module handbook master

Module Handbookfor the Master’s Program in Mathematics

at the University of Bonn

Version of 14 March 2017

The rules regulating the choice of optional subjects are contained in the Examination Regulations for theMaster’s Program.

The semester given in the tables below as the best suited for the module concerned is for orientation only. Themodules may also be taken in other semesters.

The workloads given in the tables are estimates of the amount of work for the average student. The actualamount of work required can vary greatly. Occasionally the workload is distributed over two consecutivesemesters. In the case of seminars, for example, the free period prior to the start of the semester should be usedfor preparation.

Numbering System: Every module is allotted a Module Number of the form “X iY j ”, where

• X ∈ {V,S,P,T,F} designates the module type (V=Lecture course, S=Seminar, P=Practical trainingcourse, T=Thesis, F=Foundations or Additional module),

• i ∈ {4, 5} corresponds roughly to the year of study during which the module is normally taken (i = 4:lecture courses, graduate seminars, practical training courses, i = 5: advanced lecture courses, Master’sthesis, Master’s thesis seminar),

• Y ∈ {A,B,C,D,E,F,G,X} is the area (A=Algebra, Number Theory and Logic; B=Analysis and DifferentialEquations; C=Discrete Mathematics; D=Geometry and Topology; E=Numerical Mathematics and Sci-entific Computing; F=Probability and Stochastic Analysis; G=no area assigned; X=Additional module),and

• j ∈ {1, . . . , 9} denotes consecutive numbering.

In the table of contents, the number of credit points for each module is given in rectangular brackets.

i

Contents

Compulsory Modules

T5G1 [ 30 ] Master’s Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1S5G1 [ 6 ] Master’s Thesis Seminar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Optional Modules — Foundations

F4A1 [ 9 ] Foundations in Algebra, Number Theory and Logic . . . . . . . . . . . . . . . . . . 3F4B1 [ 9 ] Foundations in Analysis and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4F4C1 [ 9 ] Foundations in Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 5F4D1 [ 9 ] Foundations in Geometry and Topology . . . . . . . . . . . . . . . . . . . . . . . . 6F4E1 [ 9 ] Foundations in Numerical Mathematics and Scientific Computing . . . . . . . . . . 7F4F1 [ 9 ] Foundations in Probability and Stochastic Analysis . . . . . . . . . . . . . . . . . . 8

Optional Modules — Lecture Courses

Area A: Algebra, Number Theory, and Logic

V4A1 [ 9 ] Algebraic Geometry I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9V4A2 [ 9 ] Algebraic Geometry II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10V4A3 [ 9 ] Representation Theory I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11V4A4 [ 9 ] Representation Theory II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12V4A5 [ 9 ] Advanced Algebra I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13V4A6 [ 9 ] Advanced Algebra II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14V4A7 [ 9 ] Advanced Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15V4A8 [ 9 ] Models of Set Theory I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16V4A9 [ 9 ] Models of Set Theory II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17V5A1 [ 7 ] Advanced Topics in Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18V5A2 [ 5 ] Selected Topics in Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19V5A3 [ 7 ] Advanced Topics in Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 20V5A4 [ 5 ] Selected Topics in Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 21V5A5 [ 7 ] Advanced Topics in Representation Theory . . . . . . . . . . . . . . . . . . . . . . 22V5A6 [ 5 ] Selected Topics in Representation Theory . . . . . . . . . . . . . . . . . . . . . . . 23V5A7 [ 7 ] Advanced Topics in Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . 24V5A8 [ 5 ] Selected Topics in Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . 25

Area B: Analysis and Differential EquationsV4B1 [ 9 ] Nonlinear Partial Differential Equations I . . . . . . . . . . . . . . . . . . . . . . . 26V4B2 [ 9 ] Nonlinear Partial Differential Equations II . . . . . . . . . . . . . . . . . . . . . . . 27V4B3 [ 9 ] Advanced Global Analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28V4B4 [ 9 ] Advanced Global Analysis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29V4B5 [ 9 ] Real and Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30V5B1 [ 7 ] Advanced Topics in Analysis and Partial Differential Equations . . . . . . . . . . . 31V5B2 [ 5 ] Selected Topics in Analysis and Partial Differential Equations . . . . . . . . . . . . 32V5B3 [ 7 ] Advanced Topics in PDE and Mathematical Models . . . . . . . . . . . . . . . . . 33V5B4 [ 5 ] Selected Topics in PDE and Mathematical Models . . . . . . . . . . . . . . . . . . 34V5B5 [ 7 ] Advanced Topics in Analysis and Calculus of Variations . . . . . . . . . . . . . . . 35V5B6 [ 5 ] Selected Topics in Analysis and Calculus of Variations . . . . . . . . . . . . . . . . 36V5B7 [ 7 ] Advanced Topics in Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37V5B8 [ 5 ] Selected Topics in Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

ii

Area C: Discrete MathematicsV4C1 [ 9 ] Combinatorial Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39V4C2 [ 9 ] Approximation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40V4C3 [ 9 ] Chip Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41V5C1 [ 7 ] Advanced Topics in Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . 42V5C2 [ 5 ] Selected Topics in Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 43

Area D: Geometry and TopologyV4D1 [ 9 ] Algebraic Topology I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44V4D2 [ 9 ] Algebraic Topology II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45V4D3 [ 9 ] Advanced Geometry I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46V4D4 [ 9 ] Advanced Geometry II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47V5D1 [ 7 ] Advanced Topics in Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48V5D2 [ 5 ] Selected Topics in Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49V5D3 [ 7 ] Advanced Topics in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50V5D4 [ 5 ] Selected Topics in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51V5D5 [ 7 ] Advanced Topics in Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . 52V5D6 [ 5 ] Selected Topics in Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . 53

Area E: Numerical Mathematics and Scientific ComputingV4E1 [ 9 ] Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54V4E2 [ 9 ] Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55V5E1 [ 7 ] Advanced Topics in Numerical Methods in Science and Technology . . . . . . . . . 56V5E2 [ 5 ] Selected Topics in Numerical Methods in Science and Technology . . . . . . . . . . 57V5E3 [ 7 ] Advanced Topics in Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . 58V5E4 [ 5 ] Selected Topics in Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . 59V5E5 [ 7 ] Advanced Topics in Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 60V5E6 [ 5 ] Selected Topics in Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 61

Area F: Probability and Stochastic AnalysisV4F1 [ 9 ] Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62V4F2 [ 9 ] Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63V5F1 [ 7 ] Advanced Topics in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 64V5F2 [ 5 ] Selected Topics in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 65V5F3 [ 7 ] Advanced Topics in Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 66V5F4 [ 5 ] Selected Topics in Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 67V5F5 [ 7 ] Advanced Topics in Applied Probability . . . . . . . . . . . . . . . . . . . . . . . . 68V5F6 [ 5 ] Selected Topics in Applied Probability . . . . . . . . . . . . . . . . . . . . . . . . . 69

Optional Modules — Graduate Seminars

S4A1 [ 6 ] Graduate Seminar on Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . 70S4A2 [ 6 ] Graduate Seminar on Representation Theory . . . . . . . . . . . . . . . . . . . . . 71S4A3 [ 6 ] Graduate Seminar on Advanced Algebra . . . . . . . . . . . . . . . . . . . . . . . . 72S4A4 [ 6 ] Graduate Seminar on Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73S4A5 [ 6 ] Graduate Seminar on Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74S4B1 [ 6 ] Graduate Seminar on Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75S4B2 [ 6 ] Graduate Seminar on Partial Differential Equations . . . . . . . . . . . . . . . . . . 76S4B3 [ 6 ] Graduate Seminar on Global Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 77S5B1 [ 6 ] Graduate Seminar on Advanced Topics in Partial Differential Equations . . . . . . 78S5B2 [ 6 ] Graduate Seminar on Partial Differential Equations in the Sciences . . . . . . . . . 79S5B3 [ 6 ] Graduate Seminar on New Developments in Partial Differential Equations . . . . . 80S5B4 [ 6 ] Graduate Seminar on Modeling and Simulation with Partial Differential Equations 81S4C1 [ 6 ] Graduate Seminar on Discrete Optimization . . . . . . . . . . . . . . . . . . . . . . 82S4C2 [ 6 ] Graduate Seminar on Chip Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 83S4D1 [ 6 ] Graduate Seminar on Differential Geometry . . . . . . . . . . . . . . . . . . . . . . 84S4D2 [ 6 ] Graduate Seminar on Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

iii

S4D3 [ 6 ] Graduate Seminar on Advanced Geometry . . . . . . . . . . . . . . . . . . . . . . . 86S4D4 [ 6 ] Graduate Seminar on Advanced Topology . . . . . . . . . . . . . . . . . . . . . . . 87S4E1 [ 6 ] Graduate Seminar on Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . 88S4E2 [ 6 ] Graduate Seminar on Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 89S5E1 [ 6 ] Graduate Seminar on Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . 90S5E2 [ 6 ] Graduate Seminar on Efficient Simulation . . . . . . . . . . . . . . . . . . . . . . . 91S4F1 [ 6 ] Graduate Seminar on Probability Theory . . . . . . . . . . . . . . . . . . . . . . . 92S4F2 [ 6 ] Graduate Seminar on Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . 93S4F3 [ 6 ] Graduate Seminar on Applied Probability . . . . . . . . . . . . . . . . . . . . . . . 94

Optional Modules — Practical Training Courses

P4G1 [ 9 ] Practical Teaching Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95P4G2 [ 9 ] External Internship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96P4A1 [ 9 ] Practical Project in Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . 97P4C1 [ 9 ] Combinatorial Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98P4C2 [ 9 ] Algorithms for Chip Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99P4E1 [ 9 ] Practical Lab Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Optional Modules — Additional Modules

F5X1 [ 6 ] Additional Graduate Seminar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101F5X2 [ 7 ] Additional Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102F5X3 [ 5 ] Additional Selected Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Optional Modules — Additional Modules in Physics (in German)

NP420 [ 9 ] Theoretische Physik III (Quantenmechanik) . . . . . . . . . . . . . . . . . . . . . . 104NP520 [ 9 ] Theoretische Physik IV (Statistische Physik) . . . . . . . . . . . . . . . . . . . . . . 105

iv

ModuleT5G1

Master’s Thesis

Credit Points: Workload: Duration: Offered:

30 900 h 12 months every semester

Person in Charge Head of the examination board

Instructors Any mathematics lecturer

Purpose Program Mode Semester

of Module Master Mathematics Compulsory module 3 - 4

Learning Targets Ability to write a scientific exposition featuring own research results.

Contents The topic can be chosen from any research area of mathematics

Prerequisites at least 30 credit points

Further RequiredQualifications

To be discussed with the supervisor. As a rule, at least three lecture courses orgraduate seminars from the corresponding area A, B, C, D, E or F are required.

Courses Type, Topic h/week Workload CP

Independent research under supervision leading to thepreparation of a Master’s thesis

- 900 30

Examination graded evaluation of the Master’s thesis

Requirements forExamination

none

More Information

1

ModuleS5G1

Master’s Thesis Seminar


6 180 h 2 semesters every semester




of Module Master Mathematics Compulsory module 3 - 4

Learning Targets Ability to present own research results and to discuss mathematical results criticallyin a wider context.

Contents Every participant of the seminar will prepare three seminar sessions on the topic of hisor her Master’s thesis. In the first talk the student will typically present the contextof his or her research work. In the second talk the student will begin to presentresearch results. In the final colloquium, which usually takes place after completionof the thesis, the research results of the thesis are presented and discussed in a widermathematical context. Particular emphasis will be placed on the ability to providea survey which allows nonspecialists to follow the talks.

Prerequisites Enrolment takes place together with the enrolment for the Master’s thesis.


none


Master’s thesis seminar 2 180 6

Examination graded final seminar talk


Before the final seminar talk, two other talks must be given. Active participationand regular attendance are required.

More Information Usually several seminars will be offered in this module. Thematically similar thesistopics are collected together in one seminar. The supervisor of the Master’s Thesiswill ensure that the student is given a place in one of these seminars.

2

ModuleF4A1

Foundations in Algebra, Number Theory and Logic


9 270 h 1 semester every term (with varying content)

Person in Charge Responsible professor for area A

Instructors Any lecturer of area A


of Module Master Mathematics Foundation course, area A 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of algebra, number theory and logic. The ability to think abstractly andthe ability to identify knowledge gaps independently and close those gaps. Confidenthandling of learning strategies leading to successful knowledge assimilation.

Contents You may choose one of the following lecture courses: “Algebra I”, “Algebra II”,“Foundations in Representation Theory” and “Set Theory”.Algebra I: groups, rings, theory of fields, Galois theory and its applications.Algebra II: Important topics of Algebra, e.g. algebraic number theory, representa-tion theory, commutative algebra, Lie algebras.Foundations in Representation Theory: basic concepts of module theory, in-troduction to classical classification problems in representation theory.Set Theory: axioms of Zermelo Fraenkel set theory; relations, functions, structures;ordinal numbers, induction, recursion theorem, arithmetics of ordinal numbers; num-ber systems: integer, rational, real numbers; axiom of choice and equivalent princi-ples; cardinal numbers and their arithmetic; sets of real numbers: Borel sets andprojective sets, regularity. Optional topics: introduction to topological set theory;paradoxical consequences of the axiom of choice; descriptive set theory; set theorywithout the axiom of choice; category theory and set theory.

Prerequisites none



lecture course with problem sessions 4+2 270 9

Examination graded oral examination


successful participation in the problem sessions

More Information Approval of the Examination Board is required: The student has to apply for thismodule to the Examination Board. He has to prove that he did not take the samecourse or a course with similar contents during his bachelor studies.

Some of the lecture courses may be taught in German.

3

ModuleF4B1

Foundations in Analysis and PDE



Person in Charge Responsible professor for area B

Instructors Any lecturer of area B


of Module Master Mathematics Foundation course, area B 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of analysis and PDEs. The ability to think abstractly and the ability toidentify knowledge gaps independently and close those gaps. Confident handling oflearning strategies leading to successful knowledge assimilation.

Contents You may choose one of the following lecture courses: “PDE and Functional Analysis”,“PDE and Modelling”, “Global Analysis I” and “Global Analysis II”.PDE and Functional Analysis: Hilbert spaces and Lax-Milgram’s theorem;Sobolev spaces as well as embedding theorems and trace theorems. weak conver-gence and completeness with respect to sequences. spectral theorem for symmetricoperators with compact inverse. elliptic differential equations without constant co-efficients: minimizing problems, calculus of variation (for Dirichlet and Neumannproblems) L2-regularity theory additional question: principle of the maximum, Har-nack’s inequality, Eigenvalue problems.PDE and Modelling: Selection of topics from PDEs in fluid dynamics, PDEsfor free boundary value problems and image processing, PDEs and mathematicalphysics, PDEs in materials science.Global Analysis I: differential forms, integration of differential forms, Stokes the-orem, vector bundles and tensor fields. Riemann manifolds, covariant derivatives,curvature, the Laplacian on forms, de Rham cohomology, Poincare’s lemma, Mayer-Vietories sequences.Global Analysis II: distributions and fourier transformation, oscillatory integrals,fourier integral operators, pseudodifferential operators, sobolev spaces on manifolds,embedding theorems, regularity theory for elliptic equations on manifolds, spectraltheorem for elliptic opertaors on closed manifolds, applications e.g. Hodge theory.

Prerequisites none









4

ModuleF4C1

Foundations in Discrete Mathematics



Person in Charge Responsible professor for area C

Instructors Any lecturer of area C


of Module Master Mathematics Foundation course, area C 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of discrete mathematics. The ability to think abstractly and the ability toidentify knowledge gaps independently and close those gaps. Confident handling oflearning strategies leading to successful knowledge assimilation.

Contents You may choose one of the following lecture courses: “Linear and integer optimiza-tion” and “Combinatorics, graphs, matroids”.Linear and integer optimization: modelling of optimization problems als (inte-ger) linear programs, polyhedra, Fourier-Motzkin-elimination, Farkas’ Lemma, dual-ity theorems, Simplex method, network Simplex method, Ellipsoid method, condi-tions for integrality of polyhedra, TDI-systems, total unimodularity, cutting planesmethods.Combinatorics, graphs, matroids: Combinatorics of finite sets, elementarycounting techniques, graphs, trees, cycles, connectivity, planarity, coloring of graphs,matroids, planar and combinatorial duality.

Prerequisites none



lecture or reading course with problem sessions 4+2 270 9






Literature:

• B. Korte, J. Vygen: Combinatorial Optimization: Theory and Algorithms.Springer, Berlin, 5th edition 2012

5

ModuleF4D1

Foundations in Geometry and Topology



Person in Charge Responsible professor for area D

Instructors Any lecturer of area D


of Module Master Mathematics Foundation course, area D 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of geometry and topology. The ability to think abstractly and the abilityto identify knowledge gaps independently and close those gaps. Confident handlingof learning strategies leading to successful knowledge assimilation.

Contents You may choose one of the following lecture courses: “Topology I”, “Topology II”,“Geometry I” and “Geometry ”.Topology I: singular homology groups, with integer and arbitrary coefficients, ho-mology theory, CW-complexes and cellular homology, calculation of homology forspheres, projective spaces and surfaces, universal coefficients theorem, Kunneth the-orem.Topology II: singular homology groups, with coefficients in commutative rings,cohomology theory, calculation of cohomology groups of spaces, DeRham cohomol-ogy, universal coefficient theorems, Kunneth theorem, Cup product, ring structureof cohomology, Poincare duality for manifolds, higher homotopy groups, Hurewicztheorem and Whitehead theorem.Geometry I: metrics, connectedness, geodetic line, curvature.Geometry II: relations between geometry and topology, symmetry.

Prerequisites none









6

ModuleF4E1

Foundations in Numerical Mathematics and Scientific Computing



Person in Charge Responsible professor for area E

Instructors Any lecturer of area E


of Module Master Mathematics Foundation course, area E 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of numerical mathematics and scientific computing. The ability to thinkabstractly and the ability to identify knowledge gaps independently and close thosegaps. Confident handling of learning strategies leading to successful knowledge as-similation.

Contents You may choose one of the following lecture courses: “Scientific Computing I” and“Scientific Computing II”.Scientific Computing I: Differential equations. mathematical modelling: firstprinciples, multiscale developments. Approximation of the model, error analysis,filtering, homogenization. Discretization: finite differences, finite elements, optional:adaptivity, error estimators, saddle point problems, multigrid.Scientific Computing II: finite dimensional optimization, numerics of parabolicand hyperbolic pde’s, fast solvers, mixed finite elements, numerical data analysis.

Prerequisites none









7

ModuleF4F1

Foundations in Probability and Stochastic Analysis



Person in Charge Responsible professor for area F

Instructors Any lecturer of area F


of Module Master Mathematics Foundation course, area F 1 or 2

Learning Targets First overview and basic understanding of propositions, relations and methods fromthe area of probability and stochastic analysis. The ability to think abstractly andthe ability to identify knowledge gaps independently and close those gaps. Confidenthandling of learning strategies leading to successful knowledge assimilation.

Contents You may choose one of the following lecture courses: “Stochastic Processes” and“Foundations in Stochastic Analysis”.

Stochastic Processes: Conditional expectations, conditional densities, stochastickernels. Markov chains: existence, Dirichlet problem, recurrence and transience,convergence to equilibrium, ergodicity. Ising Model. Reversible Markov chains andMarkov Chain Monte Carlo methods. Poisson processes and Markov chains in contin-uous time, forward- and backward equations. Brownian motion: motivation as scal-ing limit of Random Walks, marginal distributions, connection to the heat equation,Wiener-Levy construction, scale invariance and symmetries, sample path properties.Large deviations: Cramer’s theorem, Sanov’s theorem on finite sets.

Foundations in Stochastic Analysis: Martingales: stopping theorem, ruin prob-lem, discrete stochastic integrals, convergence theorems, application to Markovchains, regularity and inequalities for continuous martingales. Ito Calculus: Brow-nian motion, quadratic variation, stochastic integrals w.r.t. Brownian motions, Ito’sformula (one- and multidimensional), martingale and Levy characterization of Brow-nian motion, stochastic representations of solutions of the Dirichlet problem and theheat equation, integration w.r.t. Brownian semimartingales, Feynman-Kac-Formula,Girsanov transform.

Prerequisites none


Basic knowledge of probability theory and measure theory.








8

ModuleV4A1

Algebraic Geometry I


9 270 h 1 semester every semester one of the modules Algebraic Geom-etry I/II, Representation Theory I/II and AdvancedAlgebra I/II




of Module Master Mathematics optional module, area A 1 or 2

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of algebraic geometry. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems.

Contents algebraic varieties, commutative algebra

Prerequisites none


Knowledge of basic algebra


Lecture course “Algebraic Geometry I” with problem ses-sions

4+2 270 9




More Information Literature:

• Hartshorne, Algebraic Geometry (Springer-Verlag)

• Mumford, The red book of varieties and schemes (Springer-Verlag)

• Shafarevich, Basic Algebraic Geometry (Springer-Verlag)

9

ModuleV4A2

Algebraic Geometry II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of algebraic geometry. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents Treatment of advanced topics of algebraic geometry

Prerequisites none


Knowledge of topics covered in module “Algebraic Geometry I”


Lecture course “Algebraic Geometry II” with problem ses-sions

4+2 270 9




More Information Literature: will be announced during the course

10

ModuleV4A3

Representation Theory I







Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of representation theory. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems.

Contents chosen topics of representation theory

Prerequisites none




Lecture course “Representation Theory I” with problemsessions

4+2 270 9




More Information Literature: Will be announced during the course.

11

ModuleV4A4

Representation Theory II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of representation theory. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents Treatment of advanced topics of representation theory

Prerequisites none


Knowledge of topics covered in module “Representation Theory I”


Lecture course “Representation Theory II” with problemsessions

4+2 270 9




More Information Literature: Will be announced during the course

12

ModuleV4A5

Advanced Algebra I







Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of algebra. Competence to evaluate the scope, utility, and limits of the methodsand techniques and to independently apply abstract mathematical results to concreteproblems.

Contents chosen topics of algebra

Prerequisites none




Lecture course “Advanced Algebra I” with problem ses-sions

4+2 270 9





13

ModuleV4A6

Advanced Algebra II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of algebra. Competence to place the results in a more general mathematicalcontext. Overview of connections to other areas and ability to arrive at rigorousmathematical proofs starting from heuristic considerations.

Contents an approach of current research themes in algebra

Prerequisites none


Knowledge of topics covered in module “Advanced Algebra I”


Lecture course “Advanced Algebra II” with problem ses-sions

4+2 270 9





14

ModuleV4A7

Advanced Mathematical Logic


9 270 h 1 semester every other year




of Module Master Mathematics optional module, area A 1

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of mathematical logic. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents Introduction to an active research area of mathematical logic such as computabilitytheory, descriptive set theory, infinite combinatorics, model theory or proof theory.

Prerequisites none


Basic knowledge of mathematical logic or set theory as provided by the Bachelormodule “Introduction to mathematical logic” and the foundation module “Set The-ory” depending on the topics to be covered.


Lecture course “Advanced Mathematical Logic” with prob-lem sessions

4+2 270 9




More Information

15

ModuleV4A8

Models of Set Theory I







Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of set theory. Competence to evaluate the scope, utility, and limits of themethods and techniques and to independently apply abstract mathematical resultsto concrete problems.

Contents Transitive models of set theory. The forcing method: motivation, construction ofgeneric extensions; specific examples of forcing constructions; independence of theaxiom of choice and of the continuum hypothesis

Prerequisites none


Knowledge of topics covered in the foundation module “Set Theory”


Lecture course “Models of Set Theory I” with problemsessions

4+2 270 9





• Kunen: Set Theory. North Holland

• Jech: Set Theory. Springer

16

ModuleV4A9

Models of Set Theory II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of set theory. Competence to place the results in a more general mathemat-ical context. Overview of connections to other areas and ability to arrive at rigorousmathematical proofs starting from heuristic considerations.

Contents Godel’s model L of constructible sets; combinatorial principles in L; the fine structureof L; larger constructible models

Prerequisites none


Knowledge of topics covered in module “Models of Set Theory I”


Lecture course “Models of Set Theory II” with problemsessions

4+2 270 9





• Devlin: Constructibility. Springer

• Jech: Set Theory. Springer

17

ModuleV5A1

Advanced Topics in Algebra


7 210 h 1 semester every semester at least one of the modules V5A1,V5A2, V5A3, V5A4, V5A5 and V5A6





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof algebra. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engagein independent study of current research topics.

Contents chosen themes of an active research area of algebra

Prerequisites none


Knowledge of topics covered in the modules “Advanced Algebra I and II”


advanced lecture course with varying content 4 210 7

Examination Graded oral examination


none

More Information Literature: will be announced during the course.

18

ModuleV5A2

Selected Topics in Algebra







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof algebra. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engagein independent study of current research topics.

Contents chosen themes of an active research area of algebra

Prerequisites none


Knowledge of topics covered in the modules “Advanced Algebra I and II”





none


Students studying according to the old examination regulations from 06 June 2007(with changes from 07 January 2009 and 29 September 2010) will earn only 4 CreditPoints for this module.

19

ModuleV5A3

Advanced Topics in Algebraic Geometry







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof algebraic geometry. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents chosen themes of an active research area of algebraic geometry

Prerequisites none


Knowledge of topics covered in the modules “Algebraic Geometry I and II”





none


20

ModuleV5A4

Selected Topics in Algebraic Geometry







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof algebraic geometry. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents chosen themes of an active research area of algebraic geometry

Prerequisites none


Knowledge of topics covered in the modules “Algebraic Geometry I and II”





none



21

ModuleV5A5

Advanced Topics in Representation Theory







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof representation theory. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents chosen themes of an active research area of representation theory

Prerequisites none


Knowledge of topics covered in the modules “Representation Theory I and II”





none


22

ModuleV5A6

Selected Topics in Representation Theory







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof representation theory. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents chosen themes of an active research area of representation theory

Prerequisites none


Knowledge of topics covered in the modules “Representation Theory I and II”





none



23

ModuleV5A7

Advanced Topics in Mathematical Logic


7 210 h 1 semester every year one of the modules V4A7, V5A7 undV5A8





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof mathematical logic. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents Chosen topics in an active research area of mathematical logic such as set theory,computability theory, model theory or proof theory.

Prerequisites none


Knowledge of chosen modules in mathematical logic depending on the topics to becovered.





none

More Information

24

ModuleV5A8

Selected Topics in Mathematical Logic


5 150 h 1 semester every year one of the modules V4A7, V5A7 undV5A8





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof mathematical logic. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents Chosen topics in an active research area of mathematical logic such as set theory,computability theory, model theory or proof theory.

Prerequisites none


Knowledge of chosen modules in mathematical logic depending on the topics to becovered.





none

More Information Students studying according to the old examination regulations from 06 June 2007(with changes from 07 January 2009 and 29 September 2010) will earn only 4 CreditPoints for this module.

25

ModuleV4B1

Nonlinear Partial Differential Equations I


9 270 h 1 semester every winter semester




of Module Master Mathematics optional module, area B 1

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of nonlinear PDEs. Competence to evaluate the scope, utility, and limits of themethods and techniques and to independently apply abstract mathematical resultsto concrete problems.

Contents

• Nonlinear elliptic equations: existence (and uniqueness) of weak solutions, vari-ational methods (variational inequalities), compactness methods, Harnack in-equality, regularity theory.

• Nonlinear parabolic equations: existence (and uniqueness) of weak solutions,compactness methods.

Prerequisites none


Knowledge of linear PDEs and of the topics covered in the Bachelor’s modules“Einfuhrung in die Partiellen Differentialgleichungen” and “Partielle Differential-gleichungen und Funktionalanalysis”


Lecture course “Nonlinear Partial Differential EquationsI” with problem sessions

4+2 270 9




More Information

26

ModuleV4B2

Nonlinear Partial Differential Equations II


9 270 h 1 semester every summer semester at least one of V4B2 andV4B5





Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of nonlinear PDEs. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents

• Nonlinear hyperbolic first-order equations: method of characteristics,Hamilton-Jacobi equations (optional), Cauchy-Kowalevski theorem (optional).Scalar conservation laws (Kruzkov’s theory for entropy solutions).

• Basic properties of Schrodinger’s equation.

• One or more of the following themes:

– Viscosity solutions.

– Gradient flows.

– Advanced variational methods (for example Gamma convergence or PDE-constrained optimization.)

– Nonlinear waves.

– Advanced study of nonlinear Schrodinger equation..

– Free-boundary problems.

Prerequisites none


Knowledge of linear PDEs and of the topics covered in the Bachelor’s modules“Einfuhrung in die Partiellen Differentialgleichungen” and “Partielle Differential-gleichungen und Funktionalanalysis”


Lecture course “Nonlinear Partial Differential EquationsII” with problem sessions

4+2 270 9




More Information

27

ModuleV4B3

Advanced Global Analysis I


9 270 h 1 semester every year at least one of the modules V4B3, V4D1and V4D3




of Module Master Mathematics optional module, area B 1 or 3

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of global analysis. Competence to evaluate the scope, utility, and limits of themethods and techniques and to independently apply abstract mathematical resultsto concrete problems.

Contents The topics to be covered will be announced before course commences. Possible topicsare:

• Atiyah-Singer index theory (closed manifolds)

• spectral geometry

• local index theory

• noncommutative geometry and index theory

• representation theory and automorphic forms

Prerequisites none


Knowledge of the topics covered in the Bachelor’s modules “Globale Analysis I” and“Partielle Differentialgleichungen und Funktionalanalysis” as well as chosen modulesfrom area B depending on topics to be covered


advanced lecture course with varying content and problemsessions

4+2 270 9




More Information

28

ModuleV4B4

Advanced Global Analysis II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of global analysis. Competence to place the results in a more general math-ematical context. Overview of connections to other areas and ability to arrive atrigorous mathematical proofs starting from heuristic considerations.

Contents The topics to be covered will be announced before course commences. Possible topicsare:

• Atiyah-Patodi-Singer index theory (manifolds with boundary and singular man-ifolds)

• spectral geometry of singular manifolds ( e. g. hyperbolic surfaces with finitevolume)

• analytic torsion

• local index theorem in noncommutative geometry

• representation theory and automorphic forms

Prerequisites none




advanced lecture course with varying content and problemsessions

4+2 270 9




More Information

29

ModuleV4B5

Real and Harmonic Analysis


9 270 h 1 semester every summer semester at least one of V4B2 andV4B5





Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of real and harmonic analysis. Competence to evaluate the scope, utility, andlimits of the methods and techniques and to independently apply abstract mathe-matical results to concrete problems. Competence to place the results in a moregeneral mathematical context. Overview of connections to other areas and ability toarrive at rigorous mathematical proofs starting from heuristic considerations.

Contents

• Fourier Analysis

• Calderon-Zygmund theory

• Harmonic Analysis

Prerequisites none


Elementary knowledge of Lebesgue integration theory, Fourier series, complex anal-ysis and PDEs and of the topics covered in the Bachelor’s module “PartielleDifferentialgleichungen und Funktionalanalysis”


Lecture course “Real and Harmonic Analysis” with prob-lem sessions

4+2 270 9




More Information

30

ModuleV5B1

Advanced Topics in Analysis and Partial Differential Equations


7 210 h 1 semester every semester at least one of the modules V5B1,V5B2, V5B3, V5B4, V5B5, V5B6, V5B7 and V5B8





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof analysis and PDEs. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents Current research areas in PDEs. The topics to be covered will be announced beforecourse commences.

Prerequisites none


Knowledge of chosen modules from area B depending on topics to be covered





none

More Information

31

ModuleV5B2

Selected Topics in Analysis and Partial Differential Equations







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof analysis and PDEs. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents Current research areas in PDEs. The topics to be covered will be announced beforecourse commences.

Prerequisites none







none


32

ModuleV5B3

Advanced Topics in PDE and Mathematical Models







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof PDEs and mathematical models. Ability to verify the validity of propositions fromoriginal literature independently and to question research results critically. Compe-tence to engage in independent study of current research topics.

Contents The topics to be covered will be announced at the end of the semester prior tocommencement of the course.

Prerequisites none







none

More Information

33

ModuleV5B4

Selected Topics in PDE and Mathematical Models







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof PDEs and mathematical models. Ability to verify the validity of propositions fromoriginal literature independently and to question research results critically. Compe-tence to engage in independent study of current research topics.


Prerequisites none







none


34

ModuleV5B5

Advanced Topics in Analysis and Calculus of Variations







Learning Targets Deep understanding and detailed overview of a current research focus from the area ofanalysis and calculus of variations. Ability to verify the validity of propositions fromoriginal literature independently and to question research results critically. Compe-tence to engage in independent study of current research topics.


Prerequisites none







none

More Information

35

ModuleV5B6

Selected Topics in Analysis and Calculus of Variations







Learning Targets Deep understanding and detailed overview of a current research focus from the area ofanalysis and calculus of variations. Ability to verify the validity of propositions fromoriginal literature independently and to question research results critically. Compe-tence to engage in independent study of current research topics.


Prerequisites none







none


36

ModuleV5B7

Advanced Topics in Analysis







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof analysis. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.


Prerequisites none







none

More Information

37

ModuleV5B8

Selected Topics in Analysis







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof analysis. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.


Prerequisites none







none

More Information

38

ModuleV4C1

Combinatorial Optimization






of Module Master Mathematics optional module, area C 1 or 3

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of Combinatorial Optimization. Competence to evaluate the scope, utility, andlimits of the methods and techniques and to independently apply abstract mathe-matical results to concrete problems. Competence to place the results in a moregeneral mathematical context. Overview of connections to other areas and ability toarrive at rigorous mathematical proofs starting from heuristic considerations.

Contents Matchings, b-matchings and T -joins, optimization over matroids, minimization ofsubmodular functions, traveling salesman problem, polyhedral combinatorics, NP-hard problems

Prerequisites none


basic knowledge of linear optimization and graph algorithms


Lecture course “Combinatorial Optimization” with prob-lem sessions

4+2 270 9





• B. Korte, J. Vygen: Combinatorial Optimization: Theory and Algorithms.Springer, Berlin, 5th edition 2012 (Chapters 10 – 15 and 21)

• A. Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer2003

• W. Cook, W. Cunningham, W. Pulleyblank, A. Schrijver: Combinatorial Op-timization. Wiley 1997 (Chapters 5 – 9)

39

ModuleV4C2

Approximation Algorithms


9 270 h 1 semester every summer semester




of Module Master Mathematics optional module, area C 2

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of approximation algorithms. Competence to evaluate the scope, utility, andlimits of the methods and techniques and to independently apply abstract mathe-matical results to concrete problems. Competence to place the results in a moregeneral mathematical context. Overview of connections to other areas and ability toarrive at rigorous mathematical proofs starting from heuristic considerations.

Contents Definition of an approximation algorithm and an approximation scheme. Design andanalysis of approximation algorithms for chosen NP-hard problems, e. g. the set cov-ering and vertex covering problem, MAXSAT, TSP, knapsack, bin packing, networkdesign, facility location. Various techniques (e. g. greedy, LP-rounding, primal-dual,local search, randomization, sampling and MCMC-methods) and applications will bepresented. Analysis of approximation hardness and PCP-systems

Prerequisites none


basic knowledge of combinatorial and linear optimization


Lecture course “Approximation Algorithms” with problemsessions

4+2 270 9





• B. Korte, J. Vygen: Combinatorial Optimization: Theory and Algorithms.Springer, Berlin, 5th edition 2012 (Chapters 16 – 22)

• V.V. Vazirani: Approximation Algorithms. Springer 2001

• S. Arora, C. Lund: Hardness of Approximation. In: Approximation Algorithmsfor NP-Hard Problems (D.S. Hochbaum, ed.), PWS 1996

• M. Karpinski: Randomisierte und approximative Algorithmen fur harte Berech-nungsprobleme. Lecture Notes (4th edition), University of Bonn 2000

• D.P. Williamson, D.B. Shmoys: The Design of Approximation Algorithms.Cambridge University Press, 2011

40

ModuleV4C3

Chip Design






of Module Master Mathematics optional module, area C 2

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of chip design. Competence to evaluate the scope, utility, and limits of themethods and techniques and to independently apply abstract mathematical results toconcrete problems. Competence to place the results in a more general mathematicalcontext. Overview of connections to other areas and ability to arrive at rigorousmathematical proofs starting from heuristic considerations.

Contents Problem formulation and design flow in chip design, logic synthesis, placement, rout-ing, timing analysis and optimization, clock-tree design

Prerequisites none


Knowledge of combinatorial optimization


Lecture course “Chip Design” with problem sessions 4+2 270 9




More Information Literature: as long as no recommendable textbook is available, lecture notes will beprovided. The following two sources contain many useful references to special topics:

• C.J. Alpert, D.P. Mehta, S.S. Sapatnekar: The Handbook of Algorithms forVLSI Physical Design Automation. Taylor and Francis 2008

• B. Korte, D. Rautenbach, J. Vygen: BonnTools: mathematical innovation forlayout and timing closure of systems on a chip. Proceedings of the IEEE 95(2007), 555–572

• S. Held, B. Korte, D. Rautenbach, J. Vygen: Combinatorial optimization inVLSI design. In: ”Combinatorial Optimization: Methods and Applications”(V. Chvatal, ed.). IOS Press, Amsterdam 2011, pp 33-96.

41

ModuleV5C1

Advanced Topics in Discrete Mathematics


7 210 h 1 semester at least one of the modules V5C1 and V5C2 annually





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof discrete mathematics. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents A current research area will be treated in detail. The topic to be covered will beannounced before course commences

Prerequisites none


Knowledge of the topics covered in the module “Combinatorial Optimization” as wellas chosen modules from area C depending on topic to be covered





none

More Information

42

ModuleV5C2

Selected Topics in Discrete Mathematics


5 150 h 1 semester at least one of the modules V5C1 and V5C2 annually





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof discrete mathematics. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents A current research area of discrete optimization will be treated. The topic to becovered will be announced before course commences

Prerequisites none


Knowledge of the topics covered in the module “Combinatorial Optimization” as wellas chosen modules from area C depending on topic to be covered





none


43

ModuleV4D1

Algebraic Topology I






of Module Master Mathematics optional module, area D 1

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of algebraic topology. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems.

Contents Choice of topics:

• unstable homotopy theory

• spectra

• bordism theory

• cohomology of groups

• localization

• rational homotopy theory

• differential topology

• spectral sequences

• K-theory

• model categories

Prerequisites none


Knowledge of the topics covered in the Bachelor’s modules “Einfuhrung in Geometrieund Topologie”, “Topologie I” and “Topologie II”


Lecture course “Algebraic Topology I” with problem ses-sions

4+2 270 9




More Information The topic to be covered and the required literature will be announced at the endof the previous semester. The above-mentioned topics are covered in the books byBredon, Hatcher, Adams, Switzer, Whitehead.

44

ModuleV4D2

Algebraic Topology II






of Module Master Mathematics optional module, area D 2

Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of algebraic topology. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.


• unstable homotopy theory

• stable homotopy theory


• cohomology operations

• Steenrod algebra

• characteristic classes

Prerequisites none


Knowledge of the topics covered in the Bachelor’s modules “Einfuhrung in Geometrieund Topologie”, “Topologie I” and “Topologie II”, as well as in the module “AlgebraicTopology I”


Lecture course “Algebraic Topology II” with problem ses-sions

4+2 270 9




More Information The topic to be covered and the required literature will be announced at the end ofthe previous semester

45

ModuleV4D3

Advanced Geometry I






of Module Master Mathematics optional module, area D 1 or 3

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of geometry. Competence to evaluate the scope, utility, and limits of the methodsand techniques and to independently apply abstract mathematical results to concreteproblems.

Contents Introduction to an advanced, active research area in geometry. One of the followingtopics will be chosen on a rotational basis: geometric analysis, geometric topology,symplectic geometry, geometric group theory, complex algebraic geometry, theory ofmodule spaces

Prerequisites none


a basic knowledge of geometry


Lecture course “Advanced Geometry I” with problem ses-sions

4+2 270 9




More Information

46

ModuleV4D4

Advanced Geometry II







Learning Targets Broad overview and deep understanding of propositions, relations and methods fromthe area of geometry. Competence to place the results in a more general mathematicalcontext. Overview of connections to other areas and ability to arrive at rigorousmathematical proofs starting from heuristic considerations.

Contents a continuation and deeper treatment of the topic chosen in the module “AdvancedGeometry I”

Prerequisites none


Knowledge of topics covered in the module “Advanced Geometry I”


Lecture course “Advanced Geometry II” with problem ses-sions

4+2 270 9




More Information

47

ModuleV5D1

Advanced Topics in Topology


7 210 h 1 every semester at least one of the modules V5D1,V5D2, V5D3, V5D4, V5D5 and V5D6





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof topology. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.


• secondary cohomology operations

• nilpotency theorems

• elliptic cohomology

Prerequisites none


Knowledge of the topics covered in the Bachelor’s modules “Einfuhrung in Geome-trie und Topologie”, “Topologie I” and “Topologie II”, as well as in the modules“Algebraic Topology I” and “Algebraic Topology II”





none

More Information The topic to be covered and the required literature will be announced at the end ofthe semester prior to commencement of the course.

48

ModuleV5D2

Selected Topics in Topology


5 150 h 1 every semester at least one of the modules V5D1,V5D2, V5D3, V5D4, V5D5 and V5D6





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof topology. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.


• secondary cohomology operations

• nilpotency theorems

• elliptic cohomology

Prerequisites none


Knowledge of the topics covered in the Bachelor’s modules “Einfuhrung in Geome-trie und Topologie”, “Topologie I” and “Topologie II”, as well as in the modules“Algebraic Topology I” and “Algebraic Topology II”





none

More Information The topic to be covered and the required literature will be announced at the end ofthe semester prior to commencement of the course.


49

ModuleV5D3

Advanced Topics in Geometry


7 210 h 1 semester every semester at least one of the modules V5D1,V5D2, V5D3, V5D4, V5D5 and V5D6





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof geometry. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.

Contents The topics to be covered will be announced before course commences.

Prerequisites none


Knowledge of chosen modules from area D depending on topics to be covered





none

More Information The topic to be covered and the required literature will be announced at the end ofthe previous semester.

50

ModuleV5D4

Selected Topics in Geometry







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof geometry. Ability to verify the validity of propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.

Contents The topics to be covered will be announced before course commences.

Prerequisites none


Knowledge of chosen modules from area D depending on topics to be covered





none

More Information The topic to be covered and the required literature will be announced at the end ofthe previous semester.


51

ModuleV5D5

Advanced Topics in Differential Geometry







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof differential geometry. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents current/advanced research topics in differential geometry

Prerequisites none


Knowledge of the topics covered in the module “Advanced Geometry I” and a basicknowledge of topology





none

More Information

52

ModuleV5D6

Selected Topics in Differential Geometry







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof differential geometry. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents current/advanced research topics in differential geometry

Prerequisites none


Knowledge of the topics covered in the module “Advanced Geometry I” and a basicknowledge of topology





none


53

ModuleV4E1

Numerical Algorithms






of Module Master Mathematics optional module, area E 1

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of numerical algorithms. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents efficient numerical solution and optimization algorithms for PDEs or integral equa-tionspossible choice of

• geometric variational problems

• adaptivity and error estimators

• fast solvers and parallelization

• boundary element methods

• discontinuous Galerkin methods

• optimization allgorithms

Prerequisites none


Knowledge of topics covered in the Bachelor’s modules “Algorithmische MathematikI”, “Algorithmische Mathematik II”, and “Einfuhrung in die Grundlagen der Nu-merik”


Lecture course “Numerical algorithms” with problem ses-sions

4+2 270 9





• W. Hackbusch: Theorie und Numerik elliptischer Differentialgleichungen,Teubner

• A. Meister: Numerik linearer Gleichungssysteme, Vieweg 1999

• D. Kroner: Numerical schemes for conservation laws, Wiley-Teubner 1997

• R. J. LeVeque: Numerical methods for conservation laws, Birkhauser 1990

• V. Thomee: Galerkin finite element methods for parabolic problems, Springer1997

• W. Hackbusch: Multigrid methods and applications, Springer 1985

• A. Ern, D. Di Pietro: Mathematical aspects of discontinuous Galerkin methods,Springer 2012.

54

ModuleV4E2

Numerical Simulation






of Module Master Mathematics optional module, area E 2

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of numerical simulation. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents possible choice of

• optimization with PDEs: with and without constraints

• numerics of geometric variational problems

• efficient methods for parameter dependent PDEs

• parallelism and scalablity

Prerequisites none


Knowledge of topics covered in the Bachelor’s modules “Algorithmische MathematikI”, “Algorithmische Mathematik II”, and “Einfuhrung in die Grundlagen der Nu-merik”


Lecture course “Numerical Simulation” with problem ses-sions

4+2 270 9





• F. Troltzsch: Optimal control of partial differential equations. Theory, methodsand applications. AMS 2010.

• H. W. Engl, M. Hanke, A. Neubauer: Regularization of inverse problems.Kluwer Academic Publishers Group, 1996.

• W. Hackbusch: Theorie und Numerik elliptischer Differentialgleichungen,Teubner

• D. Kroner: Numerical schemes for conservation laws, Wiley-Teubner 1997

• R. J. LeVeque: Numerical methods for conservation laws, Birkhuser 1990

• V. Eijkhout: Introduction to high performance scientific computing, 2010.

55

ModuleV5E1

Advanced Topics in Numerical Methods in Science and Technology


7 210 h 1 semester every semester at least one of the modules V5E1,V5E2, V5E3, V5E4, V5E5 and V5E6




of Module Master Mathematics optional module, area E 2-4

Learning Targets Deep understanding of a current research focus from the area of numerical methodsin science and technology. Ability to verify the propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.

Contents Current research topics from Numerical Methods in Science and Technology. Thetopics to be covered will be announced before the course commences.

Prerequisites none


Knowledge of chosen modules from area E depending on topics to be coveredincluding “Algorithmische Mathematik I”, “Algorithmische Mathematik II”, and“Einfuhrung in die Grundlagen der Numerik”





none

More Information

56

ModuleV5E2

Selected Topics in Numerical Methods in Science and Technology







Learning Targets Deep understanding of a current research focus from the area of numerical methodsin science and technology. Ability to verify propositions from original literatureindependently and to question research results critically. Competence to engage inindependent study of current research topics.

Contents Current research topics from Numerical Methods in Science and Technology. Thetopics to be covered will be announced before the course commences.

Prerequisites none







none


57

ModuleV5E3

Advanced Topics in Scientific Computing







Learning Targets Deep understanding of a current research focus from the area of scientific computing.Ability to verify propositions from original literature independently and to questionresearch results critically. Competence to engage in independent study of currentresearch topics.

Contents Current research topics from Scientific Computing. The topics to be covered will beannounced before the course commences.

Prerequisites none







none

More Information

58

ModuleV5E4

Selected Topics in Scientific Computing







Learning Targets Deep understanding of a current research focus from the area of scientific computing.Ability to verify propositions from original literature independently and to questionresearch results critically. Competence to engage in independent study of currentresearch topics.

Contents Current research topics from Scientific Computing. The topics to be covered will beannounced before the course commences.

Prerequisites none







none


59

ModuleV5E5

Advanced Topics in Numerical Analysis







Learning Targets Deep understanding of a current research focus from the area of numerical analysis.Ability to verify propositions from original literature independently and to questionresearch results critically. Competence to engage in independent study of currentresearch topics.

Contents Current research topics from Numerical Analysis. The topics to be covered will beannounced before the course commences.

Prerequisites none







none

More Information

60

ModuleV5E6

Selected Topics in Numerical Analysis







Learning Targets Deep understanding of a current research focus from the area of numerical analysis.Ability to verify propositions from original literature independently and to questionresearch results critically. Competence to engage in independent study of currentresearch topics.

Contents Current research topics from Numerical Analysis. The topics to be covered will beannounced before the course commences.

Prerequisites none







none


61

ModuleV4F1

Stochastic Analysis


9 270 h 1 semester once a year




of Module Master Mathematics optional module, area F 1 or 2

Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of stochastic analysis. Competence to evaluate the scope, utility, and limitsof the methods and techniques and to independently apply abstract mathematicalresults to concrete problems. Competence to place the results in a more generalmathematical context. Overview of connections to other areas and ability to arriveat rigorous mathematical proofs starting from heuristic considerations.

Contents Basics : concise summary of basic results for Brownian motion and martingales,stochastic integration and Ito calculus for semimartingales, representation theorems.Stochastic Differential Equations : strong and weak solutions, linear equations(Ornstein-Uhlenbeck process, Brownian bridge), existence and uniqueness of strongsolutions, stochastic flows, Feller property (optional), Girsanov theorem and con-struction of weak solutions by change of measure or time scale.Ito Diffusions : martingale problem, connection with PDEs, boundary value prob-lems, hitting times, non-explosion criteria, local time (optional).One or several optional topics : Analysis on Wiener space (Cameron-Martintheorem, large deviations, Malliavin derivative, Clark-Ocone formula), Numericalmethods for SDE (stochastic Taylor expansions, strong and weak convergence order),Stochastic calculus for jump processes (Levy processes and Poisson point processes,Ito calculus for semimartingales with jumps).

Prerequisites none


Solid background in measure theoretic probability and stochastic processes.


Lecture course “Stochastic Analysis” with problem ses-sions

4+2 270 9




More Information Reading:

• I. Karatzas, S. Shreve : Brownian motion and stochastic calculus, Springer1991

• D. Revuz, M. Yor : Continuous martingales and Brownian motion, 3rd edition,Springer 1999

• L. C. G. Rogers, D. Williams : Diffusions, Markov processes and martingales,Vol. 2: Ito calculus, Cambridge UP 2000

• R.F. Bass : Stochastic processes, Cambridge UP 2011

• P. Protter : Stochastic integration and differential equations, 2nd ed., Springer2004

• R. Durrett : Stochastic calculus, Boca Raton 1996

62

ModuleV4F2

Markov Processes







Learning Targets Broad overview and understanding of propositions, relations and methods from thearea of Markov processes. Competence to evaluate the scope, utility, and limits of themethods and techniques and to independently apply abstract mathematical results toconcrete problems. Competence to place the results in a more general mathematicalcontext. Overview of connections to other areas and ability to arrive at rigorousmathematical proofs starting from heuristic considerations.

Contents Basics : Introduction to ergodic theory, limit theorems for stochastic processes(Prokhorov, Donsker), large deviation principles.Markov processes on discrete state spaces : construction, transition semigroupand generator, martingale problem, invariant measures, Lyapunov functions, asymp-totics, mixing times, entropy reduction, time inversion and reversibility, Dirichletforms and functional inequalities.Markov processes on general state spaces : Martingale characterization of diffu-sions, jump and Levy processes, projective limits and approximations, Kolmogorov-Centsov theorem, C0 semigroups, generators and resolvents, existence of invariantdistributions, reversible Markov processes.One or several optional topics : Spatial models (Gauß and Poisson processes,Gibbs measures and phase transitions, free energy), Selected applications (e. g.stochastic algorithms, models from statistical mechanics, mathematical biology andengineering).

Prerequisites none


Solid background in measure theoretic probability and stochastic processes.


Lecture course “Markov Processes” with problem sessions 4+2 270 9




More Information References will be announced at the beginning of the course.

63

ModuleV5F1

Advanced Topics in Probability Theory


7 210 h 1 semester every semester at least one of the modules V5F1,V5F2, V5F3, V5F4, V5F5 and V5F6





Learning Targets Deep understanding and detailed overview of a current research focus from the areaof probability theory. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.

Contents The topics to be covered will be announced at the end of the semester prior tocommencement of the course. Possible topics include:

• Limit Theorems (Large deviations, extreme value statistics)

• Random matrices and interacting particle systems

• Mathematical statistical mechanics (Phase transitions, metastability and age-ing, percolation, scaling limits, SLE, random environments)

Prerequisites none


Required background depending on topics to be covered.





none

More Information

64

ModuleV5F2

Selected Topics in Probability Theory







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof probability theory. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.


• Limit Theorems (Large deviations, extreme value statistics)

• Random matrices and interacting particle systems

• Mathematical statistical mechanics (Phase transitions, metastability and age-ing, percolation, scaling limits, SLE, random environments)

Prerequisites none


Required background depending on topics to be covered





none


65

ModuleV5F3

Advanced Topics in Stochastic Analysis







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof stochastic analysis. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.


• Analysis on probability spaces (Malliavin calculus, stochastic partial differentialequations, analysis on metric measure spaces)

• Reversible Markov processes and Dirichlet forms (Potential theory, convergenceto equilibrium)

• Optimal transport and functional inequalities

• Stochastic differential geometry (SDE on manifolds, heat kernels)

Prerequisites none







none

More Information

66

ModuleV5F4

Selected Topics in Stochastic Analysis







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof stochastic analysis. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.


• Analysis on probability spaces (Malliavin calculus, stochastic partial differentialequations, analysis on metric measure spaces)

• Reversible Markov processes and Dirichlet forms (Potential theory, convergenceto equilibrium)

• Optimal transport and functional inequalities

• Stochastic differential geometry (SDE on manifolds, heat kernels)

Prerequisites none







none


67

ModuleV5F5

Advanced Topics in Applied Probability







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof applied probability. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.


• Stochastic finance (Option pricing, econometrics, optimal stopping)

• Monte Carlo methods (Numerical methods for SDE, MCMC, filtering)

• Branching processes and models from population biology

• Probability on graphs and networks (Random graphs, models of statistical me-chanics, stochastic algorithms)

Prerequisites none







none

More Information

68

ModuleV5F6

Selected Topics in Applied Probability







Learning Targets Deep understanding and detailed overview of a current research focus from the areaof applied probability. Ability to verify the validity of propositions from originalliterature independently and to question research results critically. Competence toengage in independent study of current research topics.


• Stochastic finance (Option pricing, econometrics, optimal stopping)

• Monte Carlo methods (Numerical methods for SDE, MCMC, filtering)

• Branching processes and models from population biology

• Probability on graphs and networks (Random graphs, models of statistical me-chanics, stochastic algorithms)

Prerequisites none







none


69

ModuleS4A1

Graduate Seminar on Algebraic Geometry


6 180 h 1 semester every semester at least one of the modules S4A1,S4A2 and S4A3




of Module Master Mathematics Optional module, graduate seminar 2 or 3

Learning Targets Ability to undertake independent study of an advanced topic in algebraic geometryusing specialized literature. Assessment, evaluation and presentation of results fromalgebraic geometry. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents a current, active topic in algebraic geometry chosen on a rotational basis will betreated in depth by studying relevant textbooks or literature

Prerequisites none


Knowledge of topics covered in the modules “Advanced Algebra I” or “AlgebraicGeometry I”


graduate seminar “Algebraic Geometry” 4 180 6

Examination graded seminar talk


Active participation and regular attendance are required. A mauscript covering thecontents of the talk is demanded.

More Information the seminar theme and relevant literature, as well as the date for the initial discus-sion and allocation of talks, will be made public towards the end of the previoussemester. No further enrolments are possible after this date. The maximum numberof participants is 15

70

ModuleS4A2

Graduate Seminar on Representation Theory







Learning Targets Ability to undertake independent study of an advanced topic in representation theoryusing specialized literature. Assessment, evaluation and presentation of results fromrepresentation theory. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents a current, active topic in representation theory chosen on a rotational basis will betreated in depth by studying relevant textbooks or literature

Prerequisites none


Knowledge of topics covered in the modules “Representation Theory I” or “AdvancedAlgebra I”


graduate seminar “Representation Theory” 4 180 6





71

ModuleS4A3

Graduate Seminar on Advanced Algebra







Learning Targets Ability to undertake independent study of an advanced topic in advanced algebrausing specialized literature. Assessment, evaluation and presentation of results fromalgebra. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents a current, active topic in algebra chosen on a rotational basis will be treated in depthby studying relevant textbooks or literature

Prerequisites none


Knowledge of topics covered in the modules “Advanced Algebra I” or “AlgebraicGeometry I”


graduate seminar “Algebra” 4 180 6





72

ModuleS4A4

Graduate Seminar on Logic






of Module Master Mathematics Optional module, graduate seminar 2

Learning Targets Ability to undertake independent study of an advanced topic in logic using specializedliterature. Assessment, evaluation and presentation of results from logic. Didacticpreparation and presentation as a seminar talk and in the form of a manuscriptcovering the contents of the talk. Competence in scientific discussions.

Contents Topics from mathematical logic, usually related to set theory: e. g., applications ofinfinitary combinatorial principles, proving the relative consistency of the axiom ofchoice with the model of hereditarily ordinal definable sets, descriptive set theory

Prerequisites none


Knowledge of set theory and the basics of axiomatic set theory


graduate seminar “Logic” 4 180 6




More Information

73

ModuleS4A5

Graduate Seminar on Set Theory







Learning Targets Ability to undertake independent study of an advanced topic in set theory usingspecialized literature. Assessment, evaluation and presentation of results from settheory. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents A study of specific forcing constructions

Prerequisites none


Knowledge of topics covered in module “Models of Set Theory II”


graduate seminar “Set Theory” 4 180 6




More Information

74

ModuleS4B1

Graduate Seminar on Analysis






of Module Master Mathematics Optional module, graduate seminar 1-4

Learning Targets Ability to undertake independent study of an advanced topic in anaylsis using spe-cialized literature. Assessment, evaluation and presentation of results from analy-sis. Didactic preparation and presentation as a seminar talk and in the form of amanuscript covering the contents of the talk. Competence in scientific discussions.

Contents chosen topics of nonlinear functional analysis with applications to PDEs, based onspecialized literature

Prerequisites none


none


graduate seminar “Analysis” 4 180 6




More Information

75

ModuleS4B2

Graduate Seminar on Partial Differential Equations







Learning Targets Ability to undertake independent study of an advanced topic in PDEs using special-ized literature. Assessment, evaluation and presentation of results from PDEs. Di-dactic preparation and presentation as a seminar talk and in the form of a manuscriptcovering the contents of the talk. Competence in scientific discussions.

Contents chosen topics on nonlinear PDEs, based on specialized literature

Prerequisites none


none


graduate seminar “Partial Differential Equations” 4 180 6




More Information

76

ModuleS4B3

Graduate Seminar on Global Analysis


6 180 h 1 semester at least once a year





Learning Targets Ability to undertake independent study of an advanced topic in global analysis usingspecialized literature. Assessment, evaluation and presentation of results from globalanalysis. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents the topics to be covered will be announced before the seminar commences

Prerequisites none




graduate seminar “Global Analysis” 4 180 6




More Information

77

ModuleS5B1

Graduate Seminar on Advanced Topics in Partial Differential Equations


6 180 h 1 semester every semester





Learning Targets Ability to undertake independent study of an advanced topic in PDEs using special-ized literature. Assessment, evaluation and presentation of results from PDEs. Di-dactic preparation and presentation as a seminar talk and in the form of a manuscriptcovering the contents of the talk. Competence in scientific discussions.

Contents chosen topics on nonlinear PDEs, based on specialized literature

Prerequisites none


none


graduate seminar “Advanced topics in Partial DifferentialEquations“

4 180 6




More Information

78

ModuleS5B2

Graduate Seminar on Partial Differential Equations in the Sciences







Learning Targets Ability to undertake independent study of an advanced topic in PDEs in the sciencesusing specialized literature. Assessment, evaluation and presentation of results fromPDEs. Didactic preparation and presentation as a seminar talk and in the form of amanuscript covering the contents of the talk. Competence in scientific discussions.

Contents chosen topics on PDEs, based on specialized literature, with particular emphasis onapplications in the natural sciences

Prerequisites none


depending on the topics to be covered, chosen areas from the modules V4B1 or V4B2may be required


graduate seminar “Partial Differential Equations in theSciences”

4 180 6




More Information

79

ModuleS5B3

Graduate Seminar on New Developments in Partial Differential Equa-tions


6 180 h 1 semester evey semester





Learning Targets Ability to undertake independent study of an advanced topic in new developmentsin PDEs using specialized literature. Assessment, evaluation and presentation ofresults from PDEs. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents chosen topics on nonlinear PDEs, based on specialized literature, with particularemphasis on new research methods

Prerequisites none




graduate seminar “New Developments in Partial Differen-tial Equations”

4 180 6




More Information

80

ModuleS5B4

Graduate Seminar on Modeling and Simulation with Partial DifferentialEquations







Learning Targets Ability to undertake independent study of an advanced topic in modeling and simula-tion with PDEs using specialized literature. Assessment, evaluation and presentationof results from PDEs. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents chosen topics on PDEs, based on specialized literature, with particular emphasis onrecent applications in the natural sciences with interaction between modeling andnumerical treatment

Prerequisites none




graduate seminar “Modeling and Simulation with PartialDifferential Equations”

4 180 6




More Information

81

ModuleS4C1

Graduate Seminar on Discrete Optimization


6 180 h 1 semester at least every summer semester





Learning Targets Ability to undertake independent study of an advanced topic in discrete optimizationusing specialized literature. Assessment, evaluation and presentation of results fromdiscrete optimization. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents a current, active research topic in discrete optimization chosen on a rotational basiswill be treated in depth by studying the relevant literature

Prerequisites none


Knowledge of the topics covered in the module “Combinatorial Optimization”


graduate seminar “Discrete Optimization” 4 180 6




More Information The seminar theme and relevant literature, as well as the date for the initial dis-cussion and allocation of talks, will be made public towards the end of the previoussemester. No further enrolments are possible after this date. The maximum numberof participants is 15

82

ModuleS4C2

Graduate Seminar on Chip Design


6 180 h 1 semester at least every winter semester





Learning Targets Ability to undertake independent study of an advanced topic in chip design usingspecialized literature. Assessment, evaluation and presentation of results from chipdesign. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents a current, active research topic in chip design and related applications chosen on arotational basis will be treated in depth by studying the relevant literature

Prerequisites at least one of the modules “Combinatorial Optimization” and “Chip Design”


Knowledge of the topics covered in the modules “Combinatorial Optimization” and“Chip Design”


graduate seminar “Chip Design” 4 180 6





83

ModuleS4D1

Graduate Seminar on Differential Geometry


6 180 h 1 semester at least every other year





Learning Targets Ability to undertake independent study of an advanced topic in differential geometryusing specialized literature. Assessment, evaluation and presentation of results fromdifferential geometry. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents advanced topics in differential geometry and related areas based on recent specializedliterature

Prerequisites none


advanced knowledge of geometry, basic knowledge of topology


graduate seminar “Differential Geometry” 4 180 6




More Information

84

ModuleS4D2

Graduate Seminar on Topology







Learning Targets Ability to undertake independent study of an advanced topic in topology using spe-cialized literature. Assessment, evaluation and presentation of results from topo-logy. Didactic preparation and presentation as a seminar talk and in the form of amanuscript covering the contents of the talk. Competence in scientific discussions.

Contents an advanced topic in topology chosen on a rotational basis will be treated in depthby studying relevant textbooks or literature. Choice of topics:

• stable homotopy theory

• Postnikov systems

• Eilenberg-MacLane spaces

• characteristic classes

• simple homotopy theory

Prerequisites none




graduate seminar “Topology” 4 180 6





85

ModuleS4D3

Graduate Seminar on Advanced Geometry


6 180 h 1 semester at least every other year





Learning Targets Ability to undertake independent study of an advanced topic in advanced geometryusing specialized literature. Assessment, evaluation and presentation of results fromgeometry. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents current research topics in geometry

Prerequisites none


advanced knowledge of geometry, basic knowledge of topology


graduate seminar “Advanced Geometry” 4 180 6




More Information

86

ModuleS4D4

Graduate Seminar on Advanced Topology







Learning Targets Ability to undertake independent study of an advanced topic in advanced topologyusing specialized literature. Assessment, evaluation and presentation of results fromtopo-logy. Didactic preparation and presentation as a seminar talk and in the form ofa manuscript covering the contents of the talk. Competence in scientific discussions.

Contents an advanced topic in topology chosen on a rotational basis will be treated in depthby studying relevant textbooks or literature. Choice of topics:

• exotic spheres

• Hochschild and cyclic homology


• algebraic K-theory

Prerequisites none




graduate seminar “Advanced Topology” 4 180 6





87

ModuleS4E1

Graduate Seminar on Scientific Computing







Learning Targets Ability to undertake independent study of an advanced topic in scientific computingusing specialized literature. Assessment, evaluation and presentation of results fromscientific computing. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents selected topics in scientific computing or topics of current research interest

Prerequisites none


Knowledge of topics covered in the Bachelor’s modules “Algorithmische MathematikI”, “Algorithmische Mathematik II”, and “Einfuhrung in die Numerische Mathe-matik”


graduate seminar “Scientific Computing” 4 180 6




More Information

88

ModuleS4E2

Graduate Seminar on Numerical Simulation







Learning Targets Ability to undertake independent study of an advanced topic in numerical simulationusing specialized literature. Assessment, evaluation and presentation of results fromnumerical simulation. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents selected topics in numerical simulation or topics of current research interest

Prerequisites none




graduate seminar “Numerical Simulation” 4 180 6




More Information

89

ModuleS5E1

Graduate Seminar on Numerical Analysis







Learning Targets Ability to undertake independent study of an advanced topic in numerical analysisusing specialized literature. Assessment, evaluation and presentation of results fromnumerical analysis. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents Topics of current research interest in numerical analysis

Prerequisites none




graduate seminar “Numerical Analysis” 4 180 6




More Information

90

ModuleS5E2

Graduate Seminar on Efficient Simulation







Learning Targets Ability to undertake independent study of an advanced topic in efficient simulationusing specialized literature. Assessment, evaluation and presentation of results fromefficient simulation. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents Topics of current research interest in numerical simulation

Prerequisites none




graduate seminar “Efficient Simulation” 4 180 6




More Information

91

ModuleS4F1

Graduate Seminar on Probability Theory







Learning Targets Ability to undertake independent study of an advanced topic in probability theoryusing specialized literature. Assessment, evaluation and presentation of results fromprobability theory. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents A current, active topic in probability theory will be treated in depth by studyingrelevant literature.

Prerequisites none


Background required depending on the topics to be covered.


graduate seminar “Probability Theory” 4 180 6




More Information The seminar theme and relevant literature, as well as the time and place of a pre-liminary meeting with allocation of talks, will be made public towards the end of theprevious semester. No further enrolments are possible after this date. The maximumnumber of participants is 15.

92

ModuleS4F2

Graduate Seminar on Stochastic Analysis







Learning Targets Ability to undertake independent study of an advanced topic in stochastic analysisusing specialized literature. Assessment, evaluation and presentation of results fromstochastic analysis. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents A current, active topic in stochastic analysis will be treated in depth by studying therelevant literature.

Prerequisites none




graduate seminar “Stochastic Analysis” 4 180 6





93

ModuleS4F3

Graduate Seminar on Applied Probability







Learning Targets Ability to undertake independent study of an advanced topic in applied probabilityusing specialized literature. Assessment, evaluation and presentation of results fromapplied probability. Didactic preparation and presentation as a seminar talk and inthe form of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents A current, active topic in applied probability will be treated in depth by studyingthe relevant literature.

Prerequisites none




graduate seminar “Applied Probability” 4 180 6





94

ModuleP4G1

Practical Teaching Course


9 270 h 1-2 semesters every semester

Person in Charge Doerffel



of Module Master Mathematics practical teaching course, optional 1-4

Learning Targets Ability to assess, evaluate and explain mathematical arguments.

Contents Tutoring of problem sessions for a mathematics course, correction of homework,evaluation of students’ progress. Participation in the regular tutor meetings. Writinga final report to evaluate the own teaching experiences.

Prerequisites none


A solid background on the topics covered in the relevant course is required.


tutoring problem sessions (under supervision) 2 270 9

Examination graded project work and presentation (weighting 1:1)


More Information The student has to apply succesfully for a tutor position at one of the mathematicalinstitutes (MI, IAM, INS, DM) or for a tutor position for a mathematical module inanother department. If the tutor position is not at one of the mathematical institutes,the possibility to do the practical teaching course has to be confirmed by the personin charge of the module (see above).

95

ModuleP4G2

External Internship


9 270 h 6 weeks irregular

Person in Charge Rezny



of Module Master Mathematics practical training course, optional 1-4

Learning Targets Ability to apply mathematical methods to solve problems arising in industry

Contents Project in an external company that involves the application of mathematical meth-ods.

Prerequisites none


depends on the project


Practical training course, supervision by a representative ofthe company involved and by an examiner of the Master’sProgram in Mathematics

2 270 9



none

More Information The allocation of this module cannot be guaranteed. The student’s initiative inobtaining a suitable placing is required. This module should have a duration of aboutsix weeks full time and take place outside of the lecture period. Formal enrolmenttakes place when the examiner has confirmed that a suitable project has been found.

96

ModuleP4A1

Practical Project in Mathematical Logic


9 270 h 1 semester every other year during the summer free period




of Module Master Mathematics practical training course, optional 2

Learning Targets Ability to complete a practical programming project in one of the following areas:logical programming in the context of mathematical logic, automatic proof testingand automatic proving

Contents Acquaintance with the programming language Prolog and with the theoretical basisof logical programming. Study of established systems for proof testing and auto-matic proving. The programming projects comprise the implementation of simpleproof checkers and provers for different logics, the configuration of user interfacesfor available systems, the specification and configuration of modules for extensivesystems, etc.

Prerequisites none


Knowledge of predicate logic as covered in the Bachelor’s module “MathematischeLogik”, basic knowledge of computing


practical training course “Mathematical Logic” 4 270 9



none

More Information

97

ModuleP4C1

Combinatorial Algorithms


9 270 h 1 semester at least every summer semester

Person in Charge Responsible professor for Area C



of Module Master Mathematics practical programming course, optional 2 or 4

Learning Targets Ability to implement difficult combinatorial algorithms and to handle nontrivial datastructures, testing and documentation. Acquisition or extension of knowledge ofadvanced software techniques

Contents Combinatorial algorithms chosen on a rotational basis will be treated. The precisecontents will be explained during the initial discussion before the beginning of thesemester.

Prerequisites Module “Combinatorial Optimization”


good programming skills


practical programming course “Combinatorial Algo-rithms”, individual supervision

4 270 9



none

More Information The seminar theme and relevant literature, as well as the date for the initial dis-cussion and allocation of talks, will be made public towards the end of the previoussemester. No further enrolments are possible after this date. The maximum numberof participants is 10.

98

ModuleP4C2

Algorithms for Chip Design


9 270 h 1 semester at least every winter semester

Person in Charge Responsible professor for Area C



of Module Master Mathematics practical programming course, optional 3

Learning Targets Ability to implement algorithms for VLSI design and to handle very large instances,testing and documentation of the software efficiently. Acquisition or extension ofknowledge of advanced software techniques

Contents Algorithms for chip design chosen on a rotational basis will be treated. The precisecontents will be explained during the initial discussion before the beginning of thesemester.

Prerequisites at least three of the five modules “Combinatorial Optimization”, “ApproximationAlgorithms”, “Chip Design”, “Graduate Seminar on Discrete Optimization”, and“Combinatorial Algorithms”


Knowledge of the topics covered in the modules “Combinatorial Optimization”, “Ap-proximation Algorithms”, “Chip Design”, as well as a good programming skills


practical programming course “Algorithms for Chip De-sign”, individual supervision

4 270 9



none

More Information The seminar theme and relevant literature, as well as the date for the initial dis-cussion and allocation of talks, will be made public towards the end of the previoussemester. No further enrolments are possible after this date. The maximum numberof participants is 5.

99

ModuleP4E1

Practical Lab Numerical Simulation






of Module Master Mathematics Practical lab course, optional 1-4

Learning Targets Ability to implement numerical simulation methods.

Contents Image processing, flow mechanics, finite elements, financial mathematics

Prerequisites none




practical lab course “Numerical Simulation” 4 270 9



none

More Information

100

ModuleF5X1

Additional Graduate Seminar


6 180 h 1 semester Irregular





Learning Targets Ability to undertake independent study of an advanced topic in an area of mathemat-ics using specialized literature. Assessment, evaluation and presentation of resultsfrom this area. Didactic preparation and presentation as a seminar talk and in theform of a manuscript covering the contents of the talk. Competence in scientificdiscussions.

Contents The student can choose one of the graduate seminars from our master programme.The contents depend on the graduate seminar chosen.

Prerequisites Graduate Seminar


depending on the graduate seminar chosen


graduate seminar 4 180 6




More Information With this module the student can sign up for a second Graduate Seminar associatedwith a module that he has already taken, or is taking during the same semester. Hehas to prove that the contents of both seminars do not overlap significantly. Approvalof the examination board is required: The student has to apply for this module beforethe allocation of talks.

The seminar theme and the relevant literature, as well as the time and place of apreliminary meeting with allocation of talks, will be made public towards the endof the previous semester. No further enrolments are possible after this date. Themaximum number of participants is 15.

101

ModuleF5X2

Additional Advanced Topics






of Module Master Mathematics optional module, lecture course 3 or 4

Learning Targets Additional knowledge of an advanced active research area in Mathematics.

Contents The student can choose one of the advanced topics courses of 7 CP from our Masterprogramme. The contents of this module depend on the lecture chosen.

Prerequisites Advanced Topics


depending on the chosen lecture


advanced topics lecture course 4 210 7



none

More Information With this module the student can sign up for a second Graduate Lecture Course orAdvanced Topics Course associated with a module that he has already taken, or istaking during the same semester. He has to prove that the contents of both coursesdo not overlap significantly. Approval of the Examination Board is required: Thestudent has to apply for this module to the Examination board.

102

ModuleF5X3

Additional Selected Topics







Learning Targets Additional knowledge of an advanced active research area in Mathematics.

Contents The student can choose one of the selected topics lectures of 5 CP from our Masterprogramme. The contents of this module depend on the course chosen.

Prerequisites Selected Topics


depending on the selected topics lecture chosen


selected topics lecture course 2 150 5



none

More Information With this module the student can sign up for a second Graduate Lecture Course orAdvanced / Selected Topics Course associated with a module that he has alreadytaken, or is taking during the same semester. He has to prove that the contentsof both courses do not overlap significantly. Approval of the Examination Board isrequired: The student has to apply for this module to the Examination board.


103

ModuleNP420

Theoretische Physik III (Quantenmechanik)


9 270 h 1 Semester every summer term

Person in Charge Head of the examination board of the Bachelor study programme in physics

Instructors Lecturers from Physics



Bachelor Physik compulsory module 4

Bachelor Mathematik optional modules, secondarysubject physics

4 or 6

Learning Targets The ability to solve problems of non-relativistic quantum mechanics.

Contents Schrodinger equation, harmonic oscillator, linear operators on Hilbert spaces, uncer-tainty principle, theory of angular momentum, spherically symmetric potentials, hy-drogen atom, theory of spin, coupling of angular momentum, stationary perturbationtheory, systems with several electrons, Pauli principle, Helium atom, periodic system,time-dependent perturbation theory, electromagnetic transitions, golden rule.

Prerequisites none


Contents of the modules “Physik I,II,III” and “Theoretische Physik I,II” from theBachelor programme in Physics.


Course “Theoretische Physik III (Quantenmechanik)” withproblem classes

4+3 270 9

Examination Graded written examination


Successful participation in the problem classes.

More Information The module is usually taught in German.

104

ModuleNP520

Theoretische Physik IV (Statistische Physik)


9 270 h 1 Semester every winter term

Person in Charge Head of the examination board of the Bachelor study programme in physics

Instructors Lecturers from Physics



Bachelor Physik compulsory module 5

Bachelor Mathematik optional modules, secondarysubject physics

5

Learning Targets Kowledge of concepts and methods of statistical physics.

Contents Classical thermodynamics: Main theorems, thermodynamic potentials, entropy,ideal/real gases, thermodynamic machines, phase transitions. Classical and quantumstatistics: microcacnonical, canonical and grandcanonical ensemble, density opera-tor, density of states, distribution function, Fermi and Bose gas, Bose condensation,radiation of a black body, magnetism, Ising model, stochastic processes.

Prerequisites none


Contents of the modules “Physik I,II,III,IV” and “Theoretische Physik I,II,III” fromthe Bachelor programme in Physics.


Course “Theoretische Physik IV (Statistische Physik)”with problem classes

4+3 270 9

Examination Graded written examination


Successful participation in the problem classes.

More Information The module is usually taught in German.

105

module handbook master

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