n, innere Geometrie von Hyperflächen im R^n, Krümmung, Theorema Egregium, spezielle Klassen von Flächen, Satz von Gauss-Bonnet. Der hyperbolische Raum. Differenzierbare Mannigfaltigkeiten, Tangentialbündel, Immersionen und Einbettungen, Satz von Sard, Abbildungsgrad und Schnittzahl, Vektorbündel, Vektorfelder und Flüsse, Differentialformen, Satz von Stokes.
LernzielEinführung in die elementare Differentialgeometrie und Differentialtopologie.
Inhalt- Differentialgeometrie im R^n: Kurventheorie, Untermannigfaltigkeiten und Immersionen, innere Geometrie von Hyperflächen, Gauss-Abbildung und -Krümmung, Theorema Egregium, spezielle Klassen von Flächen, Satz von Gauss-Bonnet, Indexsatz von Poincaré.
- Der hyperbolische Raum.
- Differentialtopologie: differenzierbare Mannigfaltigkeiten, Tangentialbündel, Immersionen und Einbettungen in den R^n, Satz von Sard, Transversalität, Abbildungsgrad und Schnittzahl, Vektorbündel, Vektorfelder und Flüsse, Differentialformen, Satz von Stokes.
LiteraturDifferentialgeometrie im R^n:
- Manfredo P. do Carmo: Differentialgeometrie von Kurven und Flächen
- Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten
- Christian Bär: Elementare Differentialgeometrie
Differentialtopologie:
- Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds
- Victor Guillemin & Alan Pollack: Differential Topology
- Morris W. Hirsch: Differential Topology
401-3371-00LDynamical Systems IW10 KP4V + 1UW. Merry
KurzbeschreibungThis course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics.
LernzielMastery of the basic methods and principal themes of some aspects of dynamical systems.
InhaltTopics covered include:

1. Topological dynamics
(transitivity, attractors, chaos, structural stability)

2. Ergodic theory
(Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)

3. Low-dimensional dynamics
(Poincare rotation number, dynamical systems on [0,1])
LiteraturThe most relevant textbook for this course is

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

I will also produce full lecture notes.
Voraussetzungen / BesonderesThe material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory.
401-3001-61LAlgebraic Topology IW8 KP4GP. S. Jossen
KurzbeschreibungThis is an introductory course in algebraic topology. The course will cover the following main topics: introduction to homotopy theory, homology and cohomology of spaces.
Lernziel
Literatur1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

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

Book can be downloaded for free at:
Link

See also:
Link

3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / BesonderesGeneral topology, linear algebra.

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3132-00LCommutative Algebra Information W10 KP4V + 1UR. Pink
KurzbeschreibungThis course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. The material in this course will be assumed in the lecture course "Algebraic Geometry" in the spring semester 2017.
LernzielWe 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
LiteraturPrimary 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 / BesonderesPrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
(auch 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 (Link).
NummerTitelTypECTSUmfangDozierende
401-3601-00LProbability Theory
Das Bachelor-Kernfach 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory ist für Studierende mit einem ETH Zürich Bachelor-Abschluss in Mathematik für den Master-Studiengang Mathematik anrechenbar, falls sie im vorangegangenen Bachelor-Studium keine der drei Lerneinheiten 401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory, 401-3642-00L Brownian Motion and Stochastic Calculus bzw. 401-3602-00L Applied Stochastic Processes für den Bachelor-Abschluss anrechnen liessen.
Ausserdem ist höchstens eines der drei Fächer
401-3461-00L Funktionalanalysis I / Functional Analysis I
401-3531-00L Differentialgeometrie I / Differential Geometry I
401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory
im Master-Studiengang Mathematik anrechenbar.
W10 KP4V + 1UA.‑S. Sznitman
KurzbeschreibungBasics of probability theory and the theory of stochastic processes in discrete time
LernzielThis 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.
InhaltThis 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.
Skriptavailable, will be sold in the course
LiteraturR. 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
402-0205-00LQuantenmechanik I Information W10 KP3V + 2UT. K. Gehrmann
KurzbeschreibungEinführung 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.
LernzielEinführung 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.
InhaltStichworte: 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.
LiteraturF. Schwabl: Quantenmechanik
J.J. Sakurai: Modern Quantum Mechanics
W. Nolting: Quantenmechanik (Theoretische Physik 5.1, 5.2)
C. Cohen-Tannoudji: Quantenmechanik I
Wahlfächer
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Algebra, Topologie, diskrete Mathematik, Logik
NummerTitelTypECTSUmfangDozierende
401-3117-66LIntroduction to the Circle MethodW6 KP2V + 1UE. Kowalski
KurzbeschreibungThe circle method, invented by Hardy and Ramanujan and developped by Hardy and Littlewood and Kloosterman, is one of the most versatile methods currently available to determine the asymptotic behavior of the number of integral solutions to polynomial equations, when the number of solutions is sufficiently large.
Lernziel
InhaltThe circle method, invented by Hardy and Ramanujan and developped by Hardy and
Littlewood and Kloosterman, is one of the most versatile methods currently available
to determine the asymptotic behavior of the number of integral solutions to
polynomial equations, when the number of solutions is sufficiently large.

The lecture will present an introduction to this method. In particular, it will
present the solution of Waring's Problem concerning the representability of integers
as sums of a bounded numbers of (fixed) powers of integers.
LiteraturH. Davenport, "Analytic methods for Diophantine equations and Diophatine
inequalities", Cambridge

H. Iwaniec and E. Kowalski, "Analytic number theory", chapter 20; AMS

R. Vaughan, "The Hardy-Littlewood method", Cambridge
401-4209-66LGroup and Representation Theory: Beyond an IntroductionW8 KP3V + 1UT. H. Willwacher
KurzbeschreibungThe goal of the course is to study several classical and important (and beautiful!) topics in group and representation theory, that are otherwise often overlooked in a standard curriculum.
In particular, we plan to study reflection and Coxeter groups, classical invariant theory, and the theory of real semi simple Lie algebras and their representations.
LernzielDespite the title, the course will begin by a recollection of basic concepts of group and representation theory, in particular that of finite groups and Lie groups.
Hence the course should be accessible also for students who only had a brief exposure to representation theory, as for example in the MMP course.
401-3059-00LKombinatorik II
Findet dieses Semester nicht statt.
W4 KP2GN. Hungerbühler
KurzbeschreibungDer Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende Kombinatorik.
LernzielDie Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
InhaltInhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.
401-4145-66LReading Course: Abelian Varieties over Finite FieldsW2 KP4AJ. Fresán, P. S. Jossen
Kurzbeschreibung
Lernziel
Auswahl: Geometrie
NummerTitelTypECTSUmfangDozierende
401-4531-66LTopics in Rigidity TheoryW6 KP3GM. Burger
KurzbeschreibungThe aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups.
LernzielUnderstand the basic techniques of rigidity theory.
InhaltThis course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are:
- Describe all its proper quotients.
- Classify all its finite dimensional linear representations.
- More generally, can this group act by diffeomorphisms on "small" manifolds like the circle?
- Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure?
In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction.
Literatur- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984.
- D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv
- Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage.
- M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online.
Voraussetzungen / BesonderesFor this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques.
401-3309-66LRiemann Surfaces (Part 2) Information W4 KP2VA. Buryak
KurzbeschreibungThe program will be the following:

* Proof of the Serre duality;
* Riemann-Hurwitz formula;
* Functions and differential forms on a compact Riemann surface with prescribed principal parts;
* Weierstrass points on a compact Riemann surface;
* The Jacobian and the Picard group of a compact Riemann surface;
* Holomorphic vector bundles;
* Non-compact Riemann surfaces.
Lernziel
LiteraturO. Forster. Lectures on Riemann Surfaces.
Voraussetzungen / BesonderesThis is a continuation of 401-3308-16L Riemann Surfaces that was taught in the spring semester (FS 2016), see Link for the lecture notes. The students are also assumed to be familiar with what would generally be covered in one semester courses on general topology and on algebra.
401-3057-00LEndliche Geometrien IIW4 KP2GN. Hungerbühler
KurzbeschreibungEndliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate.
LernzielEndliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne.
InhaltEndliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne
Literatur- Max Jeger, Endliche Geometrien, ETH Skript 1988

- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983

- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press

- Dembowski: Finite Geometries.
Auswahl: Analysis
NummerTitelTypECTSUmfangDozierende
401-3536-11LGeometric Aspects of Hamiltonian DynamicsW6 KP3VP. Biran
KurzbeschreibungThe course will concentrate on the geometry of the group of Hamiltonian diffeomorphisms introduced by Hofer in the early 1990's and its relations to various topics in symplectic geometry such as capacities, Lagrangian submanifolds, holomorphic curves, as well as recent algebraic structures on the group of Hamiltonian diffeomorphisms such as quasi-morphisms.
Lernziel
LiteraturBooks:
* L. Polterovich: "The geometry of the group of symplectic diffeomorphisms"
* H. Hofer & E. Zehnder: "Symplectic invariants and Hamiltonian dynamics"
Voraussetzungen / BesonderesPrerequisites. Good knowledge of undergraduate mathematics (analysis, complex functions, topology, and differential geometry). Some knowledge of elementary algebraic topology would be useful.
401-4767-66LPartial Differential Equations (Hyperbolic PDEs)W7 KP4VD. Christodoulou
KurzbeschreibungThe course begins with characteristics, the definition of hyperbolicity,
causal structure and the domain of dependence theorem. The course then focuses
on nonlinear systems of equations in two independent variables, in particular the
Euler equations of compressible fluids with plane symmetry and the Einstein equations
of general relativity with spherical symmetry.
LernzielThe objective is to introduce students in mathematics and physics to an area of
mathematical analysis involving differential geometry which is of fundamental importance
for the development of classical macroscopic continuum physics.
InhaltThe course shall begin with the basic structure associated to hyperbolic
partial differential equations, characteristic hypersurfaces and bicharacteristics,
causal structure, and the domain of dependence theorem. The course shall then focus
on nonlinear systems of equations in two independent variables. The first topic shall be
the Euler equations of compressible fluids under plane symmetry where we shall
study the formation of shocks, and second topic shall be the Einstein equations
of general relativity under spherical symmetry where we shall study the formation of
black holes and spacetime singularities.
Voraussetzungen / BesonderesBasic real analysis and differential geometry.
401-4831-66LMathematical Themes in General Relativity IW4 KP2VA. Carlotto
KurzbeschreibungFirst part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems.
LernzielAcquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis.
InhaltLorentzian geometry; geometric review of special relativity; the Einstein equations and their basic classes of special solutions; the Einstein equations as an initial-value problem; causality theory and hyperbolicity; singularities and trapped domains; Penrose diagrams; asymptotically flat spaces: ADM invariants, positive mass theorems, Penrose inequalities, geometric properties.
SkriptLecture notes written by the instructor will be provided to all enrolled students.
Voraussetzungen / BesonderesThe content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable.
401-4497-66LFree Boundary ProblemsW4 KP2VA. Figalli
Kurzbeschreibung
Lernziel
401-4463-62LFourier Analysis in Function Space TheoryW6 KP3VT. Rivière
KurzbeschreibungIn the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
Lernziel
InhaltDuring the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces.
In the second part of the course we will study fundamental pro­perties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occa­sion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings.
In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course.
If time permits we shall present the notion of Paraproduct, Para­compositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE.
Literatur1) Elias M. Stein, "Singular Integrals and Differentiability Proper­ties of Functions" (PMS-30) Princeton University Press.
2) Javier Duoandikoetxea, "Fourier Analysis" AMS.
3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer.
4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer.
Voraussetzungen / BesonderesNotions from ETH courses in Measure Theory, Functional Analysis I and II (Fun­damental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions)
401-4475-66LPartial Differential Equations and Semigroups of Bounded Linear Operators Information W4 KP2GA. Jentzen
KurzbeschreibungIn this course we study the concept of a semigroup of bounded linear operators and we use this concept to investigate existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs) of the evolutionary type.
LernzielThe aim of this course is to teach the students a decent knowledge (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs.
InhaltThe course includes content (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. Key example PDEs that are treated in this course are heat and wave equations.
SkriptLecture Notes are available in the lecture homepage (please follow the link in the Learning materials section).
Literatur1. Amnon Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983).

2. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations. Springer-Verlag, New York (2000).
Voraussetzungen / BesonderesMandatory prerequisites: Functional analysis

Start of lectures: Friday, September 23, 2016
For more details, please follow the link in the Learning materials section.
401-3303-00LAusgewählte Themen der FunktionentheorieW6 KP3VH. Knörrer
KurzbeschreibungHypergeometrische Funktionen, Randwerte holomorpher Funktionen, Nevanlinna Theorie und andere spezielle Themen
LernzielFortgeschrittene Methoden der Funktionentheorie
LiteraturR. Remmert: Funktionentheorie II. Springer Verlag
E.Titchmarsh: The Theory of Functions. Oxford University Press
C.Caratheodory: Funktionentheorie. Birkhaeuser
E.Hille: Analytic Function Theory. AMS Chelsea Publishing
A.Gogolin:Komplexe Integration. Springer
Voraussetzungen / BesonderesFunktionentheorie
Auswahl: Weitere Gebiete
NummerTitelTypECTSUmfangDozierende
401-3502-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W2 KP4AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W3 KP6AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W4 KP9AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
Wahlfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel:
Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Auswahl: Numerische Mathematik
NummerTitelTypECTSUmfangDozierende
401-4657-00LNumerical Analysis of Stochastic Ordinary Differential Equations Information
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
W6 KP3V + 1UA. Jentzen
KurzbeschreibungCourse on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables.
LernzielThe aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.
InhaltGeneration of random numbers
Monte Carlo methods for the numerical integration of random variables
Stochastic processes and Brownian motion
Stochastic ordinary differential equations (SODEs)
Numerical approximations of SODEs
Multilevel Monte Carlo methods for SODEs
Applications to computational finance: Option valuation
SkriptLecture Notes are available in the lecture homepage (please follow the link in the Learning materials section).
LiteraturP. Glassermann:
Monte Carlo Methods in Financial Engineering.
Springer-Verlag, New York, 2004.

P. E. Kloeden and E. Platen:
Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 1992.
Voraussetzungen / BesonderesPrerequisites:

Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.

a) mandatory courses:
Elementary Probability,
Probability Theory I.

b) recommended courses:
Stochastic Processes.

Start of lectures: Wednesday, September 21, 2016
For more details, please follow the link in the Learning materials section.
401-4785-00LMathematical and Computational Methods in PhotonicsW8 KP4GH. Ammari
KurzbeschreibungThe aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces
LernzielThe field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications.

The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength.

Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures.

The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The course merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions.

In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials.
Auswahl: Wahrscheinlichkeitstheorie, Statistik
NummerTitelTypECTSUmfangDozierende
401-4604-66LTopics in Probability TheoryW4 KP2VW. Werner
KurzbeschreibungThe goal of this course is to give a sample of some basic results and features to illustrate various areas of probability theory.
LernzielThe goal of this course is to give a sample of some basic results and features to illustrate various areas of probability theory.
401-3604-66LSpecial Topics in ProbabilityW4 KP2VP. Nolin
KurzbeschreibungThe goal of this course is to present recent developments in Percolation Theory
LernzielThe goal of this course is to present recent developments in Percolation Theory
InhaltIndependent percolation is obtained by deleting randomly (and independently) the edges of a lattice, each with a given probability p between 0 and 1. One is then interested in the connectivity properties of the random subgraph so-obtained. It is arguably the simplest model from statistical mechanics that displays a phase transition, a drastic change of behavior as the parameter p varies.

We will first present classical tools and properties of percolation theory: in particular correlation inequalities, exponential decay of connection probabilities, and uniqueness of the infinite connected component. We will then discuss recent developments: for example percolation on Cayley graphs, and continuum limits in two dimensions.
LiteraturB. Bollobas, O. Riordan: Percolation, CUP 2006
G. Grimmett: Percolation 2ed, Springer 1999
Voraussetzungen / BesonderesPrerequisites:
401-2604-00L Probability and Statistics (mandatory)
401-3601-00L Probability Theory (recommended)
401-4611-66LRough Path Theory and Regularity StructuresW6 KP3VJ. Teichmann, D. Prömel
KurzbeschreibungThe course provides an introduction to the theory of controlled rough paths with focus on stochastic differential equations. In parallel, Martin Hairer's new theory of regularity structures is introduced taking controlled rough paths as guiding examples. In particular, the course demonstrates how to use the theory of regularity structures to solve singular stochastic PDEs.
LernzielThe main goal is to develop simultaneously the basic concepts of rough path theory and Hairer's regularity structures.
Literatur- Peter Friz and Martin Hairer, A Course on Rough Paths: With an Introduction to
Regularity Structures, Springer, 2014.
- Martin Hairer, Introduction to regularity structures, Braz. J. Probab. Stat. 29 (2015),
no. 2, 175-210.
- Peter Friz and Nicolas Victoir, Multidimensional stochastic processes as rough paths.
Theory and applications, Cambridge University Press, 2010.
- Martin Hairer, A theory of regularity structures, Inventiones mathematicae (2014), 1-236.
- Ajay Chandra and Hendrik Weber, Stochastic PDEs, Regularity Structures, and Inter-
acting Particle Systems, Preprint arXiv:1508.03616.
Voraussetzungen / BesonderesRequirements: Brownian Motion and Stochastic Calculus
401-3627-00LHigh-Dimensional Statistics
Findet dieses Semester nicht statt.
W4 KP2VP. 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.
LernzielKnowledge of methods and basic theory for high-dimensional statistical inference
InhaltLasso 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
LiteraturPeter 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 / BesonderesKnowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).
401-4623-00LTime Series AnalysisW6 KP3GN. Meinshausen
KurzbeschreibungStatistical 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.
LernzielUnderstanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R.
InhaltThis 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.
SkriptNot available
LiteraturA list of references will be distributed during the course.
Voraussetzungen / BesonderesBasic knowledge in probability and statistics
401-3612-00LStochastic SimulationW5 KP3GF. Sigrist
KurzbeschreibungThis course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo.
LernzielStochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R.
InhaltExamples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC).
SkriptA script will be available in English.
LiteraturP. Glasserman, Monte Carlo Methods in Financial Engineering.
Springer 2004.

B. D. Ripley. Stochastic Simulation. Wiley, 1987.

Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
Voraussetzungen / BesonderesFamiliarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.
401-3611-00LAdvanced Topics in Computational Statistics
Findet dieses Semester nicht statt.
W4 KP2VM. H. Maathuis
KurzbeschreibungThis lecture covers selected advanced topics in computational statistics, including various classification methods, the EM algorithm, clustering, handling missing data, and graphical modelling.
LernzielStudents learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.
InhaltThe course is roughly divided in three parts: (1) Supervised learning via (variations of) nearest neighbor methods, (2) the EM algorithm and clustering, (3) handling missing data and graphical models.
SkriptLecture notes.
Voraussetzungen / BesonderesWe assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.
401-0649-00LApplied Statistical Regression Information W5 KP2V + 1UM. Dettling
KurzbeschreibungThis 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.
LernzielThe students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling.
InhaltThe 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.
SkriptA script will be available.
LiteraturFaraway (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 / BesonderesThe 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.
401-0625-01LApplied Analysis of Variance and Experimental Design Information W5 KP2V + 1UL. Meier
KurzbeschreibungPrinciples of experimental design. One-way analysis of variance. Multi-factor experiments and analysis of variance. Block designs. Latin square designs. Split-plot and strip-plot designs. Random effects and mixed effects models. Full factorials and fractional designs.
LernzielParticipants 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.
InhaltPrinciples of experimental design. One-way analysis of variance. Multi-factor experiments and analysis of variance. Block designs. Latin square designs. Split-plot and strip-plot designs. Random effects and mixed effects models. Full factorials and fractional designs.
LiteraturG. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000.
Voraussetzungen / BesonderesThe 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.
Auswahl: Finanz- und Versicherungsmathematik
In den Master-Studiengängen Mathematik bzw. Angewandte 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).
NummerTitelTypECTSUmfangDozierende
401-3925-00LNon-Life Insurance: Mathematics and StatisticsW6 KP4GM. V. Wüthrich
KurzbeschreibungThe 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.
LernzielThe 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.
InhaltThe 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
SkriptM. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics
Link
Voraussetzungen / BesonderesThe 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.
401-3922-00LLife Insurance MathematicsW4 KP2VM. Koller
KurzbeschreibungThe 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
401-3929-00LFinancial Risk Management in Social and Pension InsuranceW4 KP2VP. Blum
KurzbeschreibungInvestment returns are an important source of funding for social and pension insurance, and financial risk is an important threat to stability. We study short-term and long-term financial risk and its interplay with other risk factors, and we develop methods for the measurement and management of financial risk and return in an asset/liability context with the goal of assuring sustainable funding.
LernzielUnderstand the basic asset-liability framework: essential principles and properties of social and pension insurance; cash flow matching, duration matching, valuation portfolio and loose coupling; the notion of financial risk; long-term vs. short-term risk; coherent measures of risk.

Understand the conditions for sustainable funding: derivation of required returns; interplay between return levels, contribution levels and other parameters; influence of guaranteed benefits.

Understand the notion of risk-taking capability: capital process as a random walk; measures of long-term risk and relation to capital; short-term solvency vs. long-term stability; effect of embedded options and guarantees; interplay between required return and risk-taking capability.

Be able to study empirical properties of financial assets: the Normal hypothesis and the deviations from it; statistical tools for investigating relevant risk and return properties of financial assets; time aggregation properties; be able to conduct analysis of real data for the most important asset classes.

Understand and be able to carry out portfolio construction: the concept of diversification; limitations to diversification / correlation breakdown / what happened in 2008; the Kuhn-Tucker Theorem and optimization (mean-variance, mean-downside); incorporation of constraints; sensitivity and shortcomings of optimized portfolios.

Understand and interpret the asset-liability interplay: the optimized portfolio in the asset-liability framework; short-term risk vs. long-term risk; the influence of constraints; feasible and non-feasible solutions; practical considerations.

Know about active portfolio management: practical issues when implementing an investment strategy; the notion of active management; efficient markets hypothesis and limitations to it; empirical evidence; the fundamental law of active management; Bayesian concepts and the Black-Litterman framework.

Have an overall view: see the big picture of what asset returns can and cannot contribute to social security; be aware of the most relevant outcomes; know the role of the actuary in the financial risk management process.
InhaltFor pension insurance and other forms of social insurance, investment returns are an important source of funding. In order to earn these returns, substantial financial risks must be taken, and these risks represent an important threat to financial stability, in the long term and in the short term.

Risk and return of financial assets cannot be separated from one another and, hence, asset management and risk management cannot be separated either. Managing financial risk in social and pension insurance is, therefore, the task of reconciling the contradictory dimensions of

1. Required return for a sustainable funding of the institution,
2. Risk-taking capability of the institution,
3. Returns available from financial assets in the market,
4. Risks incurred by investing in these assets.

This task must be accomplished under a number of constraints. Financial risk management in social insurance also means reconciling the long time horizon of the promised insurance benefits with the short time horizon of financial markets and financial risk.

It is not the goal of this lecture to provide the students with any cookbook recipes that can readily be applied without further reflection. The goal is rather to enable the students to develop their own understanding of the problems and possible solutions associated with the management of financial risks in social and pension insurance.

To this end, a rigorous intellectual framework will be developed and a powerful set of mathematical tools from the fields of actuarial mathematics and quantitative risk management will be applied. When analyzing the properties of financial assets, an empirical viewpoint will be taken using statistical tools and considering real-world data.
SkriptSince this is the first instance of this course, there is not yet a full script. However, to complement the blackboard notes, extensive handouts will be provided. Moreover, practical examples and data sets in Excel and Octave / Matlab will be made available to play around with and deepen the understanding of the subject matter.
Voraussetzungen / BesonderesSolid base knowledge of probability and statistics is indispensable. Specialized concepts from financial and insurance mathematics as well as quantitative risk management will be introduced in the lecture as needed, but some prior knowledge in some of these areas would be an advantage.

This course counts towards the diploma of "Aktuar SAV".

The exams ONLY take place during the official ETH examination period.
401-4947-66Lelective course <title tba>
Findet dieses Semester nicht statt.
W4 KP2VP. Cheridito
Kurzbeschreibung
Lernziel
Auswahl: Mathematische Physik, Theoretische Physik
NummerTitelTypECTSUmfangDozierende
402-0843-00LQuantum Field Theory I Information W10 KP4V + 2UC. Anastasiou
KurzbeschreibungThis course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity.
Topics include:
- Relativistic quantum mechanics
- Quantisation of bosonic and fermionic fields
- Interactions in perturbation theory
- Scattering processes and decays
- Radiative corrections
LernzielThe goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics.
402-0861-00LStatistical PhysicsW10 KP4V + 2UG. Blatter
KurzbeschreibungThis lecture covers the concepts of classical and quantum statistical physics, and some aspects of kinetic gas theory and hydrodynamics. In a more advanced part degenerate Fermions, Bose-Einstein condensation, real Bose gases, magnetism, general mean field theory and critical phenomena will be addressed.
LernzielThis lecture gives an introduction in the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.
InhaltBasics of phenomenological thermodynamics, three laws of thermodynamics.
Basics of kinetic gas theory: conservation laws, H-theorem, Boltzmann-Equations, Maxwell distribution.
Hydrodynamics.
Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons.
Degenerate fermions: Fermi gas, electrons in magnetic field.
Bosons: Bose-Einstein condensation, Bogoliubov theory, superfluidity.
Mean field and Landau theory: Ising-, XY-, Heisenberg models, Landau theory of phase transitions, fluctuations.
Critical phenomena: mean field, series expansions, scaling behavior, universality.
Renormalization group: fixed points, simple models.
SkriptLecture notes available in german.
LiteraturNo specific book is used for the course. Relevant literature will be given in the course.
402-0830-00LGeneral Relativity Information W10 KP4V + 2UP. Jetzer
KurzbeschreibungManifold, 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.
LernzielBasic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts.
LiteraturSuggested 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
402-0822-13LIntroduction to IntegrabilityW6 KP2V + 1UN. Beisert
KurzbeschreibungThis course gives an introduction to the theory of integrable systems, related symmetry algebras and efficients calculational methods.
LernzielIntegrable systems are a special class of physical models that can be solved exactly due to an exceptionally large number of symmetries. Examples of integrable models appear in many different areas of physics, including classical mechanics, condensed matter, 2d quantum field theories and lately in string- and gauge theories. They offer a unique opportunity to gain a deeper understanding of generic phenomena in a simplified, exactly solvable setting. In this course we introduce the various notions of integrability in classical mechanics, quantum mechanics and quantum field theory. We discuss efficient methods for solving such models as well as the underlying enhanced symmetries.
Inhalt* Classical Integrability
* Integrable Field Theory
* Integrable Spin Chains
* Quantum Integrability
* Integrable Statistical Mechanics
* Quantum Algebra
* Bethe Ansatz and Related Methods
* AdS/CFT Integrability
Literatur* V. Chari, A. Pressley, "A Guide to Quantum Groups", Cambridge University Press (1995).
* O. Babelon, D. Bernard, M. Talon, "Introduction to Classical Integrable Systems", Cambridge University Press (2003)
* N. Reshetikhin, "Lectures on the integrability of the 6-vertex model", Link
* L.D. Faddeev, "How Algebraic Bethe Ansatz Works for Integrable Model", Link
* D. Bernard, "An Introduction to Yangian Symmetries", Int. J. Mod. Phys. B7, 3517-3530 (1993), Link
* V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, "Quantum Inverse Scattering Method and Correlation Functions", Cambridge University Press (1997)
Auswahl: Mathematische Optimierung, Diskrete Mathematik
NummerTitelTypECTSUmfangDozierende
401-3054-14LProbabilistic Methods in Combinatorics Information W6 KP2V + 1UB. Sudakov
KurzbeschreibungThis course provides a gentle introduction to the Probabilistic Method, with an emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems.
Lernziel
InhaltThe topics covered in the class will include (but are not limited to): linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness.
Literatur- The Probabilistic Method, by N. Alon and J. H. Spencer, 3rd Edition, Wiley, 2008.
- Random Graphs, by B. Bollobás, 2nd Edition, Cambridge University Press, 2001.
- Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000.
- Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002.
Auswahl: Theoretische Informatik
NummerTitelTypECTSUmfangDozierende
252-1425-00LGeometry: Combinatorics and Algorithms Information W6 KP2V + 2U + 1AB. Gärtner, E. Welzl, M. Hoffmann, A. Pilz
KurzbeschreibungGeometric 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?)
LernzielThe 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.
InhaltPlanar 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.
Skriptyes
LiteraturMark 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 / BesonderesPrerequisites: 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.
252-1407-00LAlgorithmic Game Theory Information W7 KP3V + 2U + 1AP. Widmayer, P. Penna
KurzbeschreibungGame 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.
LernzielLearning 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.
InhaltThe 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.
SkriptNo lecture notes.
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 / BesonderesAudience: 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.
252-0417-00LRandomized Algorithms and Probabilistic MethodsW7 KP3V + 2U + 1AA. Steger, E. Welzl
KurzbeschreibungLas 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
LernzielAfter 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.
InhaltRandomized 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.
SkriptYes.
Literatur- Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995)
- Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005)
263-4655-00LLattice CryptographyW4 KP2V + 1UV. Lyubashevsky
KurzbeschreibungThe course will introduce lattice-based cryptography, which is one of the main candidates for quantum-resistant cryptography.
LernzielThe objective of the course is to bring the students up to a level where they should be able to read academic papers on state-of-the-art designs of lattice-based primitives.
InhaltIn this course, we will study lattice-based cryptography. We will cover the basic algorithms associated with integer lattices such as Gram-Schmidt orthogonalization, algorithms for finding short and near lattice vectors, as well as the critical algorithm for sampling lattice points according to a discrete Gaussian distribution. We will then proceed to build up a toolbox of lattice-based cryptographic primitives beginning from collision-resistant hash functions, then moving on to digital signatures, encryption, identity-based encryption, and fully-homomorphic encryption. Particular emphasis will be placed on concrete parameters and practical instantiations. For this purpose, we will also study cryptographic constructions based on the hardness of ideal lattices, which are ideals of polynomial rings.
Voraussetzungen / BesonderesThere are no formal mathematical pre-requisites, but students should have "mathematical maturity", which entails dealing with abstract concepts and being comfortable with doing mathematical proofs. Some previous exposure to linear algebra, abstract algebra, and cryptography would be useful.
Auswahl: Weitere Gebiete
NummerTitelTypECTSUmfangDozierende
401-3502-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W2 KP4AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W3 KP6AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-66LReading Course Belegung eingeschränkt - Details anzeigen
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.
W4 KP9AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
Anwendungsgebiet
Nur für das Master-Diplom in Angewandter Mathematik erforderlich und anrechenbar.
In der Kategorie Anwendungsgebiet für den Master in Angewandter Mathematik muss eines der zur Auswahl stehenden Anwendungsgebiete gewählt werden. Im gewählten Anwendungsgebiet müssen mindestens 8 KP erworben werden.
Atmospherical Physics
NummerTitelTypECTSUmfangDozierende
701-1221-00LDynamics of Large-Scale Atmospheric Flow Information W4 KP2V + 1UH. Wernli, S. Pfahl
KurzbeschreibungDynamische Synoptische Meteorologie
LernzielVerständnis für dynamische Prozesse in der Atmosphäre sowie deren
mathematisch-physikalische Formulierung.
InhaltDie Atmosphärenphysik II behandelt vor allem die dynamischen Prozesse in der Erdatmosphäre. Diskutiert werden die Bewegungsgesetze der Atmosphäre und die Dynamik und Wechselwirkungen von synoptischen Systemen - also den wetterbestimmenden Hoch- und Tiefdruckgebieten. Mathematische Grundlage hierfuer ist insbesondere die Theorie der quasi-geostrophischen Bewegung, die im Rahmen der Vorlesung hergeleitet und interpretiert wird.
SkriptDynamics of large-scale atmospheric flow
Literatur- Holton J.R., An introduction to Dynamic Meteorogy. Academic Press, fourth edition 2004,
- Pichler H., Dynamik der Atmosphäre, Bibliographisches Institut, 456 pp. 1997
Voraussetzungen / BesonderesVoraussetzungen: Physik I, II, Umwelt Fluiddynamik
Biology
NummerTitelTypECTSUmfangDozierende
551-0015-00LBiologie IW2 KP2VR. Glockshuber, E. Hafen
KurzbeschreibungGegenstand der Vorlesung Biologie I ist zusammen mit der Vorlesung Biologie II im folgenden Sommersemester die Einführung in die Grundlagen der Biologie für Studenten der Materialwissenschaften und andere Studenten mit Biologie als Nebenfach.
LernzielZiel der Vorlesung Biologie I ist die Vermittlung des molekularen Aufbaus der Zelle, der Grundlagen des Stoffwechsels und eines Überblicks über molekulare Genetik
InhaltDie folgenden Kapitelnummern beziehen sich auf das der Vorlesung zugrundeliegende Lehrbuch "Biology" (Campbell & Rees, 10th edition, 2015)
Kapitel 1-4 des Lehrbuchs werden als Grundwissen vorausgesetzt

1. Aufbau der Zelle

Kapitel 5: Struktur und Funktion biologischer Makromoleküle
Kapitel 6: Eine Tour durch die Zelle
Kaptiel 7: Membranstruktur und-funktion
Kapitel 8: Einführung in den Stoffwechsel
Kapitel 9: Zelluläre Atmung und Speicherung chemischer Energie
Kapitel 10: Photosynthese
Kapitel 12: Der Zellzyklus
Kapitel 17: Vom Gen zum Protein

2. Allgemeine Genetik

Kapitel 13: Meiose und Reproduktionszyklen
Kapitel 14: Mendel'sche Genetik
Kapitel 15: Die chromosomale Basis der Vererbung
Kapitel 16: Die molekulare Grundlage der Vererbung
Kapitel 18: Genetik von Bakterien und Viren
Kapitel 46: Tierische Reproduktion

Grundlagen des Stoffwechsels und eines Überblicks über molekulare Genetik
SkriptDer Vorlesungsstoff ist sehr nahe am Lehrbuch gehalten, Skripte werden ggf. durch die Dozenten zur Verfügung gestellt.
LiteraturDas folgende Lehrbuch ist Grundlage für die Vorlesungen Biologie I und II:

„Biology“, Campbell and Rees, 10th Edition, 2015, Pearson/Benjamin Cummings, ISBN 978-3-8632-6725-4
Voraussetzungen / BesonderesZur Vorlesung Biologie I gibt es während der Prüfungssessionen eine einstündige, schriftliche Prüfung. Die Vorlesung Biologie II wird separat geprüft.
636-0017-00LComputational Biology Information W4 KP3GT. Stadler, C. Magnus
KurzbeschreibungThe aim of the course is to provide up-to-date knowledge on how we can study biological processes using genetic sequencing data. Computational algorithms extracting biological information from genetic sequence data are discussed, and statistical tools to understand this information in detail are introduced.
LernzielAttendees will learn which information is contained in genetic sequencing data and how to extract information from them using computational tools. The main concepts introduced are:
* stochastic models in molecular evolution
* phylogenetic & phylodynamic inference
* maximum likelihood and Bayesian statistics
Attendees will apply these concepts to a number of applications yielding biological insight into:
* epidemiology
* pathogen evolution
* macroevolution of species
InhaltThe course consists of four parts. We first introduce modern genetic sequencing technology, and algorithms to obtain sequence alignments from the output of the sequencers. We then present methods to directly analyze this alignment (such as BLAST algorithm, GWAS approaches). Second, we introduce mechanisms and concepts of molecular evolution, i.e. we discuss how genetic sequences change over time. Third, we employ evolutionary concepts to infer ancestral relationships between organisms based on their genetic sequences, i.e. we discuss methods to infer genealogies and phylogenies. We finally introduce the field of phylodynamics. The aim of that field is to understand and quantify the population dynamic processes (such as transmission in epidemiology or speciation & extinction in macroevolution) based on a phylogeny. Throughout the class, the models and methods are illustrated on different datasets giving insight into the epidemiology and evolution of a range of infectious diseases (e.g. HIV, HCV, influenza, Ebola). Applications of the methods to the field of macroevolution provide insight into the evolution and ecology of different species clades. Students will be trained in the algorithms and their application both on paper and in silico as part of the exercises.
SkriptSlides of the lecture will be available online.
Link
LiteraturThe course is not based on any of the textbooks below, but they are excellent choices as accompanying material:
* Yang, Z. 2006. Computational Molecular Evolution.
* Felsenstein, J. 2004. Inferring Phylogenies.
* Semple, C. & Steel, M. 2003. Phylogenetics.
* Drummond, A. & Bouckaert, R. 2015. Bayesian evolutionary analysis with BEAST
Voraussetzungen / BesonderesBasic knowledge in linear algebra, analysis, and statistics will be helpful. Some programming experience will be useful for the exercises, but is not required. Programming skills will not be tested in the examination.
Computational Electromagnetics
NummerTitelTypECTSUmfangDozierende
227-2037-00LPhysical Modelling and Simulation Information W5 KP4GC. Hafner, J. Leuthold, J. Smajic
KurzbeschreibungThis module consists of (a) an introduction to fundamental equations of electromagnetics, mechanics and heat transfer, (b) a detailed overview of numerical methods for field simulations, and (c) practical examples solved in form of small projects.
LernzielBasic knowledge of the fundamental equations and effects of electromagnetics, mechanics, and heat transfer. Knowledge of the main concepts of numerical methods for physical modelling and simulation. Ability (a) to develop own simple field simulation programs, (b) to select an appropriate field solver for a given problem, (c) to perform field simulations, (d) to evaluate the obtained results, and (e) to interactively improve the models until sufficiently accurate results are obtained.
InhaltThe module begins with an introduction to the fundamental equations and effects of electromagnetics, mechanics, and heat transfer. After the introduction follows a detailed overview of the available numerical methods for solving electromagnetic, thermal and mechanical boundary value problems. This part of the course contains a general introduction into numerical methods, differential and integral forms, linear equation systems, Finite Difference Method (FDM), Boundary Element Method (BEM), Method of Moments (MoM), Multiple Multipole Program (MMP) and Finite Element Method (FEM). The theoretical part of the course finishes with a presentation of multiphysics simulations through several practical examples of HF-engineering such as coupled electromagnetic-mechanical and electromagnetic-thermal analysis of MEMS.
In the second part of the course the students will work in small groups on practical simulation problems. For solving practical problems the students can develop and use own simulation programs or chose an appropriate commercial field solver for their specific problem. This practical simulation work of the students is supervised by the lecturers.
Control and Automation
NummerTitelTypECTSUmfangDozierende
151-0563-01LDynamic Programming and Optimal Control Information W4 KP2V + 1UR. D'Andrea
KurzbeschreibungIntroduction to Dynamic Programming and Optimal Control.
LernzielCovers the fundamental concepts of Dynamic Programming & Optimal Control.
InhaltDynamic Programming Algorithm; Deterministic Systems and Shortest Path Problems; Infinite Horizon Problems, Bellman Equation; Deterministic Continuous-Time Optimal Control.
LiteraturDynamic Programming and Optimal Control by Dimitri P. Bertsekas, Vol. I, 3rd edition, 2005, 558 pages, hardcover.
Voraussetzungen / BesonderesRequirements: Knowledge of advanced calculus, introductory probability theory, and matrix-vector algebra.
Economics
NummerTitelTypECTSUmfangDozierende
363-0537-00LResource and Environmental EconomicsW3 KP2GL. Bretschger, A. Brausmann
KurzbeschreibungRelationship between economy and environment, market failure, external effects and public goods, contingent valuation, internalisation of externalities; economics of non-renewable resources, economics of renewable resources, cost-benefit analysis, sustainability, and international aspects of resource and environmental economics.
LernzielUnderstanding of the basic issues and methods in resource and environmental economics; ability to solve typical problems in the field using the appropriate tools, which are concise verbal explanations, diagrams or mathematical expressions.

Topics are:
Introduction to resource and environmental economics
Importance of resource and environmental economics
Main issues of resource and environmental economics
Normative basis
Utilitarianism
Fairness according to Rawls
Economic growth and environment
Externalities in the environmental sphere
Governmental internalisation of externalities
Private internalisation of externalities: the Coase theorem
Free rider problem and public goods
Types of public policy
Efficient level of pollution
Tax vs. permits
Command and Control Instruments
Empirical data on non-renewable natural resources
Optimal price development: the Hotelling-rule
Effects of exploration and Backstop-technology
Effects of different types of markets.
Biological growth function
Optimal depletion of renewable resources
Social inefficiency as result of over-use of open-access resources
Cost-benefit analysis and the environment
Measuring environmental benefit
Measuring costs
Concept of sustainability
Technological feasibility
Conflicts sustainability / optimality
Indicators of sustainability
Problem of climate change
Cost and benefit of climate change
Climate change as international ecological externality
International climate policy: Kyoto protocol
Implementation of the Kyoto protocol in Switzerland
InhaltEconomy and natural environment, welfare concepts and market failure, external effects and public goods, measuring externalities and contingent valuation, internalising external effects and environmental policy, economics of non-renewable resources, renewable resources, cost-benefit-analysis, sustainability issues, international aspects of resource and environmental problems, selected examples and case studies.
SkriptLearning material and script can be found here:
Link
LiteraturPerman, R., Ma, Y., McGilvray, J, Common, M.: "Natural Resource & Environmental Economics", 3d edition, Longman, Essex 2003.
363-0503-00LPrinciples of MicroeconomicsW3 KP2GM. Filippini
KurzbeschreibungThe course introduces basic principles, problems and approaches of microeconomics.
LernzielThe learning objectives of the course are:

(1) Students must be able to discuss basic principles, problems and approaches in microeconomics. (2) Students can analyse and explain simple economic principles in a market using supply and demand graphs. (3) Students can contrast different market structures and describe firm and consumer behaviour. (4) Students can identify market failures such as externalities related to market activities and illustrate how these affect the economy as a whole. (5) Students can apply simple mathematical treatment of some basic concepts and can solve utility maximization and cost minimization problems.
SkriptLecture notes, exercises and reference material can be downloaded from Moodle.
LiteraturN. Gregory Mankiw and Mark P. Taylor (2014), "Economics", 3rd edition, South-Western Cengage Learning.
The book can also be used for the course 'Principles of Macroeconomics' (Sturm)

For students taking only the course 'Principles of Microeconomics' there is a shorter version of the same book:
N. Gregory Mankiw and Mark P. Taylor (2014), "Microeconomics", 3rd edition, South-Western Cengage Learning.

Complementary:
1. R. Pindyck and D. Rubinfeld (2012), "Microeconomics", 8th edition, Pearson Education.
2. Varian, H.R. (2014), "Intermediate Microeconomics", 9th edition, Norton & Company
363-0565-00LPrinciples of MacroeconomicsW3 KP2VJ.‑E. Sturm
KurzbeschreibungThis course examines the behaviour of macroeconomic variables, such as gross domestic product, unemployment and inflation rates. It tries to answer questions like: How can we explain fluctuations of national economic activity? What can economic policy do against unemployment and inflation. What significance do international economic relations have for Switzerland?
LernzielThis lecture will introduce the fundamentals of macroeconomic theory and explain their relevance to every-day economic problems.
InhaltThis course helps you understand the world in which you live. There are many questions about the macroeconomy that might spark your curiosity. Why are living standards so meagre in many African countries? Why do some countries have high rates of inflation while others have stable prices? Why have some European countries adopted a common currency? These are just a few of the questions that this course will help you answer.
Furthermore, this course will give you a better understanding of the potential and limits of economic policy. As a voter, you help choose the policies that guide the allocation of society's resources. When deciding which policies to support, you may find yourself asking various questions about economics. What are the burdens associated with alternative forms of taxation? What are the effects of free trade with other countries? What is the best way to protect the environment? How does the government budget deficit affect the economy? These and similar questions are always on the minds of policy makers.
SkriptThe course webpage (to be found at Link) contains announcements, course information and lecture slides.
LiteraturThe set-up of the course will closely follow the book of
N. Gregory Mankiw and Mark P. Taylor (2014), Economics, Cengage Learning, Third Edition.

We advise you to also buy access to Aplia. This internet platform will support you in learning for this course. To save money, you should buy the book together with Aplia. This is sold as a bundle (ISBN: 9781473715998).

Besides this textbook, the slides and lecture notes will cover the content of the lecture and the exam questions.
363-1021-00LMonetary PolicyW3 KP2VJ.‑E. Sturm, D. Kaufmann
KurzbeschreibungThe main aim of this course is to analyse the goals of monetary policy and to review the instruments available to central banks in order to pursue these goals. It will focus on the transmission mechanisms of monetary policy and the differences between monetary policy rules and discretionary policy. It will also make connections between theoretical economic concepts and current real world issues.
LernzielThis lecture will introduce the fundamentals of monetary economics and explain the working and impact of monetary policy.
LiteraturThe course will be based on chapters of:
Mishkin, Frederic S. (2015), The Economics of Money, Banking and Financial Markets 11th edition, Pearson. ISBN 10: 1-29-209418-4 ISBN 13: 978-1-292-09418-2
Voraussetzungen / BesonderesBasic knowledge in international economics and a good background in macroeconomics. The course website can be found at: Link
Environmental Science
NummerTitelTypECTSUmfangDozierende
701-0535-00LEnvironmental Soil Physics/Vadose Zone Hydrology Information W3 KP2G + 2UD. Or
KurzbeschreibungThe course provides theoretical and practical foundations for understanding and characterizing physical and transport properties of soils/ near-surface earth materials, and quantifying hydrological processes and fluxes of mass and energy at multiple scales. Emphasis is given to land-atmosphere interactions, the role of plants on hydrological cycles, and biophysical processes in soils.
LernzielStudents are able to
- characterize quantitative knowledge needed to measure and parameterize structural, flow and transport properties of partially-saturated porous media.
- quantify driving forces and resulting fluxes of water, solute, and heat in soils.
- apply modern measurement methods and analytical tools for hydrological data collection
- conduct and interpret a limited number of experimental studies
- explain links between physical processes in the vadose-zone and major societal and environmental challenges
InhaltWeeks 1 to 3: Physical Properties of Soils and Other Porous Media – Units and dimensions, definitions and basic mass-volume relationships between the solid, liquid and gaseous phases; soil texture; particle size distributions; surface area; soil structure. Soil colloids and clay behavior

Soil Water Content and its Measurement - Definitions; measurement methods - gravimetric, neutron scattering, gamma attenuation; and time domain reflectometry; soil water storage and water balance.

Weeks 4 to 5: Soil Water Retention and Potential (Hydrostatics) - The energy state of soil water; total water potential and its components; properties of water (molecular, surface tension, and capillary rise); modern aspects of capillarity in porous media; units and calculations and measurement of equilibrium soil water potential components; soil water characteristic curves definitions and measurements; parametric models; hysteresis. Modern aspects of capillarity

Demo-Lab: Laboratory methods for determination of soil water characteristic curve (SWC), sensor pairing

Weeks 6 to 9: Water Flow in Soil - Hydrodynamics:
Part 1 - Laminar flow in tubes (Poiseuille's Law); Darcy's Law, conditions and states of flow; saturated flow; hydraulic conductivity and its measurement.

Lab #1: Measurement of saturated hydraulic conductivity in uniform and layered soil columns using the constant head method.

Part 2 - Unsaturated steady state flow; unsaturated hydraulic conductivity models and applications; non-steady flow and Richard’s Eq.; approximate solutions to infiltration (Green-Ampt, Philip); field methods for estimating soil hydraulic properties.
Midterm exam

Lab #2: Measurement of vertical infiltration into dry soil column - Green-Ampt, and Philip's approximations; infiltration rates and wetting front propagation.

Part 3 - Use of Hydrus model for simulation of unsaturated flow


Week 10 to 11: Energy Balance and Land Atmosphere Interactions - Radiation and energy balance; evapotranspiration definitions and estimation; transpiration, plant development and transpirtation coefficients – small and large scale influences on hydrological cycle; surface evaporation.

Week 12 to 13: Solute Transport in Soils – Transport mechanisms of solutes in porous media; breakthrough curves; convection-dispersion eq.; solutions for pulse and step solute application; parameter estimation; salt balance.

Lab #3: Miscible displacement and breakthrough curves for a conservative tracer through a column; data analysis and transport parameter estimation.

Additional topics:

Temperature and Heat Flow in Porous Media - Soil thermal properties; steady state heat flow; nonsteady heat flow; estimation of thermal properties; engineering applications.

Biological Processes in the Vaodse Zone – An overview of below-ground biological activity (plant roots, microbial, etc.); interplay between physical and biological processes. Focus on soil-atmosphere gaseous exchange; and challenges for bio- and phytoremediation.
SkriptClassnotes on website: Vadose Zone Hydrology, by Or D., J.M. Wraith, and M. Tuller
(available at the beginning of the semester)
Link
LiteraturSupplemental textbook (not mandatory) -Environmental Soil Physics, by: D. Hillel
Finance
NummerTitelTypECTSUmfangDozierende
401-8905-00LFinancial Engineering (University of Zurich)
Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MFOEC103

Beachten Sie die Einschreibungstermine an der UZH: Link
W4.5 KP3GUni-Dozierende
KurzbeschreibungThis lecture is intended for students who would like to learn more on equity derivatives modelling and pricing.
LernzielQuantitative models for European option pricing (including stochastic
volatility and jump models), volatility and variance derivatives,
American and exotic options.
InhaltAfter introducing fundamental
concepts of mathematical finance including no-arbitrage, portfolio
replication and risk-neutral measure, we will present the main models
that can be used for pricing and hedging European options e.g. Black-
Scholes model, stochastic and jump-diffusion models, and highlight their
assumptions and limitations. We will cover several types of derivatives
such as European and American options, Barrier options and Variance-
Swaps. Basic knowledge in probability theory and stochastic calculus is
required. Besides attending class, we strongly encourage students to
stay informed on financial matters, especially by reading daily
financial newspapers such as the Financial Times or the Wall Street
Journal.
SkriptScript.
Voraussetzungen / BesonderesBasic knowledge of probability theory and stochastic calculus.
Asset Pricing.
401-8913-00LAdvanced Corporate Finance I (University of Zurich)
Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MOEC0455

Beachten Sie die Einschreibungstermine an der UZH: Link
W6 KP4GUni-Dozierende
KurzbeschreibungThis course develops and refines tools for evaluating investments (capital budgeting), capital structure, and corporate securities. The course seeks to deepen students' understanding of the link between corporate finance theory and practice.
LernzielThis course develops and refines tools for evaluating investments
(capital budgeting), capital structure, and corporate securities. With
respect to capital structure, we start with the famous Miller and
Modigliani irrelevance proposition and then move on to study the effects
of taxes, bankruptcy costs, information asymmetries between firms and
the capital markets, and agency costs. In this context, we will also
study how leverage affects some central financial ratios that are often
used in practice to assess firms and their stock. Other topics include
corporate cash holdings, the use and pricing of convertible bonds, and
risk management. The latter two topics involve option pricing. With
respect to capital budgeting, the course pays special attention to tax
effects in valuation, including in the estimation of the cost of
capital. We will also study payout policy (dividends and share
repurchases). The course seeks to deepen students' understanding of the
link between corporate finance theory and practice. Various cases will
be assigned to help reach this objective.
InhaltTopics covered
1. Capital structure: Perfect markets and irrelevance
2. Risk, leverage, taxes, and the cost of capital
3. Leverage and financial ratios
4. Payout policy: Dividends and share repurchases
5. Capital structure: Taxes and bankruptcy costs
6. Capital structure: Information asymmetries, agency costs, cash holdings
7. Valuation: DCF, adjusted present value and WACC
8. Valuation using options
9. The use and pricing of convertible bonds
10. Corporate risk management
Voraussetzungen / BesonderesThis course replaces "Advanced Corporate Finance I" (MOEC0288), which will be discontinued from HS16.
Image Processing and Computer Vision
NummerTitelTypECTSUmfangDozierende
227-0447-00LImage Analysis and Computer Vision Information W6 KP3V + 1UL. Van Gool, O. Göksel, E. Konukoglu
KurzbeschreibungLight and perception. Digital image formation. Image enhancement and feature extraction. Unitary transformations. Color and texture. Image segmentation and deformable shape matching. Motion extraction and tracking. 3D data extraction. Invariant features. Specific object recognition and object class recognition.
LernzielOverview of the most important concepts of image formation, perception and analysis, and Computer Vision. Gaining own experience through practical computer and programming exercises.
InhaltThe first part of the course starts off from an overview of existing and emerging applications that need computer vision. It shows that the realm of image processing is no longer restricted to the factory floor, but is entering several fields of our daily life. First it is investigated how the parameters of the electromagnetic waves are related to our perception. Also the interaction of light with matter is considered. The most important hardware components of technical vision systems, such as cameras, optical devices and illumination sources are discussed. The course then turns to the steps that are necessary to arrive at the discrete images that serve as input to algorithms. The next part describes necessary preprocessing steps of image analysis, that enhance image quality and/or detect specific features. Linear and non-linear filters are introduced for that purpose. The course will continue by analyzing procedures allowing to extract additional types of basic information from multiple images, with motion and depth as two important examples. The estimation of image velocities (optical flow) will get due attention and methods for object tracking will be presented. Several techniques are discussed to extract three-dimensional information about objects and scenes. Finally, approaches for the recognition of specific objects as well as object classes will be discussed and analyzed.
SkriptCourse material Script, computer demonstrations, exercises and problem solutions
Voraussetzungen / BesonderesPrerequisites:
Basic concepts of mathematical analysis and linear algebra. The computer exercises are based on Linux and C.
The course language is English.
Information and Communication Technology
NummerTitelTypECTSUmfangDozierende
227-0427-00LSignal and Information Processing: Modeling, Filtering, LearningW6 KP4GH.‑A. Loeliger
KurzbeschreibungFundamentals in signal processing, detection/estimation, and machine learning.
I. Linear signal representation and approximation: Hilbert spaces, LMMSE estimation, regularization and sparsity.
II. Learning linear and nonlinear functions and filters: kernel methods, neural networks.
III. Structured statistical models: hidden Markov models, factor graphs, Kalman filter, parameter estimation.
LernzielThe course is an introduction to some basic topics in signal processing, detection/estimation theory, and machine learning.
InhaltPart I - Linear Signal Representation and Approximation: Hilbert spaces, least squares and LMMSE estimation, projection and estimation by linear filtering, learning linear functions and filters, L2 regularization, L1 regularization and sparsity, singular-value decomposition and pseudo-inverse, principal-components analysis.
Part II - Learning Nonlinear Functions: fundamentals of learning, neural networks, kernel methods.
Part III - Structured Statistical Models and Message Passing Algorithms: hidden Markov models, factor graphs, Gaussian message passing, Kalman filter and recursive least squares, Monte Carlo methods, parameter estimation, expectation maximisation, sparse Bayesian learning.
SkriptLecture notes.
Voraussetzungen / BesonderesPrerequisites:
- local bachelors: course "Discrete-Time and Statistical Signal Processing" (5. Sem.)
- others: solid basics in linear algebra and probability theory
227-0101-00LZeitdiskrete und statistische SignalverarbeitungW6 KP4GH.‑A. Loeliger
KurzbeschreibungDie Vorlesung vermittelt Grundlagen der digitalen Signalverarbeitung mit Betonung auf Anwendungen in der Nachrichtentechnik: zeitdiskrete lineare Filter, Egalisation, DFT, zeitdiskrete stochastische Prozesse, Grundbegriffe der Entscheidungs- und Schätztheorie, LMMSE-Schätzung und -Filterung, LMS-Algorithmus, Viterbi-Algorithmus.
LernzielDie Vorlesung vermittelt mathematische Grundlagen der digitalen Signalverarbeitung mit Betonung auf Anwendungen in der Nachrichtentechnik. Die zwei zentralen Themenkreise sind Linearität und Wahrscheinlichkeitsmodelle. Im ersten Teil wird das Verständnis von zeitdiskreten linearen Filtern vertieft. Im zweiten Teil werden zunächst die Grundlagen der Wahrscheinlichkeitsrechnung vertieft und zeitdiskrete stochastische Prozesse eingeführt. Nach einer Einführung in die Grundbegriffe der Entscheidungs- und Schätztheorie werden sodann praktische Verfahren wie LMMSE-Schätzung und -Filterung, der LMS-Algorithmus und der Viterbi-Algorithmus behandelt. Ein wiederkehrendes Leitmotiv sind Verfahren zur stabilen und robusten "Inversion" einer linearen Filterung.
Inhalt1. Zeitdiskrete lineare Systeme und Filter:
Zustandsraum-Darstellung, z-Transformation, Spektrum,
Dezimation und Interpolation, Entwurf von digitalen Filtern,
stabile Realisierungen und robuste Inversion.

2. Die diskrete Fourier-Transformation und ihre Anwendung zur digitalen Filterung.

3. Der statistische Ansatz:
Wahrscheinlichkeitsrechnung, Zufallsgrössen, zeitdiskrete stochastische Prozesse;
Entscheidungs- und Schätzprobleme: MAP, ML, Bayes, LMMSE;
Wiener-Filter, adaptive Filter (LMS), Viterbi-Algorithmus.
SkriptVorlesungsskript.
227-0417-00LInformation Theory IW6 KP4GA. Lapidoth
KurzbeschreibungThis course covers the basic concepts of information theory and of communication theory. Topics covered include the entropy rate of a source, mutual information, typical sequences, the asymptotic equi-partition property, Huffman coding, channel capacity, the channel coding theorem, the source-channel separation theorem, and feedback capacity.
LernzielThe fundamentals of Information Theory including Shannon's source coding and channel coding theorems
InhaltThe entropy rate of a source, Typical sequences, the asymptotic equi-partition property, the source coding theorem, Huffman coding, Arithmetic coding, channel capacity, the channel coding theorem, the source-channel separation theorem, feedback capacity
LiteraturT.M. Cover and J. Thomas, Elements of Information Theory (second edition)
Material Modelling and Simulation
NummerTitelTypECTSUmfangDozierende
327-1201-00LTransport Phenomena I Information W4 KP4GH. C. Öttinger
KurzbeschreibungPhenomenological approach to "Transport Phenomena" based on balance equations supplemented by thermodynamic considerations to formulate the undetermined fluxes in the local species mass, momentum, and energy balance equations; fundamentals, applications, and simulations
LernzielThe teaching goals of this course are on five different levels:
(1) Deep understanding of fundamentals: local balance equations, constitutive equations for fluxes, entropy balance, interfaces, idea of dimensionless numbers, ...
(2) Ability to use the fundamental concepts in applications
(3) Insight into the role of boundary conditions
(4) Knowledge of a number of applications
(5) Flavor of numerical techniques: finite elements, finite differences, lattice Boltzmann, Brownian dynamics, ...
InhaltApproach to Transport Phenomena
Diffusion Equation
Brownian Dynamics
Refreshing Topics in Equilibrium Thermodynamics
Balance Equations
Forces and Fluxes
Measuring Transport Coefficients
Pressure-Driven Flows
Driven Separations
Complex Fluids
SkriptA detailed manuscript is provided; this manuscript will be developed into a book entitled "A Modern Course in Transport Phenomena" by David C. Venerus and Hans Christian Öttinger
Literatur1. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd Ed. (Wiley, 2001)
2. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, 2nd Ed. (Dover, 1984)
3. W. M. Deen, Analysis of Transport Phenomena (Oxford University Press, 1998)
4. R. B. Bird, Five Decades of Transport Phenomena (Review Article), AIChE J. 50 (2004) 273-287
Voraussetzungen / BesonderesComplex numbers. Vector analysis (integrability; Gauss' divergence theorem). Laplace and Fourier transforms. Ordinary differential equations (basic ideas). Linear algebra (matrices; functions of matrices; eigenvectors and eigenvalues; eigenfunctions). Probability theory (Gaussian distributions; Poisson distributions; averages; moments; variances; random variables). Numerical mathematics (integration). Equilibrium thermodynamics (Gibbs' fundamental equation; thermodynamic potentials; Legendre transforms). Maxwell equations. Programming and simulation techniques (Matlab, Monte Carlo simulations).
Quantum Chemistry
NummerTitelTypECTSUmfangDozierende
529-0003-00LAdvanced Quantum ChemistryW7 KP3GM. Reiher, S. Knecht
KurzbeschreibungAdvanced, but fundamental topics central to the understanding of theory in chemistry and for solving actual chemical problems with a computer.
Examples are:
* Operators derived from principles of relativistic quantum mechanics
* Relativistic effects + methods of relativistic quantum chemistry
* Open-shell molecules + spin-density functional theory
* New electron-correlation theories
LernzielThe aim of the course is to provide an in-depth knowledge of theory and method development in theoretical chemistry. It will be shown that this is necessary in order to be able to solve actual chemical problems on a computer with quantum chemical methods.

The relativistic re-derivation of all concepts known from (nonrelativistic) quantum mechanics and quantum-chemistry lectures will finally explain the form of all operators in the molecular Hamiltonian - usually postulated rather than deduced. From this, we derive operators needed for molecular spectroscopy (like those required by magnetic resonance spectroscopy). Implications of other assumptions in standard non-relativistic quantum chemistry shall be analyzed and understood, too. Examples are the Born-Oppenheimer approximation and the expansion of the electronic wave function in a set of pre-defined many-electron basis functions (Slater determinants). Overcoming these concepts, which are so natural to the theory of chemistry, will provide deeper insights into many-particle quantum mechanics. Also revisiting the workhorse of quantum chemistry, namely density functional theory, with an emphasis on open-shell electronic structures (radicals, transition-metal complexes) will contribute to this endeavor. It will be shown how these insights allow us to make more accurate predictions in chemistry in practice - at the frontier of research in theoretical chemistry.
Inhalt1) Introductory lecture: basics of quantum mechanics and quantum chemistry
2) Einstein's special theory of relativity and the (classical) electromagnetic interaction of two charged particles
3) Klein-Gordon and Dirac equation; the Dirac hydrogen atom
4) Numerical methods based on the Dirac-Fock-Coulomb Hamiltonian, two-component and scalar relativistic Hamiltonians
5) Response theory and molecular properties, derivation of property operators, Breit-Pauli-Hamiltonian
6) Relativistic effects in chemistry and the emergence of spin
7) Spin in density functional theory
8) New electron-correlation theories: Tensor network and matrix product states, the density matrix renormalization group
9) Quantum chemistry without the Born-Oppenheimer approximation
SkriptA set of detailed lecture notes will be provided, which will cover the whole course.
Literatur1) M. Reiher, A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH, 2014, 2nd edition
2) F. Schwabl: Quantenmechanik für Fortgeschrittene (QM II), Springer-Verlag, 1997
[english version available: F. Schwabl, Advanced Quantum Mechanics]
3) R. McWeeny: Methods of Molecular Quantum Mechanics, Academic Press, 1992
4) C. R. Jacob, M. Reiher, Spin in Density-Functional Theory, Int. J. Quantum Chem. 112 (2012) 3661
Link
5) K. H. Marti, M. Reiher, New Electron Correlation Theories for Transition Metal Chemistry, Phys. Chem. Chem. Phys. 13 (2011) 6750
Link
6) K.H. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in Quantum Chemistry, Z. Phys. Chem. 224 (2010) 583
Link
7) E. Mátyus, J. Hutter, U. Müller-Herold, M. Reiher, On the emergence of molecular structure, Phys. Rev. A 83 2011, 052512
Link

Note also the standard textbooks:
A) A. Szabo, N.S. Ostlund. Verlag, Dover Publications
B) I. N. Levine, Quantum Chemistry, Pearson
C) T. Helgaker, P. Jorgensen, J. Olsen: Molecular Electronic-Structure Theory, Wiley, 2000
D) R.G. Parr, W. Yang: Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1994
E) R.M. Dreizler, E.K.U. Gross: Density Functional Theory, Springer-Verlag, 1990
Voraussetzungen / BesonderesStrongly recommended (preparatory) courses are: quantum mechanics and quantum chemistry
Simulation of Semiconductor Devices
NummerTitelTypECTSUmfangDozierende
227-0157-00LSemiconductor Devices: Physical Bases and SimulationW4 KP3GA. Schenk
KurzbeschreibungThe course addresses the physical principles of modern semiconductor devices and the foundations of their modeling and numerical simulation. Necessary basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided. Computer simulations of the most important devices and of interesting physical effects supplement the lectures.
LernzielThe course aims at the understanding of the principle physics of modern semiconductor devices, of the foundations in the physical modeling of transport and its numerical simulation. During the course also basic knowledge on quantum-mechanics, semiconductor physics and device physics is provided.
InhaltThe main topics are: transport models for semiconductor devices (quantum transport, Boltzmann equation, drift-diffusion model, hydrodynamic model), physical characterization of silicon (intrinsic properties, scattering processes), mobility of cold and hot carriers, recombination (Shockley-Read-Hall statistics, Auger recombination), impact ionization, metal-semiconductor contact, metal-insulator-semiconductor structure, and heterojunctions.
The exercises are focussed on the theory and the basic understanding of the operation of special devices, as single-electron transistor, resonant tunneling diode, pn-diode, bipolar transistor, MOSFET, and laser. Numerical simulations of such devices are performed with an advanced simulation package (Sentaurus-Synopsys). This enables to understand the physical effects by means of computer experiments.
SkriptThe script (in book style) can be downloaded from: Link˜schenk/vorlesung.
LiteraturThe script (in book style) is sufficient. Further reading will be recommended in the lecture.
Voraussetzungen / BesonderesQualifications: Physics I+II, Semiconductor devices (4. semester).
Systems Design
NummerTitelTypECTSUmfangDozierende
363-0541-00LSystems Dynamics and ComplexityW3 KP3GF. Schweitzer, G. Casiraghi, V. Nanumyan
KurzbeschreibungFinding solutions: what is complexity, problem solving cycle.

Implementing solutions: project management, critical path method, quality control feedback loop.

Controlling solutions: Vensim software, feedback cycles, control parameters, instabilities, chaos, oscillations and cycles, supply and demand, production functions, investment and consumption
LernzielA successful participant of the course is able to:
- understand why most real problems are not simple, but require solution methods that go beyond algorithmic and mathematical approaches
- apply the problem solving cycle as a systematic approach to identify problems and their solutions
- calculate project schedules according to the critical path method
- setup and run systems dynamics models by means of the Vensim software
- identify feedback cycles and reasons for unintended systems behavior
- analyse the stability of nonlinear dynamical systems and apply this to macroeconomic dynamics
InhaltWhy are problems not simple? Why do some systems behave in an unintended way? How can we model and control their dynamics? The course provides answers to these questions by using a broad range of methods encompassing systems oriented management, classical systems dynamics, nonlinear dynamics and macroeconomic modeling.
The course is structured along three main tasks:
1. Finding solutions
2. Implementing solutions
3. Controlling solutions

PART 1 introduces complexity as a system immanent property that cannot be simplified. It introduces the problem solving cycle, used in systems oriented management, as an approach to structure problems and to find solutions.

PART 2 discusses selected problems of project management when implementing solutions. Methods for identifying the critical path of subtasks in a project and for calculating the allocation of resources are provided. The role of quality control as an additional feedback loop and the consequences of small changes are discussed.

PART 3, by far the largest part of the course, provides more insight into the dynamics of existing systems. Examples come from biology (population dynamics), management (inventory modeling, technology adoption, production systems) and economics (supply and demand, investment and consumption). For systems dynamics models, the software program VENSIM is used to evaluate the dynamics. For economic models analytical approaches, also used in nonlinear dynamics and control theory, are applied. These together provide a systematic understanding of the role of feedback loops and instabilities in the dynamics of systems. Emphasis is on oscillating phenomena, such as business cycles and other life cycles.

Weekly self-study tasks are used to apply the concepts introduced in the lectures and to come to grips with the software program VENSIM.
SkriptThe lecture slides are provided as handouts - including notes and literature sources - to registered students only. All material is to be found on the Moodle platform. More details during the first lecture
Voraussetzungen / BesonderesSelf-study tasks (discussion exercises, Vensim exercises) are provided as home work. Weekly exercise sessions (45 min) are used to discuss selected solutions. Regular participation in the exercises is an efficient way to understand the concepts relevant for the final exam.
Theoretical Physics
Im Master-Studiengang Angewandte Mathematik ist auch 402-0205-00L Quantenmechanik I als Fach im Vertiefungsgebiet Theoretical Physics anrechenbar, aber nur unter der Bedingung, dass 402-0224-00L Theoretische Physik nicht angerechnet wurde oder wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link).
NummerTitelTypECTSUmfangDozierende
402-0809-00LIntroduction to Computational PhysicsW8 KP2V + 2UH. J. Herrmann
KurzbeschreibungDiese Vorlesung bietet eine Einführung in Computersimulationsmethoden für physikalische Probleme und deren Implementierung auf PCs und Supercomputern: klassische Bewegungsgleichungen, partielle Differentialgleichungen (Wellengleichung, Diffussionsgleichung, Maxwell-Gleichungen), Monte Carlo Simulation, Perkolation, Phasenübergänge
Lernziel
InhaltEinführung in die rechnergestützte Simulation physikalischer Probleme. Anhand einfacher Modelle aus der klassischen Mechanik, Elektrodynamik und statistischen Mechanik sowie interdisziplinären Anwendungen werden die wichtigsten objektorientierten Programmiermethoden für numerische Simulationen (überwiegend in C++) erläutert. Daneben wird eine Einführung in die Programmierung von Vektorsupercomputern und parallelen Rechnern, sowie ein Überblick über vorhandene Softwarebibliotheken für numerische Simulationen geboten.
Voraussetzungen / BesonderesVorlesung und Uebung in Englisch, Pruefung wahlweise auf Deutsch oder Englisch
402-2203-01LAllgemeine Mechanik Information W7 KP4V + 2UG. M. Graf
KurzbeschreibungBegriffliche 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
402-0861-00LStatistical PhysicsW10 KP4V + 2UG. Blatter
KurzbeschreibungThis lecture covers the concepts of classical and quantum statistical physics, and some aspects of kinetic gas theory and hydrodynamics. In a more advanced part degenerate Fermions, Bose-Einstein condensation, real Bose gases, magnetism, general mean field theory and critical phenomena will be addressed.
LernzielThis lecture gives an introduction in the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.
InhaltBasics of phenomenological thermodynamics, three laws of thermodynamics.
Basics of kinetic gas theory: conservation laws, H-theorem, Boltzmann-Equations, Maxwell distribution.
Hydrodynamics.
Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons.
Degenerate fermions: Fermi gas, electrons in magnetic field.
Bosons: Bose-Einstein condensation, Bogoliubov theory, superfluidity.
Mean field and Landau theory: Ising-, XY-, Heisenberg models, Landau theory of phase transitions, fluctuations.
Critical phenomena: mean field, series expansions, scaling behavior, universality.
Renormalization group: fixed points, simple models.
SkriptLecture notes available in german.
LiteraturNo specific book is used for the course. Relevant literature will be given in the course.
402-0843-00LQuantum Field Theory I Information W10 KP4V + 2UC. Anastasiou
KurzbeschreibungThis course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity.
Topics include:
- Relativistic quantum mechanics
- Quantisation of bosonic and fermionic fields
- Interactions in perturbation theory
- Scattering processes and decays
- Radiative corrections
LernzielThe goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics.
402-0830-00LGeneral Relativity Information W10 KP4V + 2UP. Jetzer
KurzbeschreibungManifold, 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.
LernzielBasic understanding of general relativity, its mathematical foundations, and some of the interesting phenomena it predicts.
LiteraturSuggested 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
» Wahlfächer Theoretische Physik
Transportation Science
NummerTitelTypECTSUmfangDozierende
101-0417-00LTransport Planning MethodsW6 KP4GK. W. Axhausen
KurzbeschreibungDiese Veranstaltung vermittelt das notwendige Wissen, um verkehrsplanerische Modelle zu entwerfen, welche die Lösung gegebener Planungsaufgaben unterstützen. Dabei wird das komplexe Vorhersageproblem in Teilprobleme zerlegt.
Der Kurs besteht aus einem Vorlesungsteil, in dem das theoretische Wissen vermittelt wird und einem angewandten Teil, in dem die Studierenden ein eigenes Modell erstellen.
Lernziel- Kenntnis der gängigen Verfahren der Verkehrsplanung
- Fähigkeit zur selbständigen Entwicklung eines Verkehrsmodels, welches fähig ist gestellte Aufgaben / Fragen zu lösen / zu beantworten
- Verständnis der Implementation der in der Verkehrsplanung am häufigsten verwendeten Algorithmen.
InhaltDiese Veranstaltung vermittelt das notwendige Wissen, um verkehrsplanerische Modelle zu entwerfen, welche die Lösung gegebener Planungsaufgaben unterstützen. Mögliche solche Aufgaben sind die Abschätzung des Verkehrsaufkommens, die Vorhersage der zu erwartenden Nutzung von neuen Linien des öffentlichen Verkehrs und die Beurteilung von Effekten durch Infrastrukturprojekte oder veränderte Betriebsreglemente auf z.B. die Entwicklung der Emissionen einer Stadt.

Um die Aufgabe zu lösen, wird das komplexe Vorhersageproblem in Teilprobleme zerlegt. Zur Lösung der Teilaufgaben kommen verschiedene Algorithmen zum Einsatz, wie Randausgleichsverfahren, kürzeste Wege Algorithmen und die Methode der sukzessiven Mittelwerte.

Der Kurs besteht aus einem Vorlesungsteil, in dem das theoretische Wissen vermittelt wird und einem angewandten Teil, in dem die Studierenden ein eigenes Modell erstellen. Dieser Teil findet in Form eines Tutorials statt und beinhaltet die Entwicklung eines Computerprogramms. Der Programmier-Teil ist gut geführt und ausdrücklich geeignet für Studierende mit wenig Programmiererfahrung.
SkriptDie Folien zur Vorlesung werden elektronisch zur Verfügung gestellt.
LiteraturWillumsen, P. and J. de D. Ortuzar (2003) Modelling Transport, Wiley, Chichester.

Cascetta, E. (2001) Transportation Systems Engineering: Theory and Methods, Kluwer Academic Publishers, Dordrecht.

Sheffi, Y. (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice Hall, Englewood Cliffs.

Schnabel, W. and D. Lohse (1997) Verkehrsplanung, 2. edn., vol. 2 of Grundlagen der Strassenverkehrstechnik und der Verkehrsplanung, Verlag für Bauwesen, Berlin.
Seminare und Semesterarbeiten
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.
NummerTitelTypECTSUmfangDozierende
401-4580-66LCharacteristic Classes and Elliptic Genus Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 12
W4 KP2SQ. Chen, G. Felder
KurzbeschreibungCharacteristic classes, spin structures and Dirac opertor, applications of the Atiyah-Signer index theorem, elliptic genus and modular forms.
LernzielCharacteristic classes, spin structures and Dirac opertor, applications of the Atiyah-Signer index theorem, elliptic genus and modular forms.
InhaltTentative Syllabus

1. Vector bundles and differential forms (1 lectures)
2. Basics for Chacteristic classes such as Stiefel-Whitney classes, Wu Classes, Chern Classes and Pontryagin classes (3 lectures)
3. Spin structures and Dirac operators (2 lectures)
4. Atiyah-Singer Index theorem and its application (1-2 lectures)
5. Multiplicative sequences and various genera (1 lecture)
6. Elliptic genus and modular forms (1 lecture)
7. Miraculous cancellation formulas for Hirzebruch L genus (1 lecture)
8. Miscellaneous topics (1 lecture)
Literatur1. Characteristic Classes by Milnor
2. Differential Forms in Algebraic Topology by Bott & Tu
3. Manifolds and Modular Forms by Hirzebruch, Berger and Jung
Voraussetzungen / BesonderesPrerequisite: Algebraic Topology.
401-3570-66LAlgebraic Number Theory Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 12
W4 KP2SJ. Fresán
KurzbeschreibungMuch of the progress in algebraic number theory aimed at solving concrete Diophantine equations. At the heart of the problem lies the fact that the ring of integers of a number field does not have unique factorization. The "class group" measures how much this property fails. The seminar will present basic results around this invariant, including finiteness and the relation to zeta functions.
Lernziel
InhaltThe following topics will be covered:

- The quadratic reciprocity law
- The geometry of numbers
- Integral quadratic forms
- Number fields and rings of integers
- Finiteness of the class number
- Unique factorization of ideals
- The Dedekind zeta function of a number field and the class number formula

The seminar will be (probably) followed by a more advanced course on Class Field Theory on the Spring Semester.
LiteraturOur basic reference will be chapters I and VII of Neukirch's book "Algebraic number theory" (Grundlehren Math. Wiss. 322. Springer-Verlag, Berlin, 1999). Additional references will be given at the beginning of the seminar.
Voraussetzungen / BesonderesBasic knowledge of algebraic structures (groups, rings, fields) and Galois theory, at the level of Algebra I and II. More advanced topics will be explained when needed.
401-3180-66LHomological Algebra Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 12
W4 KP2SC. Busch
KurzbeschreibungBasic concepts of homological algebra, homology and cohomology of groups.
Lernziel
LiteraturPeter J. Hilton, Urs Stammbach: A Course in Homological Algebra, Second Edition, Graduate Texts in Mathematics 4, Springer 1997
Kenneth S. Brown: Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag 1982
Charles A. Weibel: An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994
401-4600-66LStudent Seminar in Probability Belegung eingeschränkt - Details anzeigen
Limited number of participants.
Registration to the seminar will only be effective once confirmed by email from the organizers.
W4 KP2SA.‑S. Sznitman, J. Bertoin, P. Nolin, W. Werner
Kurzbeschreibung
Lernziel
InhaltThe seminar is centered around a topic in probability theory which changes each semester.
Voraussetzungen / BesonderesThe student seminar in probability is held at times at the undergraduate level (typically during the spring term) and at times at the graduate level (typically during the autumn term). The themes vary each semester.

The number of participants to the seminar is limited. Registration to the seminar will only be effective once confirmed by email from the organizers.
401-3640-66LMonte Carlo and Quasi-Monte Carlo Methods: Mathematical and Numerical Analysis Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 6
W4 KP2SC. Schwab
KurzbeschreibungIntroduction and current research topics in the theory and implementation of Monte Carlo and quasi-Monte Carlo methods and applications.
Lernziel
Voraussetzungen / BesonderesPrerequisites:
Completed courses
Numerical Analysis of Elliptic/ Parabolic PDEs,
or Numerical Analysis of Hyperbolic PDEs,
or Numerical Analysis of Stochastic ODEs,
and FAI, Probability Theory I.
401-3650-66LNumerical Analysis Seminar: Measure Theoretic Tools for Analyzing and Approximating Nonlinear PDEs Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 6
W4 KP2SF. Weber
KurzbeschreibungThe seminar covers measure theoretic tools used for the analysis and approximation of nonlinear hyperbolic partial differential equations. In particular, we will discuss Young measures, compensated compactness, weak-strong uniqueness and algorithms for the approximation of measure-valued solutions. The participants will present individual topics based on the study of research papers.
Lernziel- To learn some measure theoretic tools for the analysis and approximation of nonlinear PDEs.

- To read and understand a research paper and present it in an understandable way to other students.
InhaltPartial differential equations can be used to model an abundance of natural and physical phenomena, as well as industrial processes. Many of the more sophisticated and more realistic models involve nonlinear PDEs, among others, PDEs in fluid dynamics, astrophysics, elasticity or weather modeling. The solutions to these often exhibit complex structures, such as shocks, oscillations, singularities that are difficult to deal with mathematically and numerically. In our seminar we aim to get a better understanding of the difficulties that arise when dealing with nonlinear PDEs. In particular, we will discuss problems related to the PDEs of fluid dynamics. Solutions to these equations may exhibit shocks and oscillations, and have less regularity than what the definition of a classical solution requires. Therefore, the solution concept has to be relaxed. One way of doing this, is to look for solutions in the space of measures instead of actual functions. Our goal in this seminar is to try to understand this concept better by studying research papers related to this issue.
Specifically, we will discuss weak convergence in general, the notion of Young measures as a means to represent weak limits of nonlinear functions, and its application to compensated compactness, existence of solutions to scalar hyperbolic conservation laws, Euler equations, turbulence and statistical solutions of Navier-Stokes equations. We will also discuss algorithms to approximate solutions in the space of measures.
We are open to extend the list of topics by others that are of special interests to the attending students.
LiteraturJ. M. Ball. A version of the fundamental theorem for Young measures (1989).

Yann Brenier, Camillo De Lellis, and László Szekelyhidi, Jr. Weak-strong uniqueness for measure-valued solutions (2011).

Camillo De Lellis and László Szekelyhidi, Jr. The Euler equations as a differential inclusion (2009).

Ronald J. DiPerna. Measure-valued solutions to conservation laws (1985).

Ronald J. DiPerna and Andrew J. Majda. Concentrations in regularizations for 2-D incompressible flow (1987).

Lawrence C. Evans. Weak convergence methods for nonlinear partial differential equations.

Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor. Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws (2015).

A. Szepessy. An existence result for scalar conservation laws using measure-valued solutions (1989).
Voraussetzungen / BesonderesGood knowledge of real/functional analysis required, knowledge of hyperbolic partial differential equations and/or numerical analysis of advantage.
401-4660-66LSeminar in Applied Harmonic Analysis: Frame Theory and Phase Retrieval Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 10
W4 KP2SR. Alaifari
Kurzbeschreibung
Lernziel
401-3910-66LSeminar in Mathematical Finance: Mean Field Games Belegung eingeschränkt - Details anzeigen
Maximale Teilnehmerzahl: 15
W4 KP2SM. Burzoni, M. Soner
KurzbeschreibungThe 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.
LernzielThis course aims to give a broad understanding of the basic ideas of Mean Field Games, the main mathematical tools and the possible applications.
InhaltWe 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.
Literatur1) 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 / BesonderesBasic courses in analysis including basic knowledge of ordinary/partial differential equations.
Basic knowledge of stochastic analysis including Brownian Motion and stochastic differential equations.
Semesterarbeiten
Es gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
NummerTitelTypECTSUmfangDozierende
401-3750-01LSemesterarbeit Belegung eingeschränkt - Details anzeigen
Sie können diese Lerneinheit nicht selber in myStudies belegen, sondern müssen sich beim Studiensekretariat via Online-Anmeldeformular dafür registrieren.
Bedingungen und Anmeldeformular unter Link
(Danach erfolgt die Belegung durch das Studiensekretariat.)
W8 KP11AProfessor/innen
KurzbeschreibungSemesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / BesonderesEs gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
401-3750-02LSemesterarbeit Belegung eingeschränkt - Details anzeigen
Sie können diese Lerneinheit nicht selber in myStudies belegen, sondern müssen sich beim Studiensekretariat via Online-Anmeldeformular dafür registrieren.
Bedingungen und Anmeldeformular unter Link
(Danach erfolgt die Belegung durch das Studiensekretariat.)
W8 KP11AProfessor/innen
KurzbeschreibungSemesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / BesonderesEs gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
401-3750-03LSemesterarbeit Belegung eingeschränkt - Details anzeigen
Sie können diese Lerneinheit nicht selber in myStudies belegen, sondern müssen sich beim Studiensekretariat via Online-Anmeldeformular dafür registrieren.
Bedingungen und Anmeldeformular unter Link
(Danach erfolgt die Belegung durch das Studiensekretariat.)
W8 KP11AProfessor/innen
KurzbeschreibungSemesterarbeiten dienen der Vertiefung in einem spezifischen Fachbereich; die Themen werden den Studierenden zur individuellen Auswahl angeboten. Semesterarbeiten sollen die Fähigkeit der Studierenden zu selbständiger mathematischer Tätigkeit und zur schriftlichen Darstellung mathematischer Ergebnisse fördern.
Lernziel
Voraussetzungen / BesonderesEs gibt mehrere Lerneinheiten "Semesterarbeit", die alle gleichwertig sind. Wenn Sie im Lauf Ihres Studiums mehrere Semesterarbeiten schreiben, wählen Sie jeweils verschiedene Nummern aus, um wieder Kreditpunkte erhalten zu können.
GESS Wissenschaft im Kontext
» Empfehlungen aus dem Bereich Wissenschaft im Kontext (Typ B) für das D-MATH.
» siehe Studiengang Wissenschaft im Kontext: Typ A: Förderung allgemeiner Reflexionsfähigkeiten
» siehe Studiengang Wissenschaft im Kontext: Sprachkurse ETH/UZH
Master-Arbeit
NummerTitelTypECTSUmfangDozierende
401-2000-00LScientific Works in Mathematics Information
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
O0 KPE. Kowalski
KurzbeschreibungIntroduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)
LernzielLearn 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
SkriptMoodle of the Mathematics Library: Link
Voraussetzungen / BesonderesThis course is completed by the optional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library. For more details see: Link
401-4990-00LMaster's Thesis Belegung eingeschränkt - Details anzeigen
Zur Master-Arbeit wird nur zugelassen, wer:
a. das Bachelor-Studium erfolgreich abgeschlossen hat;
b. allfällige Auflagen für die Zulassung zum Master-Studiengang erfüllt hat.

Sie können diese Lerneinheit nicht selber in myStudies belegen, sondern müssen sich beim Studiensekretariat via Online-Anmeldeformular dafür registrieren.
Bedingungen und Anmeldeformular unter Link
(Danach erfolgt die Belegung durch das Studiensekretariat.)
O30 KP57DProfessor/innen
KurzbeschreibungDie Master-Arbeit bildet den Abschluss des Studiengangs. In der Master-Arbeit wird eine grössere mathematische Aufgabe selbständig behandelt. Sie umfasst in der Regel das Studium vorhandener Fachliteratur, die Lösung weiterer damit verbundener Fragen sowie die schriftliche Darstellung der Ergebnisse.
Lernziel
Zusätzliche Veranstaltungen
NummerTitelTypECTSUmfangDozierende
401-5000-00LZurich Colloquium in Mathematics Information E-0 KPW. Werner, P. L. Bühlmann, M. Burger, S. Mishra, R. Pandharipande, Uni-Dozierende
KurzbeschreibungThe 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
401-5990-00LZurich Graduate Colloquium (University of Zurich) Information
Der Kurs muss direkt an der UZH belegt werden.
UZH Modulkürzel: MAT075

Beachten Sie die Einschreibungstermine an der UZH: Link
E-0 KP1KUni-Dozierende
KurzbeschreibungThe 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
401-5110-00LNumber Theory Seminar Information E-0 KP1KÖ. Imamoglu, P. S. Jossen, E. Kowalski, P. D. Nelson, R. Pink, G. Wüstholz
KurzbeschreibungResearch colloquium
Lernziel
401-5350-00LAnalysis Seminar Information E-0 KP1KM. Struwe, A. Carlotto, D. Christodoulou, F. Da Lio, A. Figalli, N. Hungerbühler, T. Ilmanen, T. Kappeler, T. Rivière, D. A. Salamon
KurzbeschreibungResearch colloquium
Lernziel
401-5530-00LGeometry Seminar Information E-0 KP1KM. Burger, M. Einsiedler, U. Lang, Uni-Dozierende
KurzbeschreibungResearch colloquium
Lernziel
401-5580-00LSymplectic Geometry Seminar Information E-0 KP2KD. A. Salamon, P. Biran, A. Cannas da Silva
KurzbeschreibungResearch colloquium
Lernziel
401-5330-00LTalks in Mathematical Physics Information E-0 KP1KA. Cattaneo, G. Felder, M. Gaberdiel, G. M. Graf, H. Knörrer, T. H. Willwacher, Uni-Dozierende
KurzbeschreibungResearch colloquium
Lernziel
401-5650-00LZurich Colloquium in Applied and Computational Mathematics Information E-0 KP2KR. Abgrall, H. Ammari, R. Hiptmair, A. Jentzen, S. Mishra, S. Sauter, C. Schwab
KurzbeschreibungResearch colloquium
Lernziel
401-5600-00LSeminar on Stochastic Processes Information E-0 KP1KJ. Bertoin, A. Nikeghbali, P. Nolin, B. D. Schlein, A.‑S. Sznitman, W. Werner
KurzbeschreibungResearch colloquium
Lernziel
401-5620-00LResearch Seminar on Statistics Information E-0 KP2KP. L. Bühlmann, L. Held, T. Hothorn, D. Kozbur, M. H. Maathuis, N. Meinshausen, M. Wolf
KurzbeschreibungResearch colloquium
Lernziel
401-5640-00LZüKoSt: Seminar on Applied Statistics Information E-0 KP1KM. Kalisch, P. L. Bühlmann, R. Furrer, L. Held, T. Hothorn, M. H. Maathuis, M. Mächler, L. Meier, N. Meinshausen, M. Robinson, C. Strobl
KurzbeschreibungEtwa 5 Vorträge zur angewandten Statistik.
LernzielKennenlernen von statistischen Methoden in ihrer Anwendung in verschiedenen Anwendungsgebieten.
InhaltIn etwa 5 Einzelvorträgen pro Semester werden Methoden der Statistik einzeln oder überblicksartig vorgestellt, oder es werden Probleme und Problemtypen aus einzelnen Anwendungsgebieten besprochen.
Voraussetzungen / BesonderesDies ist keine Vorlesung. Es wird keine Prüfung durchgeführt, und es werden keine Kreditpunkte vergeben.
Nach besonderem Programm:
Link
Lehrsprache ist Englisch oder Deutsch je nach ReferentIn.
401-5910-00LTalks in Financial and Insurance Mathematics Information E-0 KP1KP. Cheridito, M. Schweizer, M. Soner, J. Teichmann, M. V. Wüthrich
KurzbeschreibungResearch colloquium
Lernziel
InhaltRegular research talks on various topics in mathematical finance and actuarial mathematics
401-5900-00LOptimization Seminar Information E-0 KP1KR. Weismantel, R. Zenklusen
KurzbeschreibungLectures on current topics in optimization
LernzielExpose graduate students to ongoing research acitivites (including applications) in the domain of otimization.
InhaltThis seminar is a forum for researchers interested in optimization theory and its applications. Speakers are expected to stimulate discussions on theoretical and applied aspects of optimization and related subjects. The focus is on efficient algorithms for continuous and discrete optimization problems, complexity analysis of algorithms and associated decision problems, approximation algorithms, mathematical modeling and solution procedures for real-world optimization problems in science, engineering, industries, public sectors etc.
401-5960-00LKolloquium über Mathematik, Informatik und Unterricht Information
Fachdidaktik für Mathematik- und Informatiklehrpersonen.
E-0 KPN. Hungerbühler, M. Akveld, J. Hromkovic, H. Klemenz
KurzbeschreibungDidaktikkolloquium
Lernziel
402-0101-00LThe Zurich Physics Colloquium Information E-0 KP1KR. Renner, G. Aeppli, C. Anastasiou, N. Beisert, G. Blatter, S. Cantalupo, M. Carollo, C. Degen, G. Dissertori, K. Ensslin, T. Esslinger, J. Faist, M. Gaberdiel, 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, K. Schawinski, T. C. Schulthess, M. Sigrist, M. Troyer, A. Vaterlaus, R. Wallny, A. Wallraff, W. Wegscheider, A. Zheludev, O. Zilberberg
KurzbeschreibungResearch colloquium
Lernziel
Voraussetzungen / BesonderesOccasionally, talks may be delivered in German.
402-0800-00LThe Zurich Theoretical Physics Colloquium Information E-0 KP1KS. Huber, C. Anastasiou, N. Beisert, G. Blatter, M. Gaberdiel, T. K. Gehrmann, G. M. Graf, P. Jetzer, L. M. Mayer, B. Moore, R. Renner, T. C. Schulthess, M. Sigrist, M. Troyer, O. Zilberberg, Uni-Dozierende
KurzbeschreibungResearch colloquium
LernzielThe 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.
251-0100-00LKolloquium für Informatik Information E-0 KP2KDozent/innen
KurzbeschreibungEingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
LernzielDas 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.
InhaltEingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.
252-4202-00LSeminar in Theoretical Computer Science Information E-2 KP2SE. Welzl, B. Gärtner, M. Hoffmann, J. Lengler, A. Steger, B. Sudakov
KurzbeschreibungPräsentation wichtiger und aktueller Arbeiten aus der theoretischen Informatik, sowie eigener Ergebnisse von Diplomanden und Doktoranden.
LernzielDas Lernziel ist, Studierende an die aktuelle Forschung heranzuführen und sie in die Lage zu versetzen, wissenschaftliche Arbeiten zu lesen, zu verstehen, und zu präsentieren.
Auflagen-Lerneinheiten
Das untenstehende Lehrangebot gilt nur für MSc Studierende mit Zulassungsauflagen.
NummerTitelTypECTSUmfangDozierende
406-2004-AALAlgebra II
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-5 KP11RR. Pink
KurzbeschreibungGalois theory and Representations of finite groups, algebras.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
LernzielIntroduction to fundamentals of Galois theory, and representation theory of finite groups and algebras
InhaltFundamentals of Galois theory
Representation theory of finite groups and algebras
SkriptFor a summary of the content and exercises with solutions of my lecture course in FS2016 see:
Link
LiteraturS. Lang, Algebra, Springer Verlag
B.L. van der Waerden: Algebra I und II, Springer Verlag
I.R. Shafarevich, Basic notions of algebra, Springer verlag
G. Mislin: Algebra I, vdf Hochschulverlag
U. Stammbach: Algebra, in der Polybuchhandlung erhältlich
I. Stewart: Galois Theory, Chapman Hall (2008)
G. Wüstholz, Algebra, vieweg-Verlag, 2004
J-P. Serre, Linear representations of finite groups, Springer Verlag
Voraussetzungen / BesonderesAlgebra I
406-2005-AALAlgebra I and II
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-12 KP26RR. Pink
KurzbeschreibungIntroduction and development of some basic algebraic structures - groups, rings, fields including Galois theory, representations of finite groups, algebras.

The precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
Lernziel
InhaltBasic notions and examples of groups;
Subgroups, Quotient groups and Homomorphisms,
Group actions and applications

Basic notions and examples of rings;
Ring Homomorphisms,
ideals, and quotient rings, rings of fractions
Euclidean domains, Principal ideal domains, Unique factorization
domains

Basic notions and examples of fields;
Field extensions, Algebraic extensions, Classical straight edge and compass constructions

Fundamentals of Galois theory
Representation theory of finite groups and algebras
SkriptFor a summary of the content and exercises with solutions of my lecture courses in HS2015 and FS2016 see:
Link
Link
LiteraturS. Lang, Algebra, Springer Verlag
B.L. van der Waerden: Algebra I und II, Springer Verlag
I.R. Shafarevich, Basic notions of algebra, Springer verlag
G. Mislin: Algebra I, vdf Hochschulverlag
U. Stammbach: Algebra, in der Polybuchhandlung erhältlich
I. Stewart: Galois Theory, Chapman Hall (2008)
G. Wüstholz, Algebra, vieweg-Verlag, 2004
J-P. Serre, Linear representations of finite groups, Springer Verlag
406-2303-AALComplex Analysis
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-6 KP13RR. Pandharipande
KurzbeschreibungComplex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, conformal mappings, Riemann mapping theorem.
Lernziel
LiteraturL. 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.

R.Remmert: Theory of Complex Functions.. Springer Verlag

E.Hille: Analytic Function Theory. AMS Chelsea Publication
406-2284-AALMeasure and Integration
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-6 KP13RF. Da Lio
KurzbeschreibungIntroduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language.
LernzielBasic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral.
Literatur1. Lecture notes by Professor Michael Struwe (Link)
2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions"
3. Walter Rudin "Real and complex analysis"
4. R. Bartle The elements of Integration and Lebesgue Measure
5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. Link
406-2554-AALTopology
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-6 KP13RW. Werner
KurzbeschreibungTopological spaces, continuous maps, connectedness, compactness, separation axioms, metric spaces, quotient spaces, homotopy, fundamental group and covering spaces, van Kampen Theorem, surfaces and manifolds.
Lernziel
LiteraturKlaus Jänich: Topologie (Springer-Verlag)
Link
James Munkres: Topology (Prentice Hall)
William Massey: Algebraic Topology: an Introduction (Springer-Verlag)
Alan Hatcher: Algebraic Topology (Cambridge University Press)
Link
Voraussetzungen / BesonderesThe precise content changes with the examiner. Candidates must therefore contact the examiner in person before studying the material.
406-2604-AALProbability and Statistics
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-7 KP15RM. Soner
KurzbeschreibungIntroduction to probability and statistics with many examples, based on chapters from the books "Probability and Random Processes" by G. Grimmett and D. Stirzaker and "Mathematical Statistics and Data Analysis" by J. Rice.
LernzielThe goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. In addition to a mathematically rigorous treatment, also an intuitive understanding and familiarity with the ideas behind the definitions are emphasized. Measure theory is not used systematically, but it should become clear why and where measure theory is needed.
InhaltProbability:
Chapters 1-5 (Probabilities and events, Discrete and continuous random variables, Generating functions) and Sections 7.1-7.5 (Convergence of random variables) from the book "Probability and Random Processes". Most of this material is also covered in Chap. 1-5 of "Mathematical Statistics and Data Analysis", on a slightly easier level.

Statistics:
Sections 8.1 - 8.5 (Estimation of parameters), 9.1 - 9.4 (Testing Hypotheses), 11.1 - 11.3 (Comparing two samples) from "Mathematical Statistics and Data Analysis".
LiteraturGeoffrey Grimmett and David Stirzaker, Probability and Random Processes.
3rd Edition. Oxford University Press, 2001.

John A. Rice, Mathematical Statistics and Data Analysis, 3rd edition.
Duxbury Press, 2006.
406-3461-AALFunctional Analysis I
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-10 KP21RM. Struwe
KurzbeschreibungBaire category; Banach spaces and linear operators; Fundamental theorems: Open Mapping Theorem, Closed Range Theorem, Uniform Boundedness Principle, Hahn-Banach Theorem; Convexity; reflexive spaces; Spectral theory.
Lernziel
SkriptLecture notes by Professor Michael Struwe (Link)
or Lecture notes by Prof. Einsiedler and Ward
(Link)
LiteraturNumerous texts in English or German
406-3621-AALFundamentals of Mathematical Statistics
Belegung ist NUR erlaubt für MSc Studierende, die diese Lerneinheit als Auflagenfach verfügt haben.

Alle andere Studierenden (u.a. auch Mobilitätsstudierende, Doktorierende) können diese Lerneinheit NICHT belegen.
E-10 KP21RF. Balabdaoui
KurzbeschreibungThe course covers the basics of inferential statistics.
Lernziel