Module Handbook - Master Mathematics

17/04/2012 Module Handbook - Master Mathematics

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Handbook of the Modules forMaster studies in Mathematics,

Economathematics, Technomathematics undMathematics International

at the University of Kaiserslautern

updated: WS 2011/12

This is a translation of the German version which can befound at

http://www.mathematik.uni-kl.de/CDstud/Module/ModHB_Mathematik_MAS.htm

Translation errors cannot be excluded. In case of doubt onlythe German version applies.

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Modules for all areas of mathematics (includingmain focus of study)

The following modules have a continuative nature and can beput into all areas of mathematics, especially the chosen mainfocus of study, considering the rules of the particular module.

3.1 Modules offered each winter semester

Computer AlgebraFinancialMathematics IIMathematicalStatisticsNon-Life Insurance

Numerical Methodsfor Hyperbolic PDEProbability andAlgorithmsStochasticDifferentialEquations

3.2 Modules offered each summer semester

Financial Numerical Methods



Mathematics ILife Insurance

for Elliptic andParabolic PDETheory ofScheduling Problems

3.3 Modules offered irregularly

3D Image AnalysisAdvanced NetworkFlows and SelfishRoutingAdvanced Topics inAlgebraic GeometryAlgebraicCombinatoricsAlgebraic GeometryII: Sheaves,Cohomology andApplicationsAlgorithmic GameTheoryAlgorithms inHomological andCommutativeAlgebraAnalytic NumberTheoryAnalytic NumberTheory - Part 1Analytic NumberTheory - Part 2Asymptotic AnalysisComputationalAlgebraic GeometryComputationalFinanceComputationalFlexible MultibodyDynamicsComputational FluidDynamicsContinuous-timePortfolioOptimizationCryptography –Geometric MethodsData Structures andAlgorithms forCombinatorialOptimizationDifferential-Algebraic EquationsFinancial TimeSeries / FinancialStatistics - Part 1

MathematicalMethods ofClassical MechanicsIIMathematicalTheory of FluidDynamicsMathematicalTheory of NeuralNetworks:Advanced TopicsMatroids - Theoryand ApplicationsMethods of ConvexAnalysis in ImageProcessingModularRepresentationTheoryModularRepresentationTheory IIMulticriteriaOptimizationNonlinear FunctionalAnalysis withApplications to PDENonparametricStatisticsNumerical Methodsin Control TheoryNumerical Methodsin FinanceNumerics ofStochasticProcessesOnline OptimizationOptimization withPDEPDE basedMultiscale Methodsand NumericalApproaches fortheir SolutionPermutation GroupsPoisson and LévyProcessesPotential Theory



Fundamentals ofValuation TheoryGeomathematicsGeometry ofSchemesGraphs andAlgorithmsGroups of Lie TypeH-infinity ControlHomogenizationIntersection TheoryIntroduction to theTheory of DirichletFormsInverse ProblemsKinetic and FluidDynamic EquationsLie AlgebrasLie TheoryLocally ConvexSpacesLocation TheoryMalliavin Calculusand ApplicationsMarkov SwitchingModels and theirapplications inFinanceMathematicalAspects ofEarthquakePredictionMathematicalMethods of ClassicalMechanics

Probability Theory IIProbability TheoryII: StochasticIntegralsRepresentationTheoryScientificComputing in SolidMechanicsSingularity TheorySingularity Theory -Part 1Sobolev SpacesSpatial StatisticsSpecial Functions ofMathematical (Geo-) PhysicsStability TheoryStochastic Controland FinancialApplicationsStochasticGeometryStochastic PartialDifferentialEquationsStochasticProcesses withApplications forInsurances /Financial Statistics- Part 2Systems andControl Theory:Advanced TopicsThe Mathematics ofArbitrageToric GeometryTropical GeometryWhite NoiseAnalysis

3.1 Modules offered each winter semester

Computer Algebra

Degree programmes: Master degree programmes of Mathematics,Technomathematics, Economathematics andMathematics International

Contacttime

Self-study

Effort Creditpoints

Semester Duration1



60 h 210 h 270 h 9 cp 1, 2 or 3 semester

1 Courses: Computer Algebra 4 lecture hours per week

2 Contents:· Normal forms and standard bases of ideals andmodules· Syzygies, free resolutions and the proof of theBuchberger criterion· Computation of normalization of Noetherian rings· Computation of primary decomposition of ideals · Hilbert function · Ext and Tor

3 Result of study / competences: The students have gained advanced knowledge of thetheory of Computer Algebra. They know how problemsfrom Commutative Algebra, Algebraic Geometry andtheir practical applications can be represented andsolved by algorithms. The students are able toprogram advanced algorithms of Computer Algebra.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Symbolic Computationand Commutative Algebra from the bachelor degreeprogramme of Mathematics.

5 Usability of this module:for master degree programmes mentioned above:

Main focus of study Algebraic Geometry and ComputerAlgebra (under the terms of MPO §4, clause 5).Areas of pure mathematics, applied mathematics orgeneral mathematics if the main focus of study wasnot Algebraic Geometry and Computer Algebra(considering the rules of the degree progamme whichmight restrict the above).

6 Award of credit points, examinations:Examination about this course

7 Frequency of occurrence: Each year (during the winter semester)

8 Intended size of class:about 10-25 students

9 Authorized representatives of module and mainlecturers: Prof. Dr. G.-M. Greuel, Prof. Dr. G. Pfister

Financial Mathematics II

Degree programmes: Master degree programmes of Mathematics,Technomathematics, Economathematics and



Mathematics International

Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Financial MathematicsII

2 lectures hours, 1 tutorialhour per week

2 Contents:· Basics of interest modelling (Bonds and linearproducts, swaps, caps and floors, bond options, rateof interest options, curves of interest structure, rateof interest (short rates and forward rates)) · Heath-Jarrow-Morton framework (easy example:Ho-Lee model, general HJM drift constraint, one- andmulti-dimensional modelling) · Short rate models (general single-coefficient-modelling, general valuation equation, affine modellingof interest rate structure, Vasicek-, Cox-Ingersoll-Ross- and further models, calibration of model) · Incomplete markets (Super- and sub-hedging,duality) · Defaultable bonds (Merton model)

3 Result of study / competences: The students have gained (additionally to Part I ofFinancial Mathematics) further basic knowledge ofcontinuous-time financial mathematics and thereforeare well grounded in basic training of continuous-timefinancial mathematics.

4 Conditions for participation concerning contentsof previous lectures: The course Probability Theory I from the bachelordegree programme of Mathematics.


Main focus of study Financial Mathematics (under theterms of MPO §4, clause 5). Areas of appliedmathematics or general mathematics if the mainfocus of study was not Financial Mathematics(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. J. Saß

10 Additional information:



This module can be combined with the moduleFinancial Mathematics I to a single module namedFinancial Mathematics I/II (13,5 cp).

Mathematical Statistics


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1 or 2

Duration1semester

1 Courses: MathematicalStatistics

4 lecture hours, 2 tutorialhours per week

2 Contents:· Asymptotic analysis of M-estimators, especially ofMaximum-Likelihood-estimators · Bayes- and Minimax-estimators · Likelihood-ratio-tests: asymptotic analysis andexamples (t-test, c²-goodness-of-fit-test) · Glivenko-Cantelli-theorem, Kolmogorov-Smirnov-test· Differentiable statistic functionals and examples ofapplications (derivation of asymptotic results,robustness) · Resampling methods on the basis of Bootstraps

3 Result of study / competences: The students know and understand classical andmodern asymptotic approaches and techniques ofproofs for mathematical statistics as well as theirusability to solve practical relevant problems. Thestudents are able to apply methods of mathematicalstatistics by themselves.

4 Conditions for participation concerning contentsof previous lectures: The course Stochastic Methods from the bachelordegree programme of Mathematics.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Financial Mathematics · StatisticsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).

6 Award of credit points, examinations:



Examination about this course7 Frequency of occurrence:

Each year (during the winter semester)


9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. H. von Weizsäcker

Non-Life Insurance

Degree programmes:Master degree programmes of Mathematics,Technomathematics, Economathematics andMathematics International

Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Non-Life Insurance 2 lecture hours per week

2 Contents:Collective risk models:· Models for claim number process· Poisson processes· Renewal processes· Total loss distribution· Aspects of reinsurance· Ruin theory and ruin probabilitiesExperience rating:· Bayes estimate· Linear Bayes estimate (Bühlmann and Bühlmann-Straub model)

3 Result of study / competences:The students have gained profound knowledge in themathematical and practical foundations of non-lifeinsurance mathematics.

4 Conditions for participation concerning contentsof previous lectures: Module "Probability Theory I"

5 Usability of this module:for master degree programmes mentioned above:Main focus of study from the following list (under theterms of MPO §4, clause 5):· Financial mathematics· Statistics

Areas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme which



might restrict the above).


8 Frequency of occurrence:Each year (in the winter semester)


9 Authorized representatives of module and mainlecturers:Prof. Dr. R. Korn, Prof. Dr. J. Saß

10 Additional information:This module can be combined with the module LifeInsurance to a single module named "„InsuranceMathematics“ (9 credit points).

Numerical Methods for Hyperbolic PDE


Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Numerical Methods forPDE II

4 lecture hours per week

2 Contents:Numerical methods to deal with hyperbolic differentialequations will be provided and analytically analysed.Especially the following contents will be dealt with: · Approximation methods for hyperbolic problems · Theory of weak solutions and entropy solutions · Consistency, stability and convergence · where appropriate approximation methods forsystems of conservation equations

3 Result of study / competences: The students know and understand basic concepts todeal with numerical hyperbolic differential equationsas well as mathematical techniques to analyse thesemethods. On the basis of concrete exercises thestudents have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought in Numerical Methods for HyperbolicPDE.

4 Conditions for participation concerning contentsof previous lectures: The module Differential Equations: Numerics of ODE &



Introduction to PDE


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Modelling and Scientific Computation · Partial Differential EquationsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau, Prof. Dr. D.Prätzel-Wolters

Probability and Algorithms


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Probability andAlgorithms


2 Contents:· Deterministic and randomized algorithms – concepts · Examples for randomized algorithms · Erdös' probabilistic method – a construction principlefor randomization · De-randomization strategies · Azuma's inequality and the tailbound trick · Probabilistic analysis of the travelling salesmanproblem · Markov couplings and flows in Markov chains –estimation of steady-state-approximation times andtheir application

3 Result of study / competences: The students got to know different types of



randomized algorithms. They are able to evaluate thecomplexity and efficiency of randomized algorithmsand apply examples on problems of mathematicaloptimization. In case studies the students havelearned to perform a probabilistic analysis ofalgorithms and apply methods based on Markovchains.

4 Conditions for participation concerning contentsof previous lectures: The courses Linear and Network Optimization andStochastic Methods from the bachelor degreeprogramme of Mathematics.


Main focus of study Optimization (under the terms of MPO §4, clause 5). Areas of pure mathematics,applied mathematics or general mathematics if themain focus of study was not Optimization (consideringthe rules of the degree progamme which mightrestrict the above).




9 Authorized representatives of module and mainlecturers: Priv.-Doz. Dr. K.-H. Küfer

Stochastic Differential Equations


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Stochastic DifferentialEquations


2 Contents:Stochastic differential equations (SDEs for shortnotation of stochastic differential equations) areused for modelling continuous-time randomphenomena.Key elements of the theory of stochastic differential



equations are covered. Also an introduction toalgorithmic issues is given. The following topics arecovered:· Brownian motion· Martingales theory· Stochastic integration (with respect to Brownianmotion)· Strong and weak solutions of SDEs· Stochastic representation of the solution of partialdifferential equations· Classical approximation

3 Result of study / competences:The students have acquired advanced knowledge inanalysis of stochastic differential equations. Theyhave also gained insights into the modelling andnumerical handling of SDEs.

4 Conditions for participation concerning contentsof previous lectures: Module Probability Theory I and basic knowledge ofFunctional Analysis.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied analysis· Modelling and scientific computation · Partial differential equations

Areas of abstract, applied or general mathematics ifthe main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).


7 Frequency of occurrence:Beginning with WS 12/13 each winter semester


9 Authorized representatives of module and mainlecturers:Prof. Dr. K. Ritter

10 Additional information:This module can not be combined with the module"Financial Mathematics I" in the final examination dueto high content overlap.

3.2 Modules offered each summer semester



Financial Mathematics I


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Financial MathematicsI


2 Contents:· Basics of stochastic analysis (Brownian motion, Itô-integral, Itô-formula, Martingale representationtheorem, Girsanov theorem, linear stochasticdifferential equations, Feynman-Kac formula) · Diffusion model for share prices and tradingstrategies · Completeness of market · Valuation of options with the duplication principle,Black-Scholes-formula · Valuation of options and partial differentialequations · Arbitrage bounds (Put-Call-parity, parity of pricesfor european and american calls) · Martingale method of Portfolio-Optimization

3 Result of study / competences: The students have gained basic knowledge ofstochastic analysis and continuous-time financialmathematics.

4 Conditions for participation concerning contentsof previous lectures: The course Probability Theory I.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Financial Mathematics · Statistics Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).


8 Frequency of occurrence: Each year (during the summer semester)





10 Additional information: This module can be combined with the moduleFinancial Mathematics II to a single module namedFinancial Mathematics I/II (13,5 cp) or with themodule Continuous-time Portfolio Optimization to asingle module named Financial Mathematics I;Continuous-time Portfolio Optimization (13,5 cp).

Life Insurance


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Life Insurance 2 lecture hours per week

2 Contents:· Elementary financial mathematics (calculation ofinterest)· Mortality· Insurance benefits· Net premiums and net actuarial reserves· Inclusion of costs· Life related insurance· Various Ausscheideursachen

3 Result of study / competences:The students have acquired fundamental knowledgein the mathematical and practical foundations ofclassical life insurance mathematics.

4 Conditions for participation concerning contentsof previous lectures: The course Stochastic methods from the bachelordegree programme of Mathematics.

5 Usability of this module:for master degree programmes mentioned above:Main focus of study from the following list (under theterms of MPO §4, clause 5):· Financial mathematics· StatisticsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above). .




8 Frequency of occurrence:Each year (during the summer semester)


9 Authorized representatives of module and mainlecturers:Prof. Dr. R. Korn, Prof. Dr. J. Saß

10 Additional information:This module can be combined with the moduleFinancial Mathematics I to a single module named"Financial Mathematics I; Life insurance" (13,5 creditpoints).

Numerical Methods for Elliptic and ParabolicPDE


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Numerical Methods forPDE I


2 Contents:Continuation of the courses Numerics of ODE andIntroduction to PDE. Numerical methods to deal withelliptic and parabolic differential equations will beprovided and analytically analysed. Especially thefollowing contents will be dealt with: · Approximation methods for elliptic problems · Theory of weak solutions · Consistency, stability and convergence · Approximation methods for parabolic problems

3 Result of study / competences: The students know and understand basic concepts todeal with numerical partial differential equations aswell as mathematical techniques to analyse thesemethods.

4 Conditions for participation concerning contentsof previous lectures: The modules Numerics of ODE and Introduction to PDE




Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and Scientific Computation · Partial Differential EquationsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).



8 Intended size of class:Lectures: about 15-50 students,Tutorials: about 15-25 students

9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau

Theory of Scheduling Problems


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Theory of SchedulingProblems


2 Contents:· Classification of scheduling problems · The link between scheduling and combinatorialoptimization problems · Single machine problems · Parallel machines · Job shop scheduling · Due-date scheduling · Time-Cost tradeoff Problems

In parts, only some of the headwords above as wellas further research topics will be covered in detailduring the lecture and tutorials. Details are to befound in the information system of the University ofKaiserslautern.

3 Result of study / competences: The students have gained a good overview of current



mathematical methods to solve scheduling orprocessing problems. The latter are of greatimportance for the oganisation of operative pocessesand computer science.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics;Integer Programming: Polyhedral Theory andAlgorithms


Main focus of study Optimization (under the terms of MPO §4, clause 5). Areas of applied mathematics orgeneral mathematics if the main focus of study wasnot Optimization (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Priv.-Doz. Dr. K.-H. Küfer

3.3 Modules offered irregularly

3D Image Analysis


Contacttime60 h

Self-study75 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1Semester

1 Courses:3D Image Analysis 2 lecture hours, 2

tutorial/practice hours perweek

2 Contents:Processing and statistical analysis of three-dimensional image data, in particular:



· Random closed sets and their characteristics· Discretization and three-dimensional context· Mathematical morphology· Methods of image processing: filtering,segmentation, Euclidean distance transformation,labeling, watershed transformation· Estimates of geometrical characteristics for randomclosed sets of image data

3 Result of study / competences:The students have learned basic classes of algorithmsfor processing and analysis of three-dimensionalimage data. Assuming that the pictured structure is arandom closed set, they can estimate geometricstructure characteristics from the image data. Theyare also able to process and to analyse the givenimage data by means of suitable image processingsoftware.

4 Conditions for participation concerning contentsof previous lectures:Course "Practical Mathematics: Stochastic methods"from the bachelor's degree programme ofMathematics. Advanced knowledge in stochastics(e.g. "Time Series Analysis" or "Probability Theory I")is an advantage, but is not necessarily required.


Main focus of study Statistics (under the terms of MPO §4, clause 5).

Areas of applied mathematics or general mathematicsif the main focus of study was not Statistics(considering the rules of the degree progamme whichmight restrict the above).


7 Frequency of occurrence:Irregular

8 Intended size of class:About 10-25 students

9 Authorized representatives of module and mainlecturers:Prof. Dr. J. Franke, Dr. C. Redenbach

Advanced Network Flows and SelfishRouting

Degree programmes: Master degree programmes of Mathematics,



Technomathematics, Economathematics andMathematics International

Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Advanced NetworkFlows and SelfishRouting


2 Contents:· Basics of network flows · Efficient methods to compute max flows (Dinic'salgorithm, Push-Relabel method) · Polynomial and strong polynomial methods for mincost flows · Dynamic network flows (dynamic max-flow-min-cuttheorem, temporally repeated flows) and networksexpanded over time · Flows with flow-depending costs, Optimality criteria · Flows in Nash-equilibrium · Price of the anarchy (lower and upper bounds) · Paradox of Braess and its consequences · Network design for selfish users · Congestion Games

3 Result of study / competences: The students are able to cope with advancedtechniques to compute network flows. The know andunderstand different accesses to network flows,esspecially those coming from game theory, withapplications in traffic planning and economic science.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics




7 Frequency of occurrence: Irregular


9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Prof. Dr. S. Krumke, Jun. Prof.Dr. S. Ruzika



Advanced Topics in Algebraic Geometry


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Advanced Topics inAlgebraic Geometry


2 Contents:A selection of topics from the following areas will behandled:· Sheaves and cohomology theory · Divisors and line bundles · Differential forms · Proj constructions and applications · Blowing up · Relative rational variaties · Hilbert schemes · Moduli spaces

3 Result of study / competences: The students have acquired advanced knowledge inthe theory of modern algebraic geometry whichenables them to understand the current work inalgebraic geometry. They are able to independentlywork on special problems in this area and solve themas a part of their thesis.

4 Conditions for participation concerning contentsof previous lectures: Module Algebraic Geometry.


Main focus of study Algebraic Geometry andComputer Algebra (under the terms of MPO §4,clause 5).

Areas of pure mathematics or general mathematics ifthe main focus of study was not Algebraic Geometryand Computer Algebra (considering the rules of thedegree progamme which might restrict the above).






9 Authorized representatives of module and mainlecturers: Prof. Dr. W. Decker, Prof. Dr. G. Trautmann

10 Additional information:: This module can not be combined with the module"Algebraic Geometry II: Sheaves, Cohomology andApplications" in the final examination due to highcontent overlap.

Algebraic Combinatorics


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: AlgebraicCombinatorics


2 Contents:· Principle of inclusion and exclusion · Formal power series and combinatoric applications · Polya-Redfield-counting · Eigenvalue-theory of graphs · Strong regular graphs · Designs and difference quantities

3 Result of study / competences: The students know and understand applications ofalgebraic structures (eigenvalue-theory, polynomialrings, formal power series, finite fields, results ofgroup theory) to combinatoric tasks.

4 Conditions for participation concerning contentsof previous lectures: The course Algebraic Structures from the bachelordegree programme of Mathematics. Selected termsand results from the course Introduction to Algebrafrom the bachelor degree programme of Mathematicsand from the course Group Theory will be sufficientlydiscussed during this lecture. Therefore, theselectures are good additions but not necessarilyrequired.


Main focus of study Algebra and Number Theory



(under the terms of MPO §4, clause 5). Areas of puremathematics or general mathematics if the main focusof study was not Algebra and Number Theory(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. U. Dempwolff

Algebraic Geometry II: Sheaves,Cohomology and Applications


Contacttime60 h

Self-study210 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Algebraic Geometry II:Sheaves, Cohomologyand Applications


2 Contents:· Sheaves and sheaf cohomology· Divisors and line bundles· Differential forms· Applications and examples (e.g. Riemann-Hurwitz-formula, Riemann-Roch theorem, projectiveembedding, blow-ups, Grassmann-varieties)

3 Result of study / competences:The students are well grounded in knowledge of thetheory of modern algebraic geometry which enablesthem to understand current works in algebraicgeometry. They particulary know and understandtypical examples and applications.

4 Conditions for participation concerning contentsof previous lectures:Module Algebraic Geometry.




Main focus of study Algebraic Geometry and ComputerAlgebra (under the terms of MPO §4, clause 5).Areas of pure mathematics, applied mathematicas orgeneral mathematics if the main focus of study wasnot Algebraic Geometry and Computer Algebra(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. W. Decker, Prof. Dr. A. Gathmann

Algorithmic Game Theory


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Algorithmic GameTheory


2 Contents:Cooperative Game Theory:

Games in characteristic function formSolution concepts, e.g., core, Shapley valueComplexity of the computation of solutionconceptsCost allocation problemsApplication, e.g., optimization problems in multi-player-situations

Non-cooperative Game Theory:

Equilibria, e.g., Nash Equilibria, dominantstrategiesComplexity of the computation of equilibriaIntroduction to mechanism design, e.g., truthfulmechanisms

3 Result of study / competences:The students know and understand different solution



concepts of cooperative and non-cooperative gametheory. They are able to evaluate the complexity ofsolutions of cooperative and non-cooperative gamesand to get the solutions with help of optimizationmethods. During the exercises students havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought duringthe lectures.

4 Conditions for participation concerning contentsof previous lectures: The course "Linear and Network Optimization" fromthe bachelor degree programme of Mathematics, themodule "Integer Programming: Polyhedral Theory andAlgorithms".


Main focus of study Optimization (under the terms of MPO §4, clause 5). Areas of applied mathematics orgeneral mathematics if the main focus was different(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. H. Hamacher, Prof. Dr. S. Krumke

Algorithms in Homological and CommutativeAlgebra


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Algorithms inHomological andCommutative Algebra


2 Contents:· Normalization, integrity bases



· Application: parameterization of rational curves · Flatness, depth and codimension, Cohen-Macaulayrings · Application: invariant rings

3 Result of study / competences:The students have learnt how to handle the mainconcepts from the field of the algorithms inhomological and commutative algebra.

4 Conditions for participation concerning contentsof previous lectures: The courses "Commutative Algebra" and "ComputerAlgebra".


Main focus of study "Algebraic Geometry andComputer Algebra" (under the terms of MPO §4,clause 5).Areas of pure mathematics or general mathematics ifthe main focus was not "Algebraic Geometry andComputer Algebra" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. W. Decker, Prof. Dr. G. Pfister

10 Additional information:: This module can not be combined with the module"Computational Algebraic Geometry" in the finalexamination due to high content overlap.

Analytic Number Theory


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Analytic NumberTheory




2 Contents:· Dirichlet series · Prime numbers in arithmetic progressions · Functional equation of the Riemann zeta function · L-functions and Gaussian sums· Binary quadratic forms · Class numbers of quadratic fields

3 Result of study / competences: The students know and understand basic methodsand results of Analytic Number Theory.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Algebra and Introductionto Complex Analysis from the bachelor degreeprogramme of Mathematics.


Main focus of study Algebra and Number Theory(under the terms of MPO §4, clause 5). Areas of puremathematics or general mathematics if the main focusof study was not Algebra and Number Theory(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. G. Malle

Analytic Number Theory - Part 1


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Analytic NumberTheory - Part 1

2 lecture hours, 1 tutorialhour per week

2 Contents:· Dirichlet series and application on asympotic



questions · L-Series and Dirichlet theorem on prime numbers inarithmetic progressions · Riemann zeta function and prime number theorem · Gamma function

3 Result of study / competences: The students know and understand basic methodsand results of Analytic Number Theory, in particularthe properties of the Riemann zeta function, theGamma function and the L-Series.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Algebra and Introductionto Complex Analysis from the bachelor degreeprogramme of Mathematics.


Main focus of study Algebra and Number Theory(under the terms of MPO §4, clause 5). Areas ofpure mathematics or general mathematics if the mainfocus of study was not Algebra and Number Theory(considering the rules of the degree progamme whichmight restrict the above).





10 Additional information:This module is part of the module Analythic NumberTheory (9 credit points).

Analytic Number Theory - Part 2


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Analytic Number 2 lecture hours, 1 tutorial



Theory - Part 2 hour per week

2 Contents:· L-functions and Gaussian sums· Binary quadratic forms · Class numbers of quadratic fields

3 Result of study / competences: The students know and understand basic methodsand results of Analytic Number Theory.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Algebra and Introductionto Complex Analysis from the bachelor degreeprogramme of Mathematics. The course AnalyticNumber Theory - Part 1.


Main focus of study Algebra and Number Theory(under the terms of MPO §4, clause 5). Areas ofpure mathematics or general mathematics if the mainfocus of study was not Algebra and Number Theory(considering the rules of the degree progamme whichmight restrict the above).





10 Additional information:This module is part of the module Analythic NumberTheory (9 credit points).

Asymptotic Analysis


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester2 or 3

Duration1semester

1 Courses: Asymptotic Analysis 2 lecture hours per week



2 Contents:Mathematical techniques of flow computation and thetheory of asymptotic expansion for differentialequations will be provided and analyzed. Especiallythe following contents will be covered: · Regular and singular disturbed problems · Scaling · Multi-scale expansions · Boundary layers on differential equations

3 Result of study / competences: The students know and understand advancedmethods of asymptotic expansions for equations,especially differential equations. Based on concreteexercises the students have developed a skilled,precise and independent handling of the terms,propositions and methods tought in AsymptoticAnalysis.

4 Conditions for participation concerning contentsof previous lectures: Module Differential Equations: Numerics of ODE &Introduction to PDE


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · Geomathematics · Modelling and scientific computation · Partial Differential Equations Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau, Prof. Dr. W.Freeden

Computational Algebraic Geometry

Degree programmes: Master degree programmes of Mathematics,Technomathematics, Economathematics and



Mathematics International

Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1Semester

1 Courses: ComputationalAlgebraic Geometry


2 Contents:· Hilbert function and projective varieties (Hilbertpolynomial, Hilbert-Poicaré series, Hilbert-Samuelfunction, theory of dimension, projective morphismsand elimination, degree of variety, local vs. globalproperties)· Homological algebra (flatness, depth, codimension,Cohen-Macaulay rings)· Solving systems of polynomial equations (resultants,triangular sets)

3 Result of study / competences: The students have learned to apply algorithms on theproblems of Algebraic Geometry solve them withcompouter algebra methods.

4 Conditions for participation concerning contentsof previous lectures: The course Algebraic Geometry, Commutative Algebraand Computer Algebra.


Main focus of study Algebraic Geometry and ComputerAlgebra (under the terms of MPO §4, clause 5). Areasof applied mathematics or general mathematics if themain focus of study was not Algebraic Geometry andComputer Algebra (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. G. Pfister

Computational Finance





Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Computational Finance 2 lecture hours per week

2 Contents:· Standard models: Black-Scholes, Heston and otherSV models, local volatility · Choice of model and calibration · Options evaluation: analytical formula, PDE, Monte-Carlo simulation, trees · Exotic options and certificats · Selected topics on Monte-Carlo simulations:generation of random variables, numerical methods forSDE, variance reduction, stochastic Taylor expansion · Selected topics on numerics of PDE in the contextof options evaluation

3 Result of study / competences: The students extend their knowledge in the area ofFinancial mathematics I and II with theoretical andpractical options evaluation.

4 Conditions for participation concerning contentsof previous lectures: The course Probability Theory I. Addditionalknowledge from Financial mathematics I is useful butnot required.


Main focus of study Financial mathematics (under theterms of MPO §4, clause 5). Areas of appliedmathematics or general mathematics if the main focusof study was not Financial mathamtics(consideringthe rules of the degree progamme which mightrestrict the above).




9 Authorized representatives of module and mainlecturers: Dr. G. Dimitroff, Prof. Dr. R. Korn

Computational Flexible Multibody Dynamics




Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1,2 or 3

Duration1semester

1 Courses: Computational FlexibleMultibody Dynamics


2 Contents:· Modeling multibody dynamics · Elastic body and floating frame of reference · Spatial discretization with Galerkin projection· Rigid mechanical system· Implicit time integration

3 Result of study / competences: The students know and understand basic conceptsfor modeling and numerical analysis of flexiblemultibody dynamics. During the exercises itegratedinto the course the students have developed askilled, precise and independent handling of theterms, propositions and methods of the subject area.

4 Conditions for participation concerning contentsof previous lectures: The course "Introduction to Ordinary DifferentialEquations" from the bachelor degree programme ofMathematics; module Numerics of ODE.


Main focus of study "Modelling and scientificcomputation" (under the terms of MPO §4, clause 5).

Areas of applied mathematics or general mathematicsif the main focus of study was not "Modelling andscientific computation" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. P. Simeon



Computational Fluid Dynamics


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester2 or 3

Duration1semester

1 Courses: Computational FluidDynamics


2 Contents:Mathematical concepts to deduce the Navier-Stokesequations by conservation principles as well asnumerical methods to find their solution will beprovided and alalyzed. Especially the followingcontents will be covered: · Derivation of Stokes and Navier-Stokes equations · Solution methods for the Stokes equation · Approximation methods for equations of fluiddynamic

3 Result of study / competences: The students know and understand advancedmethods of finding numerical solutions for fluiddynamic equations. Based on concrete exercises thestudents have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought in Asymptotic Analysis.

4 Conditions for participation concerning contentsof previous lectures:

Modules Differential Equations: Numerics of ODE &Introduction to PDE and Numerics of PDE I, desirable:Numerics of PDE II


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and scientific computation · Partial Differential Equations Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).



8 Intended size of class:



about 10-25 students9 Authorized representatives of module and main

lecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau

Continuous-time Portfolio Optimization


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Continuous-timeportfolio optimization


2 Contents:· Introduction to Portfolio-Optimization (problemstatement) · Continuous-time portfolio problem : expectedbenefit approach · Martingale method for complete markets · Stochastic control approach (HJB equation,verification theorems) · Portfolio-Optimization with restrictions (e.g. riskboundaries, transaction costs)· Alternative methods

3 Result of study / competences: The students have gained advanced knowledge ofContinuous-time Financial Mathematics. They werefamilarized with a current specialization and researchtopic.

4 Conditions for participation concerning contentsof previous lectures: Module Financial Mathematics I


Main focus of study Financial Mathematics (under theterms of MPO §4, clause 5). Areas of appliedmathematics or general mathematics if the mainfocus of study was not Financial Mathematics(considering the rules of the degree progamme whichmight restrict the above). The module forms a basisfor master theses and other research topics in thearea of Financial Mathematics.







10 Additional information: This module can be combined with the moduleFinancial Mathematics I to a single module namedFinancial Mathematics I; Continuous-time PortfolioOptimization (13,5 credit points).

Cryptography – Geometric Methods


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Cryptography –Geometric Methods

2 lectures hours, 1 tutorialhour per week

2 Contents:· Discrete logarithm, cryptographic technique of ElGamal · Projektive coordinates, plane algebraic curves,Bezout's theorem · Elliptic curves, normal forms, isogeny, complexmultiplication · Methods to identify the number of rational points · Construction of elliptic curves, modules

3 Result of study / competences: The students became acquainted with relevantapplications (elliptic curves) that modern algorithmsof Cryptography are based on. Thereby, they gainedbasic knowledge of the theory of plane algebraiccurves. Furthermore, the students have learned toabstractly compute in characteristics different tozero. By doing so, they broadend their viewinghorizon. They were sensibilized for the standardsneeded in practical situations (algorithmicimplementation) concerning mathematical theories.

4 Conditions for participation concerning contentsof previous lectures: The courses Algebraic Structure and Introduction toAlgebra from the bachelor degree programme ofMathematics.




Main focus of study Algebraic Geometry andComputer Algebra (under the terms of MPO §4,clause 5). Areas of pure mathematics, appliedmathematics or general mathematics if the mainfocus of study was not Algebraic Geometry andComputer Algebra (considering the rules of thedegree progamme which might restrict the above).



8 Intended size of class:Lectures: about 15-50 students, Tutorials: about 15-25 students

9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Gathmann, Prof. Dr. G. Pfister

10 Additional information: This module is part of the module Algebraic Curvesand Cryptography and can be combined with themodule Cryptography – Number Theoretic Methods toa single module named Cryptography – NumberTheoretic and Geometric Methods (9 credit points).

Data Structures and Algorithms forCombinatorial Optimization


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Data Structures andAlgorithms forCombinatorialOptimization


2 Contents:· Amortized analysis · Heaps (d-Heaps, Binomial-Heaps, Fibonacci-Heaps,Leftist-Heaps) · Acceleration of shortest-path-methods by priorityqueues



· Union-find-structures · Search trees, self-organized data structures (Splay-Trees) · Dynamic Trees and their applications for flowalgorithms · Parametric search

3 Result of study / competences: The students know and understand advancedtechniques to efficiently implement running time andcapacity for algorithms of combinatorial optimization.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics.




7 Frequency of occurrence: Each year


9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Prof. Dr. S. Krumke

Differential-Algebraic Equations


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Differential-AlgebraicEquations


2 Contents:The theory and numerical analysis of differential-algebraic equations will be handled, in particular:· Application fields (multibody mechanical systems and



electrical circuits)· Solution theory and index concepts · Relation with singularly perturbed problems · BDF and IRK methods

3 Result of study / competences: The students know and understand basic concepts ofthe theory and numerical analysis of differential-algebraic equations. They have developed a skilled,precise and independent handling of the terms,propositions and methods tought during the lectures.

4 Conditions for participation concerning contentsof previous lectures: Module "Numerics of ODE"


Main focus of study "Modelling and ScientificComputation" (under the terms of MPO §4, clause 5).

Areas of applied mathematics or general mathematicsif the main focus of study was not "Modelling andScientific Computation" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. B. Simeon

Financial Time Series / Financial Statistics -Part 1


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Financial Statistics 2 lecture hours per week

2 Contents:Statistics of Financial Markets: · Schemes and estimation procedures for financial



time series (ARCH, GARCH and generalizations),Value-at-Risk · Copulas and its application for risk managementbased on multivariate data

Risk in Finance and Insurance: · Statistical methods to estimate the probability ofextreme events or extreme quantiles

3 Result of study / competences: The students know and understand advancedstatistical methods to model and estimate risks,especially those concerning financial managementand insurance business.

4 Conditions for participation concerning contentsof previous lectures: The course Regression and Time Series Analysis.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Financial Mathematics · Statistics Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. J. Saß, Dr. J.-P. Stockis

10 Additional information: This module can be combined with the moduleStochastic processes with applications for insurances/ Financial statistics - part 2 to a single modulenamed Financial statistics (9 credit points).

Fundamentals of Valuation Theory


Contact Self- Credit Duration



time 30 h

study 105 h

Effort 135 h

points 4,5 cp

Semester1, 2 or 3

1semester

1 Courses: Fundamentals ofValuation Theory


2 Contents:· Valuation rings and integral closure· Extension of valuation on field extensions · Completion and henselization of valuated field · Fundamental invariants (inertia degree, ramificationindex, defect) and their characteristics · Hilbert ramification theory · Approximation theorems and divisors

3 Result of study / competences: The students know and understand the basicconcepts and methods of valuation theory. Theyhave gained basic knowledge about algebraicstructures, learnt to understand more complexstructures and to work with them confidently.

4 Conditions for participation concerning contentsof previous lectures: The course "Introduction: Algebra" from the bachelordegree programme. Knowledge of the course"Commutative algebra" is an advantage, but is notnecessarily required.


Main focus of study "Algebra and Number Theory"(under the terms of MPO §4, clause 5).

Areas of pure mathematics, applied mathematics orgeneral mathematics if the main focus of study wasnot "Algebra and Number Theory" (considering therules of the degree progamme which might restrictthe above).




9 Authorized representatives of module and mainlecturers: Dr. H. Knaf, Prof. Dr. G. Malle

Geomathematics

Degree programmes:



Master degree programmes of Mathematics,Technomathematics, Economathematics andMathematics International

Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Geomathematics 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Mathematical models, methods, and procedures aswell as different types of representation· Determination of the gravitational field of the Earth(Newton potential, gravity, gravitation, Earth'srotation, gravity anomaly, determination of geoids,normal gravity, deviation from deflections of thevertical, gravity gradients and tensorial gradiometrydata)· Determination of magnetic field of the Earth (main(dipole) field, permanent field, anomaly of magneticfield, ionospheric currents)· Deformation analysis (elastic field, behaviour ofstrain and stress, elastic waves)

3 Result of study / competences: The students have gained advanced knowledge ofapplying mathematical models to problems ofGeomathematics. They know and understand thefundamentals of the solution theory of relevantgeophysical and geodetic equations. They also havegained a deepened understanding of mathematicalmethods to model and solve analytical problems ofrelevant geoscientific areas and fields.

4 Conditions for participation concerning contentsof previous lectures: The course Constructive Approximation.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Applied Analysis · GeomathematicsAreas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).






9 Authorized representatives of module and mainlecturers: Prof. Dr. W. Freeden

Geometry of Schemes


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1Semester

1 Courses: Geometry of Schemes 2 lecture hours per week

2 Contents:· Theory of schemes (affine, projective and relativeschemes)· Structure sheaves and mudule scheaves · Flat families· Grothendieck functor

3 Result of study / competences: The students know the advanced language ofschemes, used in the modern algebraic geometry andare able to understand current works in geometry andarithmetics. Typical examples and applications areshown.

4 Conditions for participation concerning contentsof previous lectures: The course Algebraic Geometry and CommutativeAlgebra.


Main focus of study Algebraic Geometry and ComputerAlgebra (under the terms of MPO §4, clause 5). Areasof applied mathematics or general mathematics if themain focus of study was not Algebraic Geometry andComputer Algebra (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:



Prof. Dr. A. Gathmann, Priv-Doz. Dr. J. Zintl

Graphs and Algorithms


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Graphs and Algorithms 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Graphs and digraphs· Graph algorithms, basic complexity terms (P, NP),construction of graphs · Paths, cycles, connection · Eulerian and Hamiltonian cycles· Colouring and covering· Location problems on graphs· Perfect graphs, efficient algorithms for chordalgraphs · Transitive hulls, irreducible kernels · Trees, forests, matroids · Search strategies: Depth First Search, Breadth FirstSearch · Matchings · Planar graphs

3 Result of study / competences: The students are able to cope with advanced graphtheoretic methods to model and solve combinatorialproblems and their applications.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network optimization from thebachelor degree programme of Mathematics.

5 Usability of this module:for master degree programmes mentioned above:Main focus of study Optimization (under the terms of MPO §4, clause 5). Areas of pure mathematics,applied mathematics or general mathematics if themain focus of study was not Optimization (consideringthe rules of the degree progamme which mightrestrict the above).






9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Prof. Dr. S. Krumke, Jun. Prof.Dr. S. Ruzika

Groups of Lie Type


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Groups of Lie Type 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Chevalley bases for semisimple Lie algebras· Chevalley Groups· Bruhat decomposition· (B, N)-pairs and the simplicity of Chevalley Groups· Central extensions· Chevalley Groups of finite order· Automorphisms of Chevalley groups· Twisted Chevalley groups

3 Result of study / competences:The students know and understand basic methodsand principles of the theory of Lie groups. They gotto know important examples and are able to studythose by scientific methods.

4 Conditions for participation concerning contentsof previous lectures: The course "Introduction: Algebra" from the bachelordegree programme of Mathematics, module "LieAlgebras".

5 Usability of this module:for master degree programmes mentioned above:Main focus of study „Algebra and Number Theory“(under the terms of MPO §4, clause 5). Areas of puremathematics or general mathematics if the main focusof study was not „Algebra and Number Theory“(considering the rules of the degree progamme whichmight restrict the above).






9 Authorized representatives of module and mainlecturers:Prof. Dr. U. Dempwolff, Prof. Dr. G. Malle

H-infinity Control


Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: H-infinity Control 4 lecture hours per week

2 Contents:Problems of robust control will be covered, especially:· Signal spaces with related norms and operators(Lebesgue- and Hardy spaces) · Indefiniteness and robustness of models (Small GainTheorem) · Parameterisation of stabilizing controller (Youla-parameterization) · LQ- and H-infinity-controller, Riccati equations

3 Result of study / competences: The students know and understand basic principles todeal with robust controlling problems. By means ofconcrete exercises they have developed a skilled,precise and independent handling of the terms,propositions and methods tought in H-infinity Control.

4 Conditions for participation concerning contentsof previous lectures: The module Systems Theory: Systems and ControlTheory & Neural Networks


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and Scientific Computation · Systems and Control Theory<Areas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).






9 Authorized representatives of module and mainlecturers: Prof. Dr. T. Damm, Prof. Dr. D. Prätzel-Wolters

Homogenization


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Homogenization 2 lecture hours per week

2 Contents:Homogenization methods for modeling processes bypartial differential equations with rapidly oscillatingcoefficients or boundary value problems in areas withcomplex microstructure will be handled, in particular: · Variational methods for partial differential equations,· Special types of convergence in homogenization· Homogenization of boundary value problems of thesecond kind in areas with complex microstructure(especially: heat conduction equations, elasticity,viscoelasticity, fluid flow in porous media)

3 Result of study / competences: The students know and understand the advancedmethods of homogenization and their applications inprocess modeling. By means of concrete exercisesthey have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought in the lecture.

4 Conditions for participation concerning contentsof previous lectures: The course "Numerical Methods for Elliptic andParabolic PDE". Knowledge of the course "FunctionalAnalysis" is an advantage, but is not necessarilyrequired.


Main focus of study from the following list (under the



terms of MPO §4, clause 5):· Modelling and Scientific Computation · Partial Differential Equations

Areas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar

Intersection Theory


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Intersection Theory 2 lecture hours per week

2 Contents:· Algebraic cycles on algebraic varieties· Rational equivalence· Chow groups· Functoriality· Chern classes of algebraic vector bundles· Intersection pairings· Intersection theory for non-singular varieties· Chowrings of classical rational varieties· Intersection theory for singular varieties

3 Result of study / competences:The students have gained profound practicalknowledge of how to deal with general and specificvarieties. They have learned to use a variety ofmethods of algebraic geometry, which enables themto deepen in this area and its applications.

4 Conditions for participation concerning contentsof previous lectures: Courses "Algebraic Geometry" and "AlgebraicTopology".




Main focus of study "Algebraic Geometry andComputer Algebra" (under the terms of MPO §4,clause 5).

Areas of pure mathematics or general mathematics ifthe main focus of study was not "Algebraic Geometryand Computer Algebra" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. A. Gathmann, Prof. Dr. G. Trautmann

Introduction to the Theory of DirichletForms


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Introduction to theTheory of DirichletForms


2 Contents:· Resolvents, semigroups, generators (Theorem ofHille and Yosida) · Coercive bilinear forms (Stampacchia theorem,characterization by resolvents, semigroups,generators) · Closed bilinear forms · Contraction properties (Sub-Markov property,Dirichlet operators, Dirichlet forms)

3 Result of study / competences: The students have gained deepened knowledge of asubdomain of Functional Analysis with applications to



up-to-date research topics (e.g. in (Stochastic,Partial) Differential Equations or MathematicalPhysics).

4 Conditions for participation concerning contentsof previous lectures: The course Functional Analysis.


Main focus of study Applied Analysis (under the termsof MPO §4, clause 5). Areas of pure mathematics,applied mathematics or general mathematics if themain focus of study was not Applied Analysis(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. M. Grothaus

Inverse Problems


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:

Inverse Problems4 lecture hours, 2 tutorialhours per week

2 Contents:· Introductive examples: tomography, gravimetry,downward continuation of satellite data· Ill-posed operator equations · Regularization methods (singular valuedecomposition, Tikhonov-regularization, iterativemethods, multi-resolution techniques)

3 Result of study / competences: The students have gained mathematical skills in thearea of inverse problems (to get information about



inaccessible processes and features from measurableand/or observable effects). They are able to copewith stabilization and regularization techniques forrelevant geomathematical problems. They can alsojustify the theory with an experiment (clarification bymeans of examples).

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Functional Analysis fromthe bachelor degree programme of Mathematics.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · GeomathematicsAreas of applied mathematics or general mathematicsif the main focus of the study was not one of theabove (considering the rules of the degree progammewhich might restrict the above).





Kinetic and Fluid Dynamic Equations


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Kinetic and FluidDynamic Equations


2 Contents:Modeling and numerical analysis of kinetic equationsand relation of these equations to fluid dynamicequations will be handled, in particular:· Application Fields· Transport equations and the Boltzmann equation



· Scaling limits and asymptotic analysis of kineticequations

3 Result of study / competences:The students know and understand basic concepts ofasymptotic and numerical analysis of kineticequations. They have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought during the lectures.

4 Conditions for participation concerning contentsof previous lectures: Module "Numerics of ODE ", "PDE: An Introduction".


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and scientific computation · Partial Differential Equations Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. A. Klar

Lie Algebras


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Lie Algebras 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Finite root systems · Solvable and nilpotent Lie algebras · Classification of the semi-simple, complex Lie



algebras· Representation theory of the semisimple complex LieAlgebras

3 Result of study / competences: The students know and understand basic methodsand principles of the theory of Lie algebras. They gotto know important examples and are able to studythose by scientific methods.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Algebra from the bachelordegree programme of Mathematics.







Lie Theory


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Lie Theory 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Topological groups, classical groups · Exponential function, BCH-formula · Linear Lie groups and their Lie algebras · Homomorphisms and their differentials



· Universal covering groups · Local linear Lie groups, differentiable manifolds · Survey of the theory of Lie algebras (easy real Liealgebras) · Survey of the group representation of Lie algebras(maximum weights)

3 Result of study / competences: The students know and understand basic methodsand principles of the theory of Lie groups. They gotto know important examples and are able to studythose by scientific methods.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Algebra from the bachelordegree programme of Mathematics; knowledge fromthe course Introduction to Topology from thebachelor degree programme of Mathematics isdesirable but not necessarily required.






9 Authorized representatives of module and mainlecturers: Prof. Dr. U. Dempwolff , Prof. Dr. G. Malle

Locally Convex Spaces


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Locally Convex Spaces 4 lecture hours, 2 tutorial

hours per week



2 Contents:· Topological fundamental concepts (neighbourhoodfilter, base of neighbourhoods, continuous mappings,product topology), directed systems, compact sets,topological spaces (convex, absolutely convex,circular, symmetrical and absorbing sets)· Locally convex spaces (definition through a base ofneighborhoods, characterization by means ofMinkowski functionals, metrizable locally convexspaces, Fréchet spaces)· Linear maps and Hahn-Banach theorem, Mazurtheorem for locally convex spaces, Krein Milmantheorem about extreme points of compact, convexsets· Introduction into the space of distributions· Open mapping theorem for Fréchet spaces,characterization of graph-closed mappings, closedgraph theorem for Fréchet spaces· Barrelled spaces, equicontinuous mappings, Banachand Banach-Steinhaus theorems· Dual systems, weak topology, topology of the dualpair· Polar sets, bipolar theorem, Alaoglu-Bourbakitheorem· Polar topologies

3 Result of study / competences:Students know and understand mathematicalconcepts in infinite-dimensional spaces with particularemphasis on the topological aspects. They master thebasic tools which are then used in advanced coursesof infinite-dimensional analysis, i.e. in White NoiseAnalysis. During the exercises students havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought in thelectures.

4 Conditions for participation concerning contentsof previous lectures: Course "Introduction: Functional Analysis" from theBachelor program in Mathematics. Knowledge of thecourse "Introduction: Topology" from the Bachelorprogram is helpful but not required.


Main focus of study "Applied Analysis" (under theterms of MPO §4, clause 5).Areas of pure mathematics or general mathematics ifthe main focus of study was not Applied Analysis(considering the rules of the degree progamme whichmight restrict the above).


7 Frequency of occurrence:



Irregular8 Intended size of class:

about 10-25 students

9 Authorized representatives of module and mainlecturers:Prof. Dr. M. Grothaus, Prof. Dr. B. Rosenberger

Location Theory


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Location Theory 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Classification schemes for location problems · Location problems in the plane · Location problems on networks · Discrete location problemsIn parts, only some of the headwords above as wellas further research topics will be covered in detailduring the lecture and tutorials. Sometimes thelecture will be extended to two lectures covering twosemesters. Details are to be found in the informationsystem of the University of Kaiserslautern.

3 Result of study / competences: The students have gained a good survey of recentmethods to solve location problems. They are veryimportant for many applications in industry andbusiness.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics;depending on the choice of the main focus of studyalso knowledge from the course Integer Programming:Polyhedral Theory and Algorithms might be required.








9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher

Malliavin Calculus and Applications


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1,2 or 3

Duration1semester

1 Courses: Malliavin Calculus andApplications


2 Contents:Fundamentals of Malliavin calculus: · Wiener chaos decomposition · Malliavin derivative · Divergence operator and stochastic integration Applications:· Regularity of probability measures · Anticipatory stochastic differential equations · Malliavin calculus in financial mathematics · Limit theorems

3 Result of study / competences: The students have acquired basic knowledge inMalliavin calculus and its applications.

4 Conditions for participation concerning contentsof previous lectures: Basic knowledge of Stochastic analysis (e.g. from thecourses "Financial Mathematics I" or "StochasticDifferential Equations") and in Functional analysis.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · Financial Mathematics Areas of pure mathematics, applied mathematics or



general mathematics if the main focus of study wasnot chosen from the above list (considering the rulesof the degree progamme which might restrict theabove).




9 Authorized representatives of module and mainlecturers: Jun. Prof. Dr. A. Neuenkirch, Prof. Dr. K. Ritter

Markov Switching Models and theirapplications in Finance


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Markov SwitchingModels and theirapplications in Finance


2 Contents:· Choice of model among Markov switching models incontinuous time· Hidden markov models in discrete time· Filter approaches and parameter estimators,inference· Predictions of financial time series in these models- Stock price modelling- Portfolio optimization

3 Result of study / competences:Students know and understand properties of Markovswitching models that are suitable for modellingfinancial time series and their applications.

4 Conditions for participation concerning contentsof previous lectures: The course "Practical Mathematics: Stochasticmethods" from the bachelor degree programme ofMathematics. Advanced knowledge in stochastics(e.g. "Time Series Analysis" or "Financial MathematicsI") is an advantage, but is not necessarily required.




Main focus of study from the following list (under theterms of MPO §4, clause 5): · Financial Mathematics· Statistics

Areas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. J. Franke, Dr. C. Erlwein

Mathematical Aspects of EarthquakePrediction


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Mathematical Aspectsof EarthquakePrediction


2 Contents:· Mathematical basics of continuum mechanics andelasticity theory (Lagrange and Euler representation,conservation laws, stress tensor, Euler equation ofmotion) · Linearization (amongst others Hooke’s law),isotropy, Cauchy-Navier-equation · Natural oscillation (Hansen vectors, toroidal andpoloidal oscillations, spherical Bessel functions) · Basics of tensor calculations · Representation of seismic sources (utilitiesdistributions, momentum tensor, beachballrepresentation)



· Theory of seismic rays for a circular model of theEarth (eikonal equation, Snell’s law, ray parameter,Benndorf relation, Herglotz-Wiechert-formula, lowvelocity zone (LVZ) and zone of rapid velocityincrease)

3 Result of study / competences: The students know and understand the mathematicalbasics of seismology. They can cope with techniquesto model seimological phenomena and have especiallygained a practical understanding of the differenttypes of mathematical techniques.

4 Conditions for participation concerning contentsof previous lectures: The course Constructive Approximation.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · GeomathematicsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).





Mathematical Methods of ClassicalMechanics


Contacttime45 h

Self-study90 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Mathematical Methodsof Classical Mechanics




2 Contents:· Calculus of variations· Manifolds· Lagrange's equations of motion· Hamiltonian systems· Differential algebraic equations

3 Result of study / competences:The students know and understand the basicconcepts of modelling and analysis in classicalmechanics.

During the exercises students have developed askilled, precise and independent handling of theterms, propositions and methods tought during thelecture.

4 Conditions for participation concerning contentsof previous lectures: Course "Introduction to Numerical Analysis" from thebachelor's degree programme of Mathematics; module"Numerics of ODE"







9 Authorized representatives of module and mainlecturers:Prof. Dr. B. Simeon

Mathematical Methods of ClassicalMechanics II


Contact Self-Effort

CreditSemester

Duration



time30 h

study105 h

135 h points4,5 cp

1, 2 or 3 1semester

1 Courses:Mathematical Methodsof Classical Mechanics


2 Contents:· Lagrangian and Hamiltonian systems (advanced):theorems of Noether, Poincaré und Liouville· Manifolds (advanced)· Numerical methods· Multibody systems: O (n)-Multibody formalisms· Elastic bodies in examples (optional)

3 Result of study / competences:The students know and understand both advancedtheorems of classical mechanics and modern conceptsand methods of multibody systems. They havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought duringthe lecture.

4 Conditions for participation concerning contentsof previous lectures: The courses "Introduction to Ordinary DifferentialEquations", "Introduction to Numerical Analysis" fromthe bachelor's degree programme of Mathematics;modules "Numerics of ODE", "Mathematical Methods ofClassical Mechanics"








Mathematical Theory of Fluid Dynamics

Degree programmes:



Master degree programmes of Mathematics,Technomathematics, Economathematics andMathematics International

Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1Semester

1 Courses:Mathematical Theoryof Fluid Dynamics


2 Contents:It deals with mathematical concepts for deriving theNavier-Stokes equations from conservation principlesand with the consequent results in fluid dynamics.Specifically, the following contents will be taught:· Derivation of the Stokes and Navier-Stokesequations· Potential flows, vortex formation· Circulation theorems· Turbulence

3 Result of study / competences:The students know and understand advancedmathematical methods to explore fluid dynamicequations. Based on concrete tasks, they havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought duringthe lectures.

4 Conditions for participation concerning contentsof previous lectures: Module "Differential Equations: Numerics of ODE &Introduction to PDE"


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and scientific computing· Partial differential equations

Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).

6 Award of credit points, examinations:Examination about this course.


8 Intended size of class:About 10-25 students

9 Authorized representatives of module and mainlecturers:Prof. Dr. A. Klar, Prof. Dr. R. Pinnau



Mathematical Theory of Neural Networks:Advanced Topics


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1Semester

1 Courses: Neural Networks (2ndhalf of the lecture)

2 lecture hours per week, 1hours tutorials per week

2 Contents:Advanced topics from the theory of neural networksas well as their applications will be discussed.Especially the following contents will be covered: · Vector support machines · Capacity of perceptrons · Recurrent neural networks· Neural networks with radial basis functions

3 Result of study / competences: The students know and understand a basic conceptto describe dynamic systems as well as mathematicaltechniques to analyse these systems. They alsoknow the possibilities where this can be applied,resulting from the usage of mathematical ControlTheory.

During the exercises students have developed askilled, precise and independent handling of theterms, propositions and methods tought in NeuralNetworks.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Numerics andIntroduction to ODE from the bachelor degreeprogramme of Mathematics. The first half of thecourse Neural networks - Introduction to Neuralnetworks.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and scientific computing · System and control theory Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degree



progamme which might restrict the above).

6 Award of credit points, examinations:Examination about this course.




10 Additional information:The module is part of the module Neural_Networks.

Matroids - Theory and Applications


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1Semester

1 Courses: Matroids - Theory andApplications


2 Contents: · Basic concepts of matroid theory (e.g. independentsystem, circle, cycle, basis, rang function, closure),matroids as generalization of terms and concepts fromlinear algebra and graph theorie. · Equivalent definition and axiom systems for matroids· Matroid duality, MINORENBILDUNG, reprentability,matrix matroides, regular and binary matroids, graphicand co-graphic matroids, uniform matroids, transversematroids, algebraic matroids · Algorithmic aspects of matroids, Greedy algorithms · Application of matroids in the combinatorialoptimization, e.g. matroid polyheder and theirdescription, wight-minimal (fundamental) cycle basisproblems in binary, graphic and co-graphic matroids,complexity observations and algorithms, polyhederobservations, fomulations as integer linear programs,relaxations, heuristics. · Applications and relations of matroid theory to otherareas (e.g. coding theory, electrotechnic,informatics).

3 Result of study / competences: The students master the basic terms, ideas and



concepts of matroid theory. They understand themeaning and the application of matroid theory in thecombinatorial optimization and know the relations ofmatroid theory to other mathematical areas andapplications of the theory of practical problems.During the exercises students have developed askilled, precise and independent handling of theterms, propositions and methods from the lecture.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics.Knowledge fromn the course Integer Programming:Polyhedral Theory and Algorithms are usefull but notnecessarily required.

5 Usability of this module: for master degree programmes mentioned above:

Main focus of study Optimization (under the termsof MPO §4, clause 5). Areas of abstract, applied orgeneral mathematics if the main focus of study wasnot chosen Optimization (considering the rules of thedegree progamme which might restrict the above).

6 Award of credit points, examinations: Examination about this course.


8 Intended size of class: about 10-25 students

9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Dr. F. Bunke

Methods of Convex Analysis in ImageProcessing


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1Semester

1 Courses: Methods of ConvexAnalysis in ImageProcessing


2 Contents: · Motivation: basic problems of image processing



(image restoration: denoising, removal of blur,inpainting, segmentation) · Convex sets (basic concepts, convex cone,projection and separation theorems)· Convex functions (basic concepts, continuity ofconvex functions, convex optimization problems) · Subgradients (basic concepts, subdifferentialcalculus, set-valued mappings) · Duality (Legendre-Fenchel conjugate, Lagrangefunctions, saddle point problems)· Numerical optimization methods with application toimage processing problems

3 Result of study / competences: The students know the basic terms and structures ofconvex analysis. They are familiar with the numericalalgorithms and can apply them to various problems ofdigital image processing.

During the exercises students have developed askilled, precise and independent handling of theterms, propositions and methods from the lecture andhave gone through different implementations of thealgorithms of image processing.

4 Conditions for participation concerning contentsof previous lectures: The course "Introduction to Numerical Analysis" fromthe bachelor degree programme of Mathematics.

5 Usability of this module: for master degree programmes mentioned above:

Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied analysis· Modelling and scientific computation

Areas of applied or general mathematics if the mainfocus of study was not chosen from the above list(considering the rules of the degree progamme whichmight restrict the above).

6 Award of credit points, examinations: Examination about this course


8 Intended size of class: about 10-25 students

9 Authorized representatives of module and mainlecturers: Prof. Dr. G. Steidl

Modular Representation Theory




Contacttime45 h

Self-study90 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:ModularRepresentationTheory


2 Contents:· Brauer characters and blocks· Projective characters, decomposition matrices· Group theoretical applications

3 Result of study / competences:The students know and understand basic propositionsof Brauer characters and decomposition numbers ofgroups. During the exercises students have developeda skilled, precise and independent handling of theterms, propositions and methods tought in ModularRepresentation Theory.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Algebra from the bachelordegree programme of Mathematics; Module"Representation Theory".






9 Authorized representatives of module and mainlecturers:Prof. Dr. G. Malle

Modular Representation Theory II




Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:ModularRepresentationTheory II


2 Contents:· Brauer's main theorems· Group theoretical applications

3 Result of study / competences:The students know and understand the Brauer's maintheorems of modular representation theory and theirapplications.

4 Conditions for participation concerning contentsof previous lectures: The course "Introduction to Algebra" from thebachelor degree programme of Mathematics; module"Representation Theory" and "Modular RepresentationTheory"


Main focus of study "Algebra and Number Theory"(under the terms of MPO §4, clause 5).

Areas of pure mathematics or general mathematics ifthe main focus of study was not Algebra and NumberTheory (considering the rules of the degree progammewhich might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. G. Malle

Multicriteria Optimization





Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: MulticriteriaOptimization


2 Contents:· Necessity of modelling with more than one objectivefunction · Structures of order and concept of optimality · Characterization of efficient and non-dominatingsolutions· Scalarization methods · Multicriteria linear programmes · Multicriteria combinatorial optimization

In parts, only some of the headwords above as wellas further research topics will be covered in detailduring the lecture and tutorials. Details are to befound in the information system of the University ofKaiserslautern.

3 Result of study / competences: The students are able to cope with advancedmethods and algorithms to solve multicriteriaoptimization problems. They can also model and solvereal problems of scientific, technical and physicalresearch areas via mathematical methods.

4 Conditions for participation concerning contentsof previous lectures: The course Linear and Network Optimization from thebachelor degree programme of Mathematics;depending on the choice of the main focus of studyalso knowledge from the course IntegerProgramming: Polyhedral Theory and Algorithms mightbe required.






9 Authorized representatives of module and main



lecturers: Prof. Dr. H. Hamacher, Jun. Prof. Dr. S. Ruzika

Nonlinear Functional Analysis withApplications to PDE


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester2 or 3

Duration1semester

1 Courses: Nonlinear FunctionalAnalysis withApplications to PDE


2 Contents:Methods and techniques from the large area ofnonlinear functional analysis will be handled, inparticular those which play central role in the studyof nonlinear elliptic and parabolic partial differentialequations. Especially the following will be discussed:· Fixed point theorems · Integration and differentiation in Banach spaces · The theory of monotone operators and theirapplications in the study of nonlinear elliptic andparabolic partial differential equations

3 Result of study / competences: The students are able to cope with the advancedmethods and techniques which play central role in theanalysis and solving of nonlinear elliptic and parabolicpartial differential equations.They have also gained adeepened knowledge of the interaction and mutualinfluence of the theory and applications.

By means of concrete exercises students havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought duringthe lectures.

4 Conditions for participation concerning contentsof previous lectures: Module "Differential Equations: Numerics of ODE &Introduction to PDE", "Functional Analysis", desirable:"Numerical Methods for Elliptic and Parabolic PDE"


Main focus of study Applied Analysis (under the termsof MPO §4, clause 5).



Areas of applied, pure or general mathematics if themain focus of study was not Applied Analysis(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau

Nonparametric Statistics


Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: NonparametricStatistics


2 Contents:Smoothing Methods for Estimating Functions: · Smoothing methods for estimating functions (kernelestimator, local polynomial estimator, next-neighbour-estimator, smoothing splines) and their asymptoticanalysis · Application in regression and image analysis · Data controlled choice of smoothing parameters withcross-validation · Decomposition of spectra and their estimators forstationary time series

Nonparametric Regression and Classification: · Analysis of regression in higher dimensions on thebasis of Boosting · General sieve estimator for functions and theirasymptotic analysis · Trees of regression and classification, neuronalnetworks, expansion of orthogonal series andwavelets for example · Applications of estimating functions and solvingclassification problems with high dimensional causalvariables



3 Result of study / competences: The students have gained a good overview of modernstatistic methods which can go without restrictiveassumptions of the model but require a great amountof data and high computing time. They have alsogained basic knowledge of the theory and algorithmsof nonparametric statistics.

4 Conditions for participation concerning contentsof previous lectures: The course Regression and Time Series Analysis

5 Usability of this module:for master degree programmes mentioned above/i>:Main focus of study from the following list (under theterms of MPO §4, clause 5): · Financial Mathematics· StatisticsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above)




9 Authorized representatives of module and mainlecturers: Dr. J.-P. Stockis

Numerical Methods in Control Theory


Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Numerical Methods inControl Theory


2 Contents:Numerical methods to deal with problems of linearcontrol theory will be covered, especially: · General and structured eigenvalue problems, normalforms · Matrix equations (e.g. Lyapunov and Riccati) and



their numerical solutions · General theory of huge systems of equations · Reduction of model (especially based on singularvalue decompositions and Krylov subspace method)

3 Result of study / competences: The students know and understand basic concepts todeal with numerical problems of control theory as wellas mathematical techniques to analyse thesemethods. On the basis of concrete exercises thestudents have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought in Numerical Methods in ControlTheory.

4 Conditions for participation concerning contentsof previous lectures: The modules Einführung in die Numerik and SystemsTheory: Systems and Control Theory & NeuralNetworks from the bachelor degree programme ofMathematics.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Modelling and Scientific Computation · Systems and Control TheoryAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. T. Damm, Prof. Dr. A. Klar, Prof. Dr. R.Pinnau, Prof. Dr. D. Prätzel-Wolters

Numerical Methods in Finance


Contacttime

Self-study Effort

Creditpoints Semester

Duration1



90 h 180 h 270 h 9 cp 1, 2 or 3 semester

1 Courses: Numerical Methods inFinance


2 Contents:· Basic portfolio optimization · Binomial options pricing model for option valuation · Black-Scholes-equation and its solutions · Transformation of random numbers · Monte-Carlo-integration · Option valuation by Monte-Carlo-simulation · Variance-reduction-techniques · Quasi-random numbers and quasi-Monte-Carlo-simulation · Introduction to the finite difference method · Explicit and implicit finite difference schemata · Crank-Nicolson-method · Free boundary value problems for american options

3 Result of study / competences: The students have gained basic knowledge ofnumerical algorithms which are often used in financialmathematics. They are able to write softwareprogrammes of these algorithms especially for theirapplication in option valuation.

4 Conditions for participation concerning contentsof previous lectures: The courses Stochastic Methods and Introduction toScientific Programming from the bachelor degreeprogramme of Mathematics.


Main focus of study Financial Mathematics (under theterms of MPO §4, clause 5). Areas of appliedmathematics or general mathematics if the main focusof study was not Financial Mathematics (consideringthe rules of the degree progamme which mightrestrict the above).





Numerics of Stochastic Processes




Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Numerics ofStochastic Processes


2 Contents:Stochastic processes with a multidimensionalcontinuous parameter space, called random fields, areused to model random spatial (and temporal)phenomena. Topics of the course are:· Classic examples of random fields: Brownian sheetand Lévy's Brownian motion· Hilbert spaces with reproducing kernel· Isotropy and stationarity of random fields· Regularity and approximation of random fields, seriesrepresentations· Sparse grid and Smolyak's algorithms

3 Result of study / competences: The students have gained advanced knowledge of thetheory, approximation and simulation of random fields.They have also gained an exemplary insight intoapplications (geostatistics, fluid dynamics).

4 Conditions for participation concerning contentsof previous lectures: Modules "Probability Theory I", "Functional Analysis".


Main focus of study Applied Analysis. (under theterms of MPO §4, clause 5).

Areas of pure, applied or general mathematics if themain focus of study was not Applied Analysis(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Jun. Prof. Dr. A. Neuenkirch, Prof. Dr. K. Ritter



Online Optimization


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Online Optimization 4 lecture hours, 2 tutorial

hours per week

2 Contents:· Competitive analysis for deterministic andrandomized algorithms· Opponent concepts, adaptive and non-adaptiveopponents of randomized algorithms · Amortized costs, potential method for costs analysis· Competitive algorithms for paging / caching· Metrical Task Systems and Request-Answer Gamesas more general online problems · Design of competitive algorithms for particular onlineproblems (e.g. K-server problem, network routing,packing and covering problems, scheduling) · Principle of Yao as a means for calculating lowerbounds · Alternative approaches for analyzing online problems

3 Result of study / competences: The students know and understand online problemsand the principle of competitive analysis. They havelearned to analyze online problems on the existenceof competitive algorithms and to to calculate lowerbounds for deterministic and randomized algorithms.

By means of concrete exercises students havedeveloped a skilled, precise and independent handlingof the terms, propositions and methods tought in thelecture.

4 Conditions for participation concerning contentsof previous lectures: The course "Linear and Network Optimization " fromthe bachelor degree programme of Mathematics; "Integer Programming: Polyhedral Theory andAlgorithms".


Main focus of study Optimization (under the terms of MPO §4, clause 5).



Areas of applied mathematics or general mathematicsif the main focus of study was not Optimization(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. H. Hamacher, Prof. Dr. S. Krumke

Optimization with PDE


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester2 or 3

Duration1semester

1 Courses: Optimization with PDE 2 lecture hours per week

2 Contents:· Mathematical concepts to deal with optimizationproblems with differential equation constraints will beprovided and analysed. Especially the followingcontents will be dealt with: · Non-linear theory of operators · Adjoint calculus · Approximation methods to numerically solveconstrained optimization problems

3 Result of study / competences: The students are able to cope with the theory andwith numerical methods to analyse and solveconstrained optimization problems. By means ofconcrete exercises students have developed a skilled,precise and independent handling of the terms,propositions and methods tought in Optimization withPDE.

4 Conditions for participation concerning contentsof previous lectures: The modules Differential Equations: Numerics of ODE& Introduction to PDE and Functional Analysis”, desirable: Numerical Methods for Elliptic and ParabolicPDE




Main focus of study from the following list (under theterms of MPO §4, clause 5): · Modelling and Scientific Computation · Partial Differential EquationsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. A. Klar, Prof. Dr. R. Pinnau, Prof. Dr. D.Prätzel-Wolters

PDE based Multiscale Methods andNumerical Approaches for their Solution


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester2 or 3

Duration1semester

1 Courses:PDE based MultiscaleMethods andNumerical Approachesfor their Solution


2 Contents:An introduction will be given to PDEs based multiscaleproblems and to approaches for their treatment.Special attention will be given to the following topics:

• Homogenization of elliptic equations with oscillatingcoefficients• Classification of multiscale problems• Advanced numerical algorithms for PDEs andsystems of PDEs with oscillating coefficients(including Multiscale Finite Element Method, MultiscaleFinite Volume Method, Heterogeneous MultiscaleMethod, Subgrid approach)



• Numerical approaches for stochastic elliptic PDE.

3 Result of study / competences:The students are able to cope with the theory andnumerical methods to analyse and solve PDEs basedmultiscale problems. By means of concrete exercisesstudents have developed a skilled, precise andindependent handling of the terms, propositions andmethods tought during the lectures.

4 Conditions for participation concerning contentsof previous lectures: Module „Differential Equations: Numerics of ODE &Introduction to PDE“, desirable: „Numerics of PDE I“.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and Scientific Computation· Partial Differential EquationsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers:PD Dr. O. Iliev, Prof. Dr. A. Klar

Permutation Groups


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester2 or 3

Duration1semester

1 Courses: Permutation Groups 2 lecture hours per week

2 Contents:· Primitive and multiply transitive permutation groups · Theorem of Scott O'Nan



· Classification theorems (near-fields, doublytransitive permutation groups)

3 Result of study / competences: The students know and understand basic methodsand statements of the theory of Permutation groups.They got to know important examples and are able tostudy those by scientific methods.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Algebra from the bachelordegree programme of Mathematics. Initial knowledgefrom the course Group Theory (up to the Sylowschentheorems) are desireable but not necessarily required.







Poisson and Lévy Processes


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1,2 or 3

Duration1semester

1 Courses: Poisson and LévyProcesses


2 Contents:During the lecture the fundamental theory of thestructure of Lévy processes is developed andexamples with non-mathematical applications are



provided. The following will be adressed in details:· Poisson point processes · Characteristics of Lévy process and Itôrepresentation via Poisson processes· Symmetric stable processes

3 Result of study / competences: The students have gained advanced knowledge of thetheory of stochastic processes. They are able toscientifically work in this area of mathematics.

4 Conditions for participation concerning contentsof previous lectures: The course "Probability Theory I"


Main focus of study Applied Analysis (under the termsof MPO §4, clause 5).

Areas of pure, general or applied mathematics if themain focus of study was not Applied Analysis(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. H. von Weizsäcker

Potential Theory


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Potential Theory 4 lecture hours, 2 tutorial

hours per week

2 Contents: · Laplace equation, harmonic function, representationtheorem, mean value theorem of Gauß, min/max-principle



· Volume- and surface potentials, limit- and jumprelations · Boundary value problems of potential theory,propositions about existence and uniqueness, Green'sfunction, examples · Boundary integral equation method· Runge-Walsh-approximation with harmonicpolynomials and kernel function structures (e.g.splines, wavelets) for geo-relevant geometries (suchas sphere, ellipsoid, geoid, (actual) Earth's surface)

3 Result of study / competences: The students know and understand the mathematicalbasics of classical potential theory. They are able tocope with the targeted determination of gravitationalfields by methods and techniques of modern geo-engineering. The students have gained advancedunderstanding of boundary value problems on geo-relevant surfaces (e.g. sphere, ellipsoid, geoid,telluroid, (actual) surface of the Earth).

4 Conditions for participation concerning contentsof previous lectures: The course Vector Analysis from the bachelor degreeprogramme of Mathematics.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · GeomathematicsAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).





Probability Theory II




Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Probability Theory II 4 lecture hours per week

2 Contents:· Stochastic processes: Kolmogorov's consistencytheorem · Gaussian processes, especially fractional Brownianmotion· Martingales in continuous time· Poisson processes · Prokhorov theorem, Donskers invariance principle · Lévy processes

3 Result of study / competences: The students have gained advanced knowledge of thetheory of stochastic processes. They are able toscientifically work in this area of mathematics.

4 Conditions for participation concerning contentsof previous lectures: The course Probability Theory I


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis, · Financial MathematicsAreas of pure mathematics, applied mathematics orgeneral mathematics if the main focus of study wasnot one of the above (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. H. von Weizsäcker

Probability Theory II: Stochastic integrals




Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Probability Theory II:Stochastic Integrals

4 lecture hours per week, 2tutorial hours per week

2 Contents:· Stopping times and martingales in continuous time · Localization and approximation· Stochastic integrals for regular integrands· Previsibility and semimartingales · Ito-calculus · Wiener chaos and martingale representation· Girsanov theorem and Feynman-Kac formula · Strong and weak solutions of stochastic differentialequations

3 Result of study / competences: The students have gained advanced knowledge ofthe proof and application techniques of stochasticanalysis.

4 Conditions for participation concerning contentsof previous lectures: The course "Probability Theory I"


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis, · Financial MathematicsAreas of pure mathematics, applied mathematics orgeneral mathematics if the main focus of study wasnot one of the above (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. H. von Weizsäcker

10 Additional information:: This module can not be combined with the module"Stochastic Differential Equations" in the finalexamination due to high content overlap.



Representation Theory


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: RepresentationTheory


2 Contents:· Semisimple algebras, theorem of Wedderburn · Ordinary characters of groups · Tensor products and induced characters · Group theoretical applications· Brauer characters and blocks

3 Result of study / competences: The students know and understand basic propositionsof ordinary characters and character theory ofgroups. They were introduced to modularrepresentation theory. During the exercises studentshave developed a skilled, precise and independenthandling of the terms, propositions and methodstought in Representation Theory.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Algebra from the bachelordegree programme of Mathematics.






9 Authorized representatives of module and mainlecturers: Prof. Dr. U. Dempwolff , Prof. Dr. G. Malle



Scientific Computing in Solid Mechanics


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Scientific Computingin Solid Mechanics


2 Contents:Mathematical modelling, numerical methods andsoftware on the topics:· Elastic bodies· Special cases of beams and plane strain/stressstate· Finite element space discretization· Special time integration scheme· Inelastic deformation

3 Result of study / competences:The students know and understand the basicconcepts for modelling and numerical handling ofproblems of solid mechanics. Furthermore, they canuse appropriate software and implement their ownextensions.

During the exercises students have developed askilled, precise and independent handling of theterms, propositions and methods tought during thelecture.

4 Conditions for participation concerning contentsof previous lectures: Course "Introduction to Numerical Analysis" from thebachelor's degree programme of Mathematics, module"Differential Equations: Numerics of ODE &Introduction to PDE"


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Modelling and Scientific Computation· Partial Differential Equations

Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).







Singularity Theory


Contacttime 60 h

Self-study 210 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Singularity Theory 4 lecture hours per week

2 Contents:· Power series, Theorems of Weierstrass · Analytical algebras · Elementary theory of coherent sheaves · Germ of a complex variety · Local compactness theorem for morphisms · Invariants of hyper surface singularities · Finite determination · Deformation theory of complete intersections · Classification of simple hyper surface singularities

3 Result of study / competences: The students have gained basic knowledge ofSingularity Theory (local analytical and algebraicgeometry). Using the example of hyper surfacesingularities they know and understand how thetheory was developed further by applying modernmethods of algebraic geometry. The studentsespecially know and understand the classification ofsimple or ADE-singularities which appear in manydifferent areas of mathematics and theoreticalphysics.

4 Conditions for participation concerning contentsof previous lectures: The courses Commutative Algebra and AlgebraicGeometry


Main focus of study Algebraic Geometry and ComputerAlgebra (under the terms of MPO §4, clause 5).



Areas of pure mathematics or general mathematics ifthe main focus of study was not Algebraic Geometryand Computer Algebra (considering the rules of thedegree progamme which might restrict the above).





Singularity Theory - Part 1


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Singularity Theory -Part 1


2 Contents:· Power series, Theorems of Weierstrass · Analytical algebras · Invariants of hyper surface singularities · Finite determination · Deformation theory of complete intersections

3 Result of study / competences: The students have gained basic knowledge ofSingularity Theory (local analytical and algebraicgeometry). Using the example of hyper surfacesingularities they know and understand how thetheory was further developed by applying modernmethods of algebraic geometry. The studentsespecially know and understand the classification ofsimple or ADE-singularities which appear in manydifferent areas of mathematics and theoreticalphysics.

4 Conditions for participation concerning contentsof previous lectures: The courses Commutative Algebra




Main focus of study Algebraic Geometry andComputer Algebra (under the terms of MPO §4,clause 5). Areas of pure mathematics or generalmathematics if the main focus of study was notAlgebraic Geometry and Computer Algebra(considering the rules of the degree progamme whichmight restrict the above).





10 Additional information: This module is part of the module Singularity Theory(9 credit points).

Sobolev Spaces


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Sobolev Spaces 2 lecture hours per week

2 Contents:· Advanced integration theory (Convergence

theorems, Lp-spaces, integration by parts)· Construction of Sobolev spaces · Analysis in Sobolev spaces (convolution, Diracsequences, dense function sets, Fourier transform)· Applications to PDE (Poincaré inequality,fundamental lemma of calculus of variations, weakboundary problems)

3 Result of study / competences: The students have gained deepened knowledge of asubdomain of functional analysis with applications toPDE.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Functional Analysis from



the Bachelor programme of Mathematics.







Spatial Statistics


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Spatial Statistics 2 lecture hours per week

2 Contents:

· Spatial point processes (in R2 and R3) · Point process models (Poisson process, Hard-coreand cluster processes, Gibbs processes) · Statistical methods for pint processes · Marked point processes and particle processes

3 Result of study / competences: The students know the basics of tje point processtheory as well as common point process models andare able to statistically analyse and modell pointpatterns.

4 Conditions for participation concerning contentsof previous lectures: The course Stochastic Methods from the bachelordegree programme of Mathematics. Additionalknowledge in Stochastics (e.g. Regression and TimeSeries Analysis or Probability Theory I) is usefull but



not necessarily required.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Financial Mathematics · Statistics Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Dr. C. Redenbach

Special Functions of Mathematical (Geo-)Physics


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Special Functions ofMathematical (Geo-)Physics


2 Contents:· Special solution systems of scalar Laplace equation

in R3, in particular (scalar) sperical harmonics· Special solution systems of vector and tensor

Laplace equation in R3, in particular vector and tensorspherical harmonics· Special solution systems of (time-harmonic) Maxwell

equation in R3, in particular Bessel and Hankelfunctions (in combination with spherical harmonics)· Special solution systems of elasticity theory, einparticular solutions of the Cauchy-Navier equation(using ansatz of vector/tensor spherical harmonics)



· Special solution systems of the Navier-Stokes

equation in R3, in particular solutions for shallowwater equation and geostrophic flow (using ansatz ofvectorial/tensor spherical harmonics).

3 Result of study / competences: The students know and understand the solutiontheory of partial differential equations based onseparation of the variables. They have gainedadvanced competences to apply geo-scientificallyrelevant function systems and basic knowledge ofmodelling of vector and tensor data, in particular ofmodern satellite technique (e.g. Satellite-to-Satellite-Tracking (SST), Satellite-Gravity-Gradiometry (SGG)).

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Ordinary DifferentialEquations from the bachelor degree programme ofMathematics.

5 Usability of this module:for master degree programmes mentioned above:Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied Analysis · Geomathematics Areas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).





Stability Theory


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:



Stability Theory 2 lecture hours, 1 tutorialhour per week

2 Contents:Qualitative analysis of dynamical systems (amongstothers non-linear ordinary differential equations) willbe covered, especially: · Stability terms · Parameter dependence of solutions · Lyapunov functions · Invariant manifolds · Periodic solutions and floquet theory · Structural stability and normal forms

3 Result of study / competences: The students have learned to give qualitativepropositions about dynamical systems which do notnecessarily result from numerical computations. Theirawareness of sensitive dependence is raised. Methodstought are fundamental for boundary value analysis inother disciplines like non-linear partial differentialequations of non-linear control theory.

4 Conditions for participation concerning contentsof previous lectures: The course Introduction to Ordinary DifferentialEquations” from the bachelor degree programme ofMathematics.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Modelling and Scientific Computation · Systems and Control TheoryAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).





Stochastic Control and FinancialApplications




Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Stochastic Controland FinancialApplications


2 Contents:· Stochastic optimization in discrete time· Stochastic control theory in continuous time· Viscosity solutions· Stochastic optimization with duality methods

3 Result of study / competences:The students know and understand the fundamantalmethods of solving stochastic control problems andcan apply them to concrete problems.

4 Conditions for participation concerning contentsof previous lectures: Module "Probability Theory I", basic knowledge ofstochastic analysis, e.g. from the module "StochasticDifferential Equations" or from the module "FinancialMathematics I"


Main focus of study Financial Mathematics (under theterms of MPO §4, clause 5).

Areas of general or applied mathematics if the mainfocus of study was not Financial Mathematics(considering the rules of the degree progamme whichmight restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. R. Korn, Prof. Dr. J. Saß, Jun. Prof. Dr. F.Seifried

10 Additional information:This module can not be combined with the module"Continuous Time Portfolio Optimization" in the finalexamination.



Stochastic Geometry


Contacttime45 h

Self-study90 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Stochastic Geometry 2 lecture hours, 1 tutorial

hour per week

2 Contents:· Basic concepts of the theory of spatial pointprocesses (marking, intensity measure, ...)· Multidimensional Poisson process, Poisson clusterprocesses· Basic concepts of the theory of random closed sets · Germ-grain models, in particular the Boolean model· Random mosaics

3 Result of study / competences:The students have learnt the most importantconcepts and models of the theory of spatial pointprocesses and theory of random closed sets.

4 Conditions for participation concerning contentsof previous lectures: Module Probability Theory I.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied analysis· Statistics

Areas of pure, applied or general mathematics if themain focus of study was not chosen from the abovelist (considering the rules of the degree progammewhich might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. J. Franke, Dr. C. Redenbach, Dr. J. Kampf



Stochastic Partial Differential Equations


Contacttime90 h

Self-study180 h

Effort270 h

Creditpoints9 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Stochastic PDE 4 lecture hours, 2 tutorial

hours per week

2 Contents:Infinite-dimensional Wiener processes,Integration for operator-valued processes,Mild solutions of stochastic PDE (semigroupapproach),Approximation methods.

3 Result of study / competences:The students have acquired a deeper understandingof important aspects (modeling, solution andregularity theory, approximation) of the section of thearea of stochastic analysis.

4 Conditions for participation concerning contentsof previous lectures: Knowledge of stochastic differential equations (e.g.from the courses "Stochastic Differential Equations" or"Financial Mathematics I") and basic knowledge ofFunctional Analysis.


Main focus of study from the following list (under theterms of MPO §4, clause 5): · Applied analysis· Partial differential equations

Areas of pure, applied or general mathematics if themain focus of study was not chosen from the abovelist (considering the rules of the degree progammewhich might restrict the above).




9 Authorized representatives of module and main



lecturers:Prof. Dr. K. Ritter

Stochastic Processes with Applications forInsurances/Financial Statistics - Part 2


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Stochastic Processeswith Applications forInsurances


2 Contents:· Point processes · Modelling risk processes · Ruin theory · Concepts of extreme value theory

3 Result of study / competences: The students know and understand properties ofstatistical methods suitable for modelling in the areaof property insurances, as well as their applications.

4 Conditions for participation concerning contentsof previous lectures: The course Stochstic Methods from the Bachelordegree programme of Mathematics. Additionalknowledge in stochastics (e.g. Regression and TimeSeries Analysis or Probability Theory I) is usefull, butnot necessarily required.


Main focus of study from the following list (under theterms of MPO §4, clause 5):· Financial Mathematics · Statistics Areas of applied mathematics or general mathematicsif the main focus of study was not chosen from theabove list (considering the rules of the degreeprogamme which might restrict the above).






9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. J. Saß, Dr. J.-P. Stockis

10 Additional information: This module can be combined with the moduleFinancial Time Series / Financial statistics - part 2 toa single module named Financial statistics (9 creditpoints).

Systems and Control Theory: AdvancedTopics


Contacttime 45 h

Self-study 90 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: Systems and ControlTheory (second partof the course)

2 lecture hours, 1 hourtutorial per week

2 Contents:Deepening of topics concerning control theory as wellas applications will be dealt with. Especially thefollowing contents will be covered:· Extended state space and behaviour theory · Control concepts · Transfer functions and realization theory · Polynomial system models

3 Result of study / competences: The students know and understand advancedconcepts to describe dynamic systems as well asmathematical techniques to analyse these systemsand to formulate rules. Furthermore, the knowapplication possibilities which result from the usage ofmathematical control theory.

4 Conditions for participation concerning contentsof previous lectures: The courses Introduction to Numerics andIntroduction to Ordinary Differential Equations fromthe bachelor degree programme of Mathematics, firstpart of the course Systems and Control Theory




Main focus of study from the following list (under theterms of MPO §4, clause 5): · Modelling and Scientific Computation · Systems and Control TheoryAreas of applied mathematics or general mathematicsif the main focus of study was not one of the above(considering the rules of the degree progamme whichmight restrict the above).





10 Additional information: This module is part of the module Systems andControl Theory.

The Mathematics of Arbitrage


Contacttime 30 h

Self-study 105 h

Effort 135 h

Creditpoints 4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses: The Mathematics ofArbitrage


2 Contents:· First and second fundamental theorems of assetpricing: finite probability spaces · No-arbitrage and separation theorems: the Kreps-Yan theorem · No-arbitrage in discrete time: the Dalang-Morton-Willinger theorem · The general financial market model: a crash coursein the theory of semimartingales · No-arbitrage in continuous time: The Delbaen,Schachermayer theorem

3 Result of study / competences: The students have learned how the general economicprinciple of no-arbitrage can be expressed in precisemathematical terms. In the proof of the Fundamental



Theorem of Asset Pricing they have learned anexample of how the methods of the Theory ofSemimartingales and Functional Analysis can comeinto play in Financial Mathematics.

4 Conditions for participation concerning contentsof previous lectures: Module "Probability Theory I"; good knowledge ofstochastic processes, e.g. from the module"Probability Theory II"; basic knowledge of functionalanalysis, e.g. from the module "Functional Analysis".


Main focus of study Financial Mathematics (under theterms of MPO §4, clause 5).

Areas of applied mathematics or general mathematicsif the main focus of study was not FinancialMathematics(considering the rules of the degreeprogamme which might restrict the above).




9 Authorized representatives of module and mainlecturers: Prof. Dr. R. Korn, Prof. Dr. J. Saß, Jun. Prof. Dr. F.Seifried

Toric Geometry


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Toric Geometry 2 lecture hours per week

2 Contents:· Convex cone and toric varieties· Group rings and orbits· Toric divisors and line bundles· Chow groups and rings of toric varieties· Desingularization· Associated homogeneous coordinate rings



· Toric sheaves

3 Result of study / competences:On the example of toric varieties students getprofound practical knowledge of how to deal withparticular varieties. They learn how to use differentmethods of algebraic geometry, which enables themto deepen in this area and its applications. During thediscussion of the homogeneous coordinate rings theyget to know areas of algebra, which forms thetheoretical basis for implementation of computeralgebra.

4 Conditions for participation concerning contentsof previous lectures: The course Algebraic Geometry


Main focus of study "Algebraic Geometry andComputer Algebra" (under the terms of MPO §4,clause 5). Areas of pure mathematics or general mathematics ifthe main focus of study was not "Algebraic Geometryand Computer Algebra" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. A. Gathmann, Prof. Dr. G. Trautmann

Tropical Geometry


Contacttime30 h

Self-study105 h

Effort135 h

Creditpoints4,5 cp

Semester1, 2 or 3

Duration1semester

1 Courses:Tropical Geometry 2 lecture hours per week

2 Contents:· Puiseux body· Convex geometry



· Tropicalization and tropical varieties· Tropical moduli spaces and intersection theory· Selected results from tropical enumerative geometry

3 Result of study / competences:On the example of tropical varieties the studentshave gained profound practical knowledge of how todeal with particular varieties. They have learned touse a variety of methods of algebraic geometry,which enables them to deepen in this area and itsapplications.

4 Conditions for participation concerning contentsof previous lectures: Module "Algebraic Geometry", knowledge of thecourse "Algebraic Geometry II: Sheaves, Cohomologyand Applications" would be helpful but not mandatory.


Main focus of study "Algebraic Geometry andComputer Algebra" (under the terms of MPO §4,clause 5). Areas of pure mathematics or general mathematics ifthe main focus of study was not "Algebraic Geometryand Computer Algebra" (considering the rules of thedegree progamme which might restrict the above).




9 Authorized representatives of module and mainlecturers:Prof. Dr. A. Gathmann

White Noise Analysis


Contacttime 90 h

Self-study 180 h

Effort 270 h

Creditpoints 9 cp

Semester1, 2 or 3

Duration1semester

1 Courses: White Noise Analysis 4 lecture hours, 2 tutorial

hours per week

2 Contents:



· Introduction to the basics in distribution theory withspecial focus on temperated distributions· Construction of the white noise space (Minlostheorem, chaos-decomposition, T-transform, S-transform, Ito-Wiener-Segal isomorphism)· Introduction of test function spaces and spaces ofgeneralized functions of White Noise Analysis (Hidaand Kondratiev spaces)· Applications to Feyman path integrals andstochastic PDE

3 Result of study / competences: The students have gained deepened knowledge of asubdomain of Functional Analysis with applications toFeyman path integrals and stochastic PDE.

4 Conditions for participation concerning contentsof previous lectures: The course Functional Analysis.







[University of Kaiserslautern] [Department of Mathematics][Kaiserslautern Graduate School]

Documents

Module Handbook - Master Mathematics