Modules

176 results

A1.3 Mathematics Modules on Special Topics176

Algebraic TopologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamental algebraic invariants of topological spaces and how to apply them. The focus is on singular homology and cohomology as well as their computational methods and applications (e.g. Poincaré duality for manifolds). In the end you can compute these theories on concrete examples and solve typical problems with them.9 ECTSruns this semesterCIT413039Computational Methods for Single-cell BiologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramIn this module you will learn modern computational methods for the analysis of single-cell data and apply them practically. After lectures on important topics in single-cell biology and data analysis, you will work eight weeks in an ICB laboratory on a concrete project to gain practical experience.6 ECTSruns this semesterMA5617Computational Plasma PhysicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn numerical methods for solving and applying models from plasma physics. The focus is on the discretization of partial differential equations (e.g., Poisson, conservation and kinetic equations) and the implementation of these procedures in Python. In the end you will be able to derive an appropriate numerical method for a concrete model, implement it, and test and verify the code.5 ECTSruns this semesterMA4304Financial Mathematics 1No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of mathematical finance in discrete time: pricing and valuation of derivatives in one- and multi-period models, concepts of arbitrage and completeness, as well as basics of portfolio optimization. By the end you will be able to understand price models and implement them numerically, and analyze and optimize portfolios according to risk-return criteria.9 ECTSruns this semesterMA3407Fundamentals of Mathematical StatisticsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamental principles and methods of statistical inference, including likelihood approaches, Bayesian methods and minimax estimation. You will deal with asymptotic theory for large samples and apply it in particular to the study and application of maximum-likelihood techniques.9 ECTSruns this semesterMA5441
171 more in A1.3 Mathematics Modules on Special TopicsInsurance Mathematics 1No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamental stochastic methods of damage/non-life insurance. The focus areas are tariff calculation, capital allocation, individual and collective model, claims provisions, as well as reinsurance. In the end you will be able to understand and apply the most important models for premium calculation, reserving and risk sharing.runs this semesterMA3405Large DeviationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theory of large deviations, which describes how unlikely events occur in probability models and how quickly their probabilities decay. In the end you will be able to apply fundamental techniques of large-deviation theory and understand applications such as for sums of random variables and empirical distributions.5 ECTSruns this semesterMA5417Mathematical EcologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn to apply mathematical methods and models to ecological questions. In the end you can describe, analyze, and ecologically interpret populations, their interactions, spatial spread, and structure with suitable mathematical models.9 ECTSruns this semesterMA5602Mathematical Models in BiologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn how dynamic and stochastic models are constructed and applied in biology. The module covers, among other things, linear compartment models, Markov chains, birth–death and branching processes, as well as age- and space-structured populations. In the end you will be able to compare different modeling approaches and choose the appropriate model level for a biological system.9 ECTSruns this semesterMA3601Mathematische KontinuumsmechanikNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamental conservation laws of continuum mechanics (mass, momentum, energy) as well as the mathematical methods with which these laws are formulated and applied. After the module you can apply these methods to specific models such as fluid dynamics, chemical/biological reactions, self-gravitation, or liquid crystals, and understand elasticity theory as well as distributions.9 ECTSruns this semesterMA5019Multilinear AlgebraNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn concepts and tools of multilinear algebra and apply them, in particular in connection with matrix analysis and matrix algebra. In the end you will be able to understand tensor products, Kronecker operations and related structures and use them in mathematical contexts.5 ECTSruns this semesterCIT413063Nonconvex Global OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods of global optimization for nonconvex and possibly nonsmooth objective functions and how to apply them. Beginning with (stochastic) gradient methods and simulated annealing, you walk toward multi‑particle methods such as Particle Swarm Optimization (PSO) and Consensus‑Based Optimization (CBO) and understand their global convergence properties.6 ECTSruns this semesterCIT4130019Probabilistische Techniken und Algorithmen in der DatenanalyseNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn probabilistic techniques and algorithms used in data analysis for dimensionality reduction and data recovery from incomplete information. In the end, you will understand the basics of random matrices, assess randomized algorithms, and apply and analyze methods such as JL embeddings, compressed sensing, and randomness-based matrix/tensor reconstruction.6 ECTSruns this semesterMA4803Projektive Geometrie 1No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of projective geometry: projective spaces, their transformation groups and algebraic representations of geometric structures. In the end you will be able to apply projective viewpoints to different geometries and understand the role of transformations and algebraic methods in this context.9 ECTSruns this semesterMA3203Random Graphs and NetworksNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn models of random graphs and how they describe real networks. The focuses are Erdős–Rényi models, generalized random graphs, the configuration model and preferential-attachment models as well as proof techniques for the number of components, degree distribution and distances.5 ECTSruns this semesterMA5436Statistical LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the basics and advanced methods of supervised and unsupervised statistical learning. In the end you will be able to understand and apply models for regression and classification, use techniques for dimensionality reduction and cluster analysis, and follow probabilistic formulations of learning problems while designing new algorithms.6 ECTSruns this semesterMA4802ZinsmärkteNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will gain a solid introduction to interest rate markets and models for describing term structures. At the end you will be able to value interest-rate instruments and interest rate derivatives as well as measure and manage interest rate risks.5 ECTSruns this semesterMA3703Abelian VarietiesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamental concepts and constructions of abelian varieties within algebraic geometry. In the end you will be able to derive their most important properties and apply the learned concepts to abelian varieties in a broader geometric context.5 ECTSno date this semesterMA5125Advanced Finite ElementsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn advanced finite element techniques and apply them to concrete applications. In the end, you will be able to independently explore current research topics, work on Fachliteratur, and use modern open-source frameworks to solve complex PDE problems.7 ECTSno date this semesterMA5337Advanced Numerical Linear AlgebraNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn methods of numerical linear algebra with applications in data assimilation, model reduction and the solution of matrix equations. In the end you will be able to formulate and interpret variational and statistical data assimilation problems, understand and implement common model reduction procedures for linear systems, and apply theory and numerical methods for Lyapunov and Sylvester equations.1 ECTSno date this semesterMA5922Advanced Topics in Algebraic TopologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn algebraic methods for the study of smooth manifolds, in particular De‑Rham theory, cohomology and characteristic classes. By the end you will be able to understand, prove and apply the techniques for calculating examples of central theorems (e.g. Poincaré duality, Stokes, Poincaré–Hopf).5 ECTSno date this semesterMA5123Advanced Topics in Uncertainty QuantificationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods for uncertainties quantification for sensitivity analyses, estimation of rare events, and the Bayesian approach to inverse problems. In the end you will be able to select appropriate numerical and statistical solution procedures and critically assess their foundations as well as their limitations.6 ECTSno date this semesterCIT4130021Algebraic GeometryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn central terms and techniques of algebraic geometry: dimension theory of schemes, products of schemes, separated and genuine morphisms, coherent sheaves as well as differentials and smoothness. In the end you will be able to apply these algebraic and geometric concepts to fundamental questions of algebraic geometry.9 ECTSno date this semesterMA5107Algebraic Number Theory 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn p-adic numbers Q_p and their finite field extensions (local fields) and their parallels to global algebraic number theory. A focus is the local class field theory, which relates abelian finite Galois extensions of a local field to suitable subgroups of the field. In the end you can apply properties of extensions of local fields (ramified, unramified, tam) and use simple statements of local class field theory.5 ECTSno date this semesterCIT4130023Algebraic Number Theory 3No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of class field theory, both locally and globally. The focus is on local class field theory (including the invariant map, reciprocity law and existence theorem) and applications such as the local Kronecker–Weber theorem; in addition you will get an introduction to global class field theory and selected applications. In the end you will be able to apply the central theorems and methods of class field theory and transfer them to concrete problems in number theory.5 ECTSno date this semesterCIT413025Algebraic SurfacesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with algebraic curves and surfaces, their classifications and the birational geometry of surfaces. In the end you know the classification (Kodaira–Enriques), minimal models, blow-ups, and important surfaces classes such as rational, elliptic, del Pezzo, K3 and Enriques surfaces; you can apply theoretical statements and compute invariants in concrete examples.9 ECTSno date this semesterMA5132Algebraische ZahlentheorieNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the basic concepts and main theorems of algebraic number theory for number fields and their rings of integers. In the end you will be able to compute the norm, trace and discriminant, investigate the structure of ideals and the class group, and apply the theory to concrete examples such as quadratic and cyclotomic number fields.6 ECTSno date this semesterMA5110An Introduction to the Regularity Theory of Elliptic Partial Differential EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will get an introduction to the regularity theory of elliptic partial differential equations. You will learn classical and modern methods for deriving regularity results for linear and nonlinear problems as well as for systems, and you will be able to apply the most important proof techniques, e.g. De Giorgi's solution of Hilbert's XIX. In the end you will understand the basics of Nirenberg's methods, Schauder theory and partial regularity for systems.5 ECTSno date this semesterMA5081An Introduction to the Theory of Functions of Bounded VariationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the theory of functions of bounded variation (BV functions) and important tools from geometric measure theory. In the end you will be able to apply structural and compactness properties, traces and extensions as well as fine point properties of BV functions and sets with finite perimeter.5 ECTSno date this semesterMA5948Applied Introduction to Differential GeometryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou receive an application-oriented introduction to the fundamentals of differential geometry. By the end you can recognise manifolds and vector bundles and compute with simple geometric objects (Lie derivatives, exterior derivatives, integration of differential forms) as well as apply Lie groups to describe symmetries in physical systems.5 ECTSno date this semesterMA5228Applied Risk ManagementNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou analyze five historical risk management cases from different asset classes (Fixed Income, Equity, Commodities, Credit, Hedge funds). You will learn to apply the underlying historical events, the mathematical concepts, and their implementation in simple risk management tools in standard software.5 ECTSno date this semesterMA5730Approximation AlgorithmsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to design and analyze efficient approximation algorithms for combinatorial optimization problems. In the end you can assess the running time and approximation guarantees of algorithms, apply known techniques (e.g. Greedy, LP-Rounding, Primal-Dual), and prove limits of approximability.9 ECTSno date this semesterCIT4100003Asymptotic Kinetic Theories for Magnetized PlasmasNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods for asymptotic, multi-scale reduction of kinetic equations for magnetized plasmas. In the end you will be able to apply non-canonical Hamiltonian and Lagrangian formulations, identify suitable small parameters for a given regime, construct an asymptotic dynamical reduction, and explain the meaning of these formulations for Particle‑In‑Cell models.5 ECTSno date this semesterMA5360Bordisms and Topological Field TheoriesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theory of bordisms (Cobordisms) and their algebraic structures as well as the formulation of topological field theories as functors according to Atiyah and Segal. In the end you will be able to justify basic results, compute bordism rings, and follow and understand the connections to topological field theories as well as simple examples.6 ECTSno date this semesterMA5133Branching Random WalksNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the basics and advanced aspects of branching random walks. In the end you will know the classical theory of branching processes and Galton‑Watson trees as well as the connections to partial differential equations and relevant application areas.3 ECTSno date this semesterCIT413035Calculus of VariationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods of variational calculus, both indirect and direct approaches for scalar and vector-valued problems. This includes relaxation theory and Gamma-convergence as well as their application to model problems of nonlinear continuum mechanics. In the end you can apply direct and indirect methods and investigate various linear and nonlinear optimization problems.9 ECTSno date this semesterMA5074Case Studies in Scientific ComputingNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou work in small, interdisciplinary teams on concrete application problems from research or industry. In doing so, you create the entire solution chain from modeling through analysis and numerical solution to presentation, and you can ultimately develop mathematical models, select and implement suitable numerical methods, and present the results scientifically.7 ECTSno date this semesterCIT4130015Case Studies in Scientific ComputingNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou work in small interdisciplinary teams on real application problems from science or industry. You model, analyze and solve these problems with modern numerical methods and present the results scientifically (poster, talk, short paper). In the end you can mathematically model complex applications, select appropriate algorithms, implement them and communicate scientifically.10 ECTSno date this semesterCIT413058Catastrophe TheoryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theory of critical points and their local classifications, in particular nondegenerate and certain degenerate singularities such as fold and cusp. In the end you will be familiar with classification theorems for germs and with stable unfoldings (unfoldings) up to codimension 4.5 ECTSno date this semesterCIT4130016Category TheoryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the basic concepts of category theory: categories, functors and natural transformations as well as central concepts such as universal properties, representability and the Yoneda lemma. You will also become familiar with limits, colimits and adjunctions and will subsequently be able to analyze properties of categories from different mathematical areas.5 ECTSno date this semesterMA5144Code-based CryptographyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of coded cryptography: relevant terms from algebraic coding theory, cryptographic basic concepts and complexity theory as well as central problems and systems such as Syndrome Decoding, Information Set Decoding, Code Equivalence, McEliece and Niederreiter. In the end you will be able to name several code-based cryptosystems and attacks, analyze a system and estimate the costs of an attack, as well as evaluate the use cases and apply the knowledge to examples.9 ECTSno date this semesterCIT413062Complex Function Theory 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with the continuation of holomorphic functions, meromorphic functions on the Riemann sphere, and fundamental constructions in function theory (infinite products, Weierstrass and Mittag-Leffler theorems). You will also learn descriptive Riemann surfaces, homology and holomorphic differential forms, and apply the Riemann mapping theorem. In the end you will be able to construct holomorphic or meromorphic functions with prescribed zeros/poles and practically use the Riemann mapping.5 ECTSno date this semesterMA5005Compressed SensingNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to reconstruct sparse signals from a few structured linear measurements and which measurement matrices guarantee reliable recovery. In the end you can formulate important recovery statements, apply the Restricted Isometry Property (RIP) to random matrices, and use the underlying probabilistic proof techniques.5 ECTSno date this semesterMA5352Computational Complexity in OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the fundamentals of complexity theory with a focus on optimization problems: formal models, classes such as P and NP, NP-completeness and typical NP-hard optimization tasks. In the end you will be able to assess the difficulty of optimization problems, apply appropriate modeling guidelines, and analyze practical examples.5 ECTSno date this semesterMA5222Computational Convexity - Optimal ContainmentNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou deal with algorithmic questions on convex problems in arbitrary dimensions and, to some extent, in generalized normed spaces. In the module you will learn typical problems such as optimal containment, the underlying concepts of convex analysis and techniques from linear optimization, as well as their algorithmic solution and analysis.9 ECTSno date this semesterMA5206Computational Integer ProgrammingNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you learn the computational methods for solving mixed-integer optimization problems (MIP). You understand fundamental algorithms such as Simplex, Branch-and-Bound and Cutting-Plane-Separation as well as practical improvements and heuristics that make MIP solvers applicable to real problems. By the end you can explain these procedures, justify their correctness, and use your modeling knowledge to improve MIP models.3 ECTSno date this semesterMA8034Computational Inverse ProblemsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical and numerical methods for solving mainly linear inverse problems. In the end you will know the theory (ill-posedness, regularization, SVD, generalized Tikhonov) and common numerical procedures (direct and iterative regularization) as well as criteria for stability, convergence and stopping.6 ECTSno date this semesterMA4302Computational Risk Management of Equity-Linked InsuranceNo ratings for this module yet.A1.3.2 Additional to the Study ProgramThe module covers the modeling and risk management of equity-linked insurance products, which combine elements of life insurance and financial derivatives. You will learn how to model and numerically treat complex guarantees, dynamic insurance behavior, and the interaction of mortality and financial risks. In the end, you will be able to perform dynamic hedging, apply numerically demanding methods, and assess and classify regulatory capital requirements.3 ECTSno date this semesterMA5721Computational TopologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the basics of algebraic topology with a focus on homology and its computation as well as concepts of persistent homology. In the end you will be able to compute the homology of simplicial complexes, work with coefficients, apply the fundamental exact sequences, and understand and sketch the key results of persistent homology.6 ECTSno date this semesterMA5224Concentration of MeasureNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamental inequalities and techniques of Concentration of Measure (e.g., Chernoff, Hoeffding, Bernstein inequalities, isoperimetry, Johnson–Lindenstrauss, Martingale methods, Poincaré and Gromov–Milman theorems). In the end you will be able to understand these inequalities and apply them to simple examples from applications.5 ECTSno date this semesterCIT4100004Conformal Mapping and ProbabilityNo ratings for this module yet.A1.3.2 Additional to the Study ProgramThe module covers conformal mappings and their connection with stochastic processes in the plane. You will learn how conformal methods can solve probabilistic problems, and gain insight into two-dimensional critical systems and their scaling limits (e.g., percolation, loop-erased random walks, Ising model) as well as SLE and loop ensembles.5 ECTSno date this semesterMA5432Convex Duality and Applications in Mass Transport and Calculus of VariationsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn foundations of convex duality and how they are applied in variational problems and in the theory of optimal mass transport. In the end you will be able to understand Legendre/Fenchel duality, dual formulations of optimization problems and central results of mass transport theory as well as explain simple numerical procedures for it.3 ECTSno date this semesterMA5910Copulas: Inference and ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn modern Copula models to describe dependencies, both static and dynamic variants, with a focus on estimation and testing procedures. In the end you can assess parametric, semiparametric and nonparametric approaches and apply them in applications of the finance and insurance industry (e.g., portfolio risk, multi-name pricing, risk management).3 ECTSno date this semesterMA5728Coxeter GroupsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the definition and fundamental properties of Coxeter groups as well as central theorems and methods of the theory. In the second, shorter part you apply the theory to a concrete topic (e.g. finite reflection groups, Iwahori–Hecke algebras and knot theory, root systems and Lie theory, or computations for Coxeter groups). At the end you can explain the terms, apply the theorems to examples and work on new examples.3 ECTSno date this semesterMA5138Credit-Equity ModelingNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn models for credit and equity risks and deepen the fundamentals of Credit-Equity modeling. At the end you will be able to apply theoretical models (e.g., defaultable Markov-diffusion processes, reduced-form and firm-value models) and numerically value derivative products such as convertible bonds within these models.5 ECTSno date this semesterCIT413034CryptologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with fundamental concepts of modern cryptology: secrecy, complexity-theoretic foundations and concrete encryption methods such as RSA, Diffie–Hellman, ElGamal, Rabin as well as elliptic curves. In the end you will be able to describe the presented systems, discuss their advantages and disadvantages and apply the necessary algebraic foundations.5 ECTSno date this semesterMA5104Delay Differential Equations with ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theory and numerics of delay differential equations (in particular with constant delays). You can formulate models with delays, analyze their existence, uniqueness and stability properties, as well as apply suitable numerical procedures and interpret results.5 ECTSno date this semesterMA5062Dependence Models Generated via Line Integrals and Actuarial ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn how multivariate lifetime models can be constructed with the help of line integrals and hazard vector fields. In the end you will be able to generate such models, adapt them, and apply them in actuarial applications, as well as understand discrete and continuous cases with physical interpretation.5 ECTSno date this semesterMA5734Differential FormsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the foundations and tools of differential forms on smooth manifolds: construction, exterior product, exterior derivative, integration and Stokes’ theorem. In the end you can differentiate, integrate and apply differential forms in contexts of geometry, topology, analysis and physics.5 ECTSno date this semesterCIT4130022Differential Forms 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn how differential forms are applied in algebraic and differential topology as well as in differential geometry. In the end you will be able to apply and interrelate concepts such as de-Rham cohomology, Poincaré duality, Mayer-Vietoris, the Künneth formula, fixed-point and degree theory as well as basics on fiber bundles, characteristic classes, Lie groups and Riemannian manifolds.5 ECTSno date this semesterCIT413033Direct Methods in the Calculus of VariationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with the fundamentals and modern methods of the calculus of variations. By the end you can understand central terms such as coercivity, sublevelness, relaxation and Γ-convergence and apply them to problems in e.g. mathematical physics and materials science.9 ECTSno date this semesterMA5917Discontinuous Galerkin MethodsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will obtain an introduction to Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations. In the end you will be able to derive DG formulations for elliptic, parabolic and hyperbolic problems, evaluate their stability and convergence properties, and practically implement DG schemes.9 ECTSno date this semesterMA5343Discrete Harmonic AnalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with harmonic analysis in discrete, finite structures. The module covers Fourier transforms on finite (abelian and non-abelian) groups, discrete Fourier and fast Fourier transforms, and applications e.g. in signal processing, graph theory, and number theory. In the end you will be able to formulate the fundamental methods of discrete harmonic analysis and apply them to examples and applications.6 ECTSno date this semesterMA5911Diskrete FlächentheorieNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will become familiar with discrete models of surfaces in R^3 and R^4 as well as in projective and Möbius geometry. The module introduces discrete (integrable and non-integrable) surface theory, discrete curvature concepts and transformation theories, so that you can practically work with discrete differential geometry methods afterwards.5 ECTSno date this semesterMA5229Diskrete Geometrie: GitterpolytopeNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou deal with convex, rational polyhedra and their relation to lattices. In the module you will learn central results such as Ehrhart theory (the connection between volume and the number of lattice points) as well as aspects of the geometry of numbers and unimodular triangulations, and you can apply these methods to examples from combinatorial geometry, integer optimization, and algebraic geometry.9 ECTSno date this semesterMA5215Dynamics of Democratic ElectionsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn models from socio-physics and mathematical social sciences that describe the behavior of voters and the dynamics of democratic elections. In the end you will be able to explain these models, apply them, and perform and interpret simple data-based analyses of voting processes.6 ECTSno date this semesterMA5619Dynamics of Infectious DiseasesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn standard models and central results on the spread and control of infectious diseases. You will set up classical models (e.g., Kermack–McKendrick) and variants, estimate the basic reproduction number R0, consider stochastic modeling, contact tracing, control measures, and evolutionary aspects of pathogens. In the end you will be able to adapt models to concrete disease cases and recognize the driving evolutionary forces and their consequences.6 ECTSno date this semesterCIT413056Einführung in die Algebraische ZahlentheorieNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the foundations of algebraic number theory: number fields, rings of algebraic numbers and their ideal theory. In the end you can understand central concepts such as norm, trace, integrality, discriminant and units and apply the fundamental theorems to concrete problems and simple diophantine questions.5 ECTSno date this semesterMA5129Elements of Harmonic AnalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of harmonic analysis for different groups: finite groups, the torus and the real line. Topics include Fourier analysis in more abstract settings, convolution operations, representation theory of finite type, as well as central theorems such as Plancherel, inversion, Bochner and duality theorems; at the end you will be able to understand classical results of Fourier analysis in the context of locally compact (abelian) groups and to give elementary proofs.5 ECTSno date this semesterMA5021Elements of the Theory of DistributionsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of distribution theory: test functions, the delta distribution, distributions on D and S, as well as operations such as convolution, derivatives, integration and Fourier transforms of distributions. By the end you can perform calculations with distributions and apply them in various applications.5 ECTSno date this semesterMA5319Elliptic CurvesNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations of the theory of elliptic curves: from algebraic curves and their invariants to the arithmetic structure of the points on elliptic curves. In the end you will know definitions, important theorems (e.g. Bezout, Mordell, Hasse) and you can compute invariants, differentials, the genus and the group structure for curves in Weierstrass form.9 ECTSno date this semesterMA5114Fallstudien in Risikomanagement, Finanz- und VersicherungsmathematikNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou work on practice-oriented questions from financial mathematics, actuarial science and risk management. You model suitable stochastic approaches, implement numerical algorithms and adapt models to market data; in the end you can evaluate solutions and present your results.6 ECTSno date this semesterMA5727Fallstudien VersicherungsmathematikNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn how life insurance products are designed and their risks controlled. You implement numerical simulation algorithms, evaluate risk and return of different contract forms, and apply models to real actuarial and financial data. In the end you can analyze contracts, guarantees, and longevity risks and present your results appropriately.5 ECTSno date this semesterMA5726Financial Market VolatilityNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn how volatility of financial markets is modeled, estimated and forecasted – both under the physical and under the risk-neutral measure. The course covers discrete procedures (e.g. GARCH, moving averages, Realized Volatility), stochastic models in continuous time (e.g. Heston, local volatility) and practical applications such as hedging, risk assessment, portfolio allocation and trading of volatility products (Variance Swaps, VIX futures, ETFs/ETNs). At the end you can calibrate models, make forecasts and implement the methods in software such as Matlab.5 ECTSno date this semesterMA5719Financial Mathematics 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn stochastic processes and Ito calculus for modeling financial markets and understand the concepts of arbitrage, completeness and risk-neutral valuation. You can price derivatives (including exotic options) in continuous models, especially Black-Scholes and its generalizations, and implement various numerical methods.9 ECTSno date this semesterMA3408Fine Properties of Sobolev FunctionsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamentals and further properties of Sobolev functions. Starting with weak derivatives, you will treat approximations by smooth functions, traces and extension theorems, as well as compact embeddings. In addition, finer properties such as precise representatives, quasi-continuity and differentiability on lines are shown using measure-theoretic techniques (e.g. Hausdorff measures and capacity). At the end you will be able to apply and explain these concepts and proof methods.3 ECTSno date this semesterMA5067First Order Mean Field GamesNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations and current methods of First Order Mean Field Games (MFG) with a focus on variational principles, congestion or congestion models and connections to dynamic optimal transport. By the end you will understand existence and uniqueness questions, regularity tools and variants such as dense- or time-minimal constrained MFG.3 ECTSno date this semesterMA5912First Order Primal-Dual Optimization MethodsNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn modern First-Order Primal-Dual optimization methods. You will understand how simple iterative schemes with primal-dual decompositions can be combined to design efficient, structure-exploiting algorithms for large-scale problems, and you will be able to apply and further investigate these methods.5 ECTSno date this semesterCIT413065Foundations of Data AnalysisNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn mathematical methods of data analysis, in particular linear and nonlinear procedures for data dimensionality reduction as well as techniques for reconstructive restoration of structured signals. By the end you will be able to apply and assess singular value decomposition, random matrices, compressive methods and manifold-based procedures.8 ECTSno date this semesterMA4800Fourier- und Laplace-TransformationNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the theory and applications of Fourier and Laplace transforms: Fourier series, Fourier analysis (integral transform), Gabor transform and Laplace transform. In the end you can understand the interaction between a function and its Fourier transform and use it for optimized approximations.9 ECTSno date this semesterMA5039FourieranalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of Fourier analysis on Euclidean spaces: Fourier series on period intervals and the Fourier transform on R^n, including generalizations to L^2 and distributions. In the end you will be able to assess convergence and regularity questions and apply Fourier techniques to applications such as PDEs, signal processing and sampling.5 ECTSno date this semesterMA4064Fraktale GeometrieNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn how to construct fractal sets with iterated function systems, analyze their fundamental properties and dimensions, and relate them to dynamical systems. You will also learn how to create approximation models with fractal functions and fractal surfaces.5 ECTSno date this semesterMA5207Functions of Bounded Variations and ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals and important tools of the theory of functions of bounded variation as well as geometric measure theory. You will understand measures such as Hausdorff measures, area and Coarea formulas, concepts of sets of finite perimeter and special BV functions. In the end you will be able to apply these terms to the treatment of relaxation problems in W^{1,1} and understand the mathematics behind the Mumford–Shah functional.9 ECTSno date this semesterCIT4130001Fundamentals of Optimization for Machine LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn fundamentals and advanced techniques of optimization, both convex and nonconvex as well as combinatorial and continuous, with a focus on applications in machine learning. In the end you will be able to understand optimization problems from ML research and approach research questions in this area.5 ECTSno date this semesterCIT413031Funktional-DifferentialgleichungenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will be introduced to the theory of functional differential equations, in particular delay differential equations and further types (e.g., mixed type). At the end you will be able to understand and assess fundamental questions on existence, uniqueness, and stability of solutions.3 ECTSno date this semesterMA5006Generalized Linear ModelsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn methods of regression for non-normally distributed target variables (e.g., binary, count data, nominal, positive values). In addition to classical GLMs such as logistic, Probit-, Poisson-, Gamma-, and log-linear models, extensions (e.g., overdispersion, random effects) are covered. By the end you will be able to estimate models, validate them, and analyze and interpret the results with R.9 ECTSno date this semesterMA3403Geometric Continuum MechanicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of a differential-geometric approach to continuum mechanics. With the help of differential forms and integration on manifolds you will formulate and apply laws such as balance principles and stress theory in a metric- and coordinate-free language.6 ECTSno date this semesterMA5339Geometric Measure Theory and ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of Geometric Measure Theory and how to transfer classical geometric properties of smooth sets to non-smooth situations. In the end you will be able to understand and apply central concepts such as tangent measures, Hausdorff measures, rectifiability, as well as area- and co-area-formulas, and follow the modern analysis of the Plateau problem.no date this semesterMA5925Geometric Methods for Physics of Magnetized PlasmasNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn to apply geometric methods (Hamiltonian and Lagrangian formalism) for the systematic reduction of complex multi-scale dynamical systems based on magnetized plasmas. You understand perturbative Lie-Transform methods, the derivation of field and kinetic equations from variational principles as well as the associated conservation laws; in addition you will see how continuous variational descriptions can be transferred to discrete formulations and implemented numerically, e.g. in Particle-In-Cell Monte-Carlo simulations.5 ECTSno date this semesterMA5333Geometric Numerical Integration 1No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of geometric (structure-preserving) numerical integration for ordinary differential equations. In the end you can recognize different geometric structures in differential equations and select and apply suitable numerical methods that preserve these structures.5 ECTSno date this semesterMA5341Geometrie und Topologie für die DatenanalyseNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn fundamental methods of geometric and topological data analysis. This includes constructions such as Voronoi and Delaunay diagrams, Alpha Shapes, as well as concepts from topology such as simplicial complexes and homology and their application to filtrations and persistent homology to investigate the topological properties of point sets.6 ECTSno date this semesterMA4804Geometrische Numerische Verfahren für gewöhnliche DifferentialgleichungenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn fundamental techniques of geometric (structure-preserving) numerics for ordinary differential equations. In the end you recognize geometric structures in ODEs, know modern integrators that preserve these structures, and can select and implement appropriate procedures.6 ECTSno date this semesterMA5329Geometry of Finite-dimensional Normed SpacesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with the geometry of finite-dimensional normed spaces and convex bodies. You learn fundamental theorems of convex and discrete geometry as well as tools (e.g. Helly, Carathéodory, Radon, John ellipsoid, Banach–Mazur distance) and apply these to concrete problems in finite normed spaces.3 ECTSno date this semesterCIT413057Gitterfreie VerfahrenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods for mesh-free approximation and interpolation of multivariate, scattered data, in particular Radial Basis Functions (RBF) and Moving Least Squares (MLS). In the end you can apply the fundamental principles, analyze them and implement the algorithms in Matlab.5 ECTSno date this semesterMA5324Gradient Flows in Metric SpacesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the analytical and geometric foundations of the metric theory of gradient flows. In the end you will be able to investigate existence, regularity and long-term behavior of such flows and qualitatively analyze solutions of the associated class of partial differential equations.5 ECTSno date this semesterMA5059Graphical Models in StatisticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn statistical models whose conditional independencies are described by graphs. The focus is on continuous distributions (in particular the multivariate normal distribution), undirected graphical models (Gaussian models) and directed acyclic graphs (Bayesian networks). By the end you will be able to represent dependency structures in data with graphical models and select appropriate model classes.9 ECTSno date this semesterMA5439Harmonic AnalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn core techniques of harmonic analysis such as Fourier transformation, singular integrals and function spaces. In the end you will be able to apply these methods to the study of partial differential equations, in the theory of function spaces and in applications such as signal processing.3 ECTSno date this semesterMA5954Harmonic Analysis on Commutative SpacesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamentals of harmonic analysis on topological groups and on their coset spaces. You will practice integration and convolution on groups, fundamentals of representation theory and specific techniques for Abelian, compact as well as homogeneous spaces and double coset spaces. In the end you will be able to analyze functions on these spaces and understand the connections to geometric analysis.9 ECTSno date this semesterCIT413026High-dimensional StatisticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn methods for the analysis of data where the number of variables is large. After the module you can apply procedures for controlling false discoveries in large-scale multiple testing, use sparse regression methods (e.g., Lasso) and methods for group-specific or low-dimensional structured signals, as well as use regularization approaches for matrix parameters and graph models.5 ECTSno date this semesterMA5442HomogenizationNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn methods of homogenization for partial differential equations with rapidly oscillating coefficients as well as for PDEs in domains with periodically distributed small holes. In the end you will be able to apply fundamental homogenization techniques such as asymptotic expansions and two-scale convergence and describe the concepts behind stochastic homogenization problems.3 ECTSno date this semesterMA5921Homological AlgebraNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the fundamental tools of homological algebra, which are applied in topology, geometry and representation theory. In the end you can work with categorical vocabulary, compute with chain complexes and homology, and compute derived functors such as Ext and Tor.6 ECTSno date this semesterMA5140Identification of Artificial Neural Networks: from the Analysis of one Neuron to Deep Neural NetworksNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical methods for the analysis and identification of feed-forward neural networks — from individual neurons to flat networks up to deep networks. The focus is on linear algebra, probability (in particular concentration inequalities) and optimization to determine weights and activation functions efficiently and robustly.6 ECTSno date this semesterMA5929Inequalities in Operator AlgebrasNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of operator algebras and deepen key inequalities and structures that play a role in quantum mechanics. In the end you will be able to apply and understand important concepts such as complete positivity, operator monotonicity/-convexity, Tomita–Takesaki theory, as well as central results on entropy and relaxation in noncommutative systems.3 ECTSno date this semesterMA5930Insurance Mathematics 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of personal insurance: life insurance, occupational pension provision, and health insurance. You can calculate expected values and present values of insurance benefits, determine premiums and reserves according to classical and modern models, and apply valuation methods for pension obligations and specific questions of private health insurance.9 ECTSno date this semesterMA3406Introduction to Conservation LawsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals and properties of hyperbolic conservation laws, in particular the occurrence and handling of discontinuities (e.g. shock waves). By the end you can solve the Riemann problems for simple systems, apply the entropy condition to select physically relevant weak solutions, and assess and develop basic finite-difference and finite-volume methods along with modern Riemann solvers for compressible flows.5 ECTSno date this semesterMA5935Introduction to Group Representation TheoryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will receive an introduction to the representation theory of groups and algebras with a view to applications in mathematics and the natural sciences. In the end you will be able to apply fundamental techniques such as complete reducibility, character theory and the representation of symmetric groups (or alternatively Hecke algebras and braid groups) and to explore further areas.5 ECTSno date this semesterMA5134Introduction to Stochastic Differential Equations: Theory and NumericsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn the fundamentals of stochastic differential equations (SDEs) as well as their numerical treatment. By the end you will know the most important stochastic processes, the construction of stochastic integrals, and solution methods for SDEs, and you can analytically solve simple SDEs and apply numerical schemes to approximate general SDEs.3 ECTSno date this semesterMA5950Introduction to Variational and Level Set Methods for Geometric FlowsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn analytical and variational methods for the study of geometric flows. The focus is on Level-Set methods, viscosity solutions of degenerate parabolic equations, and variational formulations of the Mean-Curvature Flow. In the end you will be able to understand existence and uniqueness statements for generalized Level-Set Flows and cope with concepts such as Weak–Strong-Uniqueness, Fattening, and Minimising Movements.5 ECTSno date this semesterCIT413066Invariant TheoryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the foundations and methods of invariant theory: behavior of orbits, generation and structure of invariants as well as classical and modern results for finite and reductive groups. In the end you can work on examples, connect algebraic and geometric perspectives, and practically apply methods to concrete problems.6 ECTSno date this semesterMA5131Investment StrategiesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will gain an overview of important static and dynamic investment strategies and their mathematical foundations. The module is complemented by an introduction to stochastic control methods and utility maximization. In the end you will be able to develop new investment strategies, calculate their net present value, and analyze their risk.5 ECTSno date this semesterMA5709Isogeometric Analysis: Theory and PracticeNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn B-Splines/NURBS from the CAD side through approximation theory to application in isogeometric finite element methods. In the end you will be able to derive weak formulations for elliptic, Maxwell, and Stokes problems, explain their properties, and implement corresponding finite-element solvers in Python.5 ECTSno date this semesterMA5938Lattice Boltzmann methodsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the Lattice-Boltzmann Method (LBM) as an alternative to classical CFD methods: how the method is derived from kinetic models, how streaming and collision steps work, and which advantages LBM offers for complex geometries, microscopic interactions, and parallelization. In the end you will be able to theoretically classify, implement, and apply the method to selected model problems from science and engineering.9 ECTSno date this semesterCIT4130009Lattices and CodesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamental types of lattices (e.g., even/odd, unimodular, root lattices) and important properties of root lattices as well as the connection between (binary) linear codes and certain integer lattices. In the end you will be able to geometrically construct codes and derive their properties from the lattice structure and associated theta functions or modular forms.6 ECTSno date this semesterCIT4130014Linear Algebraic GroupsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theory of linear algebraic groups: structure, core concepts and central structural results such as connected components, Lie algebras, quotients and Jordan decomposition. In the end you will be able to apply these results and analyze examples as well as transfer the concepts to connections with representation theory, arithmetic geometry and invariant theory.9 ECTSno date this semesterCIT4130018Low Rank ApproximationNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods for low-rank approximation of matrices and tensors, including singular value decomposition and various tensor factorizations. In the end you will be able to assess which approximation is suitable for a given application and apply and implement it for data compression or analysis.3 ECTSno date this semesterMA5328Markov ProcessesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn theory and methods of continuous-time Markov processes: continuous-time Markov chains, the Markov property, Feller processes as well as properties of transition semigroups and their generators. In the end you can analyze the long-term behavior of processes, apply ergodic theorems and use them in examples (e.g. queueing theory, interacting particle systems, time series).9 ECTSno date this semesterMA4408Mathematical Foundations of ImagingNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical foundations and models of imaging — from Fourier series and -transforms through frame theory and time-frequency analysis to wavelet transforms and variational methods. In the end you can compare different imaging models, apply their mathematical analysis, and employ suitable tools to solve concrete imaging problems.9 ECTSno date this semesterMA5063Mathematical Foundations of Machine LearningNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the mathematical foundations of modern machine learning methods: structure, learning algorithms and approximation properties of neural networks, theory and application of kernel methods in reproducing kernel Hilbert spaces as well as qualitative aspects such as loss functions, risk and complexity questions. By the end you will be able to construct networks, discuss their approximation properties, apply kernel methods and assess the statistical efficiency of procedures.6 ECTSno date this semesterMA4801Mathematical Introduction to Quantum Information ProcessingNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the mathematical foundations of quantum information theory. The module covers abstract foundations of quantum mechanics, measurement and development theory, quantum statistics and tomography, as well as fundamental concepts regarding limits and resources of quantum information processing.9 ECTSno date this semesterMA5057Mathematics of Reinforcement LearningNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the mathematical foundations of reinforcement learning: Markov decision processes (MDPs), tabular RL methods such as Monte Carlo, Temporal Difference, SARSA and Q-Learning, as well as basics of stochastic approximation to analyze the convergence of these algorithms. In the end you can formulate dynamic decision problems under uncertainty as MDPs and solve them with tabular RL algorithms.6 ECTSno date this semesterCIT413036Mathematische Einführung in die MagnetohydrodynamikNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will receive a mathematically focused introduction to magnetohydrodynamics (MHD) — the theory of electrically conducting fluids in a magnetic field. By the end you will be able to understand the fundamental MHD equations, their derivation from multi-fluid models, central conservation laws, as well as reduced and variational/Hamiltonian formulations, and place them in the scholarly literature.3 ECTSno date this semesterMA5902Mathematische Grundlagen der Neuronalen NetzeNo ratings for this module yet.A1.3.1 Related to the Study ProgramThe module conveys selected mathematical foundations for the analysis of artificial neural networks. You will learn how approximation properties, stability with respect to input perturbations, and the learnability of networks are studied using various mathematical tools. In the end you will understand the central theoretical results and the analytical methods used.6 ECTSno date this semesterMA5913Mechanics and SymmetryNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn mathematical foundations of mechanics with a focus on symmetries: multilinear algebra, differential geometry and Lie groups, variational principles and reduction (Euler–Poincaré), as well as related structures such as symplectic geometry and Lie–Poisson brackets. In the end you will be able to apply variational principles, perform reduction for symmetric systems, and assess conservation laws using momentum maps/Noether arguments.3 ECTSno date this semesterMA5940Meromorphic Functions on the Riemann SphereNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with meromorphic functions on the Riemann sphere, with covering mappings and Möbius transformations as well as with elliptic functions and their topology. In the end you will be able to construct meromorphic continuations, specify coverings and you will know the fundamentals of Riemann surfaces.5 ECTSno date this semesterMA5942Methods for Inverse ProblemsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical foundations and numerical methods for solving inverse problems. The focus is on regularization techniques for linear and nonlinear problems as well as iterative solution methods (gradient-, Newton-, and Kaczmarz-type) and practical aspects such as adaptive discretization in PDE-based reconstruction tasks.3 ECTSno date this semesterMA5931Models and Numerical Methods for Eulerian and Lagrangian Hyperbolic EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramIn this module you will learn mathematical models and numerical methods for hyperbolic conservation laws in Eulerian and Lagrangian representations. You will engage with fundamentals such as Burgers’, shallow-water and gas dynamics, learn concepts such as weak solutions, shock and rarefaction structure as well as entropy conditions, and implement robust finite-volume methods, also for moving meshes.3 ECTSno date this semesterMA5928Models for Material Defects and Grain BoundariesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou are dealing with variational “semi-discrete” models for defects in metals and their significance for plastic deformation. The course covers multiscale analyses that lead to the formulation of line-energy models for dislocations and the modeling of grain boundaries. In the end you can apply mathematical techniques to analyze such models and understand the underlying energy estimates and asymptotics.3 ECTSno date this semesterMA5944Modern Approximation TheoryNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn tools of modern approximation theory: s-numbers (such as approximation, Gelfand and Kolmogorov numbers), entropy numbers as well as methods and bounds for sparse reconstruction (e.g., Prony methods, Restricted Isometry Property, iterative hard thresholding, CoSaMP). In the end you can apply these concepts to tasks for the approximation of vectors, functions and operators and you will recognize fundamental lower bounds in sampling theory.6 ECTSno date this semesterMA5952Moderne Methoden der Nichtlinearen Optimierung (2)No ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn selected modern methods of nonlinear optimization, such as convex methods, nonsmooth optimization, interior-point methods, semidefinite programming, robustness approaches, duality, or globalization techniques. In the end, you will be able to read current research articles on the treated topics and follow the presented methods independently further.5 ECTSno date this semesterMA4505Nonlinear AnalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamental methods of nonlinear analysis in infinite dimensions: Schauder degree theory regarding the question whether equations have zeros; generalized versions of inverse and implicit function theorems as well as Lagrange multipliers in the infinite-dimensional context; and bifurcation theory for the study of parameter-dependent solutions and their stability. In the end you will be able to apply these concepts to problems from partial and ordinary differential equations.9 ECTSno date this semesterMA5923Nonsmooth OptimizationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn methods and concepts of nonsmooth optimization and how to apply them. The module conveys fundamentals of nonsmooth analysis, numerical procedures for minimization and for handling nonsmooth equations, as well as their convergence properties. At the end you will be able to select appropriate methods for concrete nonsmooth models and to assess their behavior theoretically.5 ECTSno date this semesterCIT4130020Numerical Analysis for High-dimensional Quantum DynamicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn mathematical foundations and numerical methods for simulating quantum-dynamical systems in high dimensions. In the module you treat grid-based methods, the Truncated-Wigner Approximation, the time-dependent variational principle methodology, and the Gaussian wave packet approximation. In the end you will be able to understand and apply central approximations in phase-space formulation and methods from the time-dependent variational principle.6 ECTSno date this semesterCIT413027Numerical Methods for Hyperbolic and Kinetic EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn numerical methods for hyperbolic and kinetic partial differential equations. The module covers fundamental discretization procedures (e.g., finite differences, finite volumes, semi-Lagrangian) for hyperbolic conservation laws and methods for high-dimensional kinetic equations as well as asymptotic-preserving methods and applications in control and uncertainty quantification. In the end you will understand the connections between mathematical structure and practical implementation and will be able to apply selected methods to applications.3 ECTSno date this semesterMA5932Numerical Methods for Hyperbolic SystemsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with numerical methods for hyperbolic conservation laws as they occur in fluid and plasma physics. You will learn finite-volume and discontinuous-Galerkin methods in 1D and extend them to linear and nonlinear systems. In the end you can derive and implement a corresponding numerical scheme.5 ECTSno date this semesterMA5090Numerical Methods for Uncertainty QuantificationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn how to formulate, analyze, and numerically approximate elliptic boundary value problems with random coefficients. This includes modeling and sampling of random fields as well as numerical methods such as Monte Carlo, stochastic collocation, and stochastic Galerkin methods.6 ECTSno date this semesterMA5348Operator TheoryNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the fundamentals of spectral theory of operators, in particular spectra of operators, spectral theory for normal operators and functional calculus for different classes of operators. In the end you will be able to analyze and apply these tools to integral and differential operators.9 ECTSno date this semesterMA5012Operatoralgebraische Methoden in der QuanteninformationstheorieNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn central operator-algebraic methods used in quantum information theory. After the module you can understand and apply basic concepts of quantum mechanics, channels and entropy, and you can independently read and present current research results.3 ECTSno date this semesterMA5920Optimal TransportNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn the foundations of the mathematical theory of Optimal Transport (OT). By the end you will understand central concepts such as Monge and Kantorovich formulations, the Wasserstein distances and their role in analysis, physics, economics and machine learning, as well as methods for handling high-dimensional, multi-criterion problems.9 ECTSno date this semesterMA5934Optimal Transport, Numerics and SamplingNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn this module you will learn the theory of optimal transport and methods for its numerical treatment. You will understand the fundamental models (Monge, Kantorovich), important properties of Wasserstein spaces and gain an overview of numerical procedures and applications in Data Science.3 ECTSno date this semesterMA5933Optimale Steuerung gewöhnlicher Differentialgleichungen 1No ratings for this module yet.A1.3.1 Related to the Study ProgramYou learn fundamental concepts and methods of optimal control for ordinary differential equations. In the end you will be able to formulate necessary optimality conditions (e.g., Euler–Lagrange, Legendre–Clebsch), distinguish different types of constraints and control restrictions, and convert control problems into boundary-value forms suitable for numerical treatment.5 ECTSno date this semesterMA3312Partial Differential Equations 2 - Nonlinear Evolution EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with qualitative properties of nonlinear evolution equations and learn methods to prove properties of solutions (e.g., regularity gain, convergence to stationary or self-similar profiles, contractivity). The main focuses are variational methods, Hamilton–Jacobi–Bellman equations, systems of conservation laws and degenerate parabolic equations.9 ECTSno date this semesterCIT4130024Partial Differential Equations 2 - Nonlinear Parabolic Evolution EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods of semigroup theory and gradient flows and apply them to investigate both linear and nonlinear parabolic evolution equations. By the end of the module you can analyze existence and qualitative properties of solutions using these techniques.5 ECTSno date this semesterMA5918PDE2 - Nonlinear Partial Differential EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn methods for existence and qualitative analysis of nonlinear partial differential equations. The focus is on variational techniques (calculus of variations, Euler–Lagrange, existence of minimizers, regularity) as well as nonvariational methods (monotone, fixed-point iterations, sub- and supersolutions, methods for proving non-existence). In the end you can apply these tools to semilinear elliptic problems.9 ECTSno date this semesterMA5077PDE2: Dynamics of Nonlinear Evolution EquationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with nonlinear time-dependent partial differential equations (du/dt = Au + N(u)) and learn techniques to analyze their local and global behavior. In the end you will classify prototypes such as Burgers, nonlinear Schrödinger, KdV, or Nagumo equations and apply methods such as analytic semigroups, variation-of-constants (mild solutions), stability and invariant manifold theory.9 ECTSno date this semesterMA5946Population GeneticsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn mathematical models of population genetics (e.g., Wright–Fisher, Moran, Kingman coalescent) and how evolutionary and ecological forces shape genomic variation. In the end you will be able to formulate models, analyze them, and interpret results as well as patterns in genomic polymorphism, including interspecific interactions such as cooperation or coevolution.6 ECTSno date this semesterMA5615Probability on GraphsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn stochastic models on graphs, e.g. Random Walks, percolation and random graphs. In the end you will be able to analyse Random Walks on networks, use the connection to electrical networks to show recurrence/transience, and understand phase transitions and random spanning trees.5 ECTSno date this semesterMA4406Projektive Geometrie 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou explore selected topics of projective geometry and learn to apply algebraic and combinatorial methods to geometric problems. In the end you can, for example, purposefully use tensor or Clifford algebra, apply combinatorial tools such as oriented matroids or polytope theory, and explain the interplay between geometry, combinatorics and algebra in concrete examples.5 ECTSno date this semesterMA3204Quantum Dynamics 2No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou are dealing with mathematical methods for describing and approximating quantum-dynamical systems. By the end of the module you will understand concepts such as unbounded self-adjoint operators, one- and two-parameter groups, coherent states, WKB approximations, as well as time-dependent density matrices and path integrals, and you will be able to apply these methods to concrete models and implement them numerically.5 ECTSno date this semesterMA5025Quantum Dynamics 3No ratings for this module yet.A1.3.2 Additional to the Study ProgramYou deal with mathematical methods for the analysis and numerical treatment of quantum-dynamical systems. In the module you will learn advanced approaches such as variational principles, Galerkin approximations and semiclassical models, and you can apply these to concrete model systems of medium to moderately high dimension.5 ECTSno date this semesterMA5926Quantum Tradeoff RelationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with various tradeoff relations in quantum mechanics and their applications in statistics, metrology and information theory. Starting with a short mathematical overview of density operators, POVMs, completely positive maps and instruments, you will learn to analyze and apply uncertainty relations, information-/disturbance as well as energy-/temperature tradeoffs and tradeoffs between speed, precision and sample size.5 ECTSno date this semesterCIT4130012Random Matrices: Theory, Numerical Methods, and ApplicationNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will receive an introduction to large random matrices, their eigenvalue distributions and eigenvalue spacings. At the end of the module you will know the fundamental methods (e.g. Stieltjes transform, orthogonal polynomials, Fredholm determinants, free probability) and you will be able to compute limit distributions and apply them to models in applications.3 ECTSno date this semesterMA5306Representation Theory of Finite GroupsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn fundamental concepts and methods of representation theory of finite groups. The topic covers representations, group algebras, module theory, semisimple algebras and modules, as well as characters and character tables; you can apply the developed statements and techniques to concrete problems.9 ECTSno date this semesterCIT413032Representations of Compact GroupsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the representation theory of compact groups. The module conveys the relationships between Lie groups and their Lie algebras, central results such as the Peter–Weyl theorem, the structure of maximal tori, roots, weights and the Weyl character formula as well as representations of the classical groups. In the end you will be able to construct, decompose and analyze representations from algebraic, differential-geometric and analytic perspectives.5 ECTSno date this semesterMA5054Scheduling: Theory and AlgorithmsNo ratings for this module yet.A1.3.1 Related to the Study ProgramYou will learn models and algorithms for the assignment of tasks to scarce resources. The module covers classical and modern scheduling problems (including stochastic, online, robust) as well as methods for their modeling, analysis and solution, so that you can, in the end, assess complexity, design exact or approximate algorithms and prove their quality.5 ECTSno date this semesterCIT413053Selected Chapters from the Mathematical Continuum MechanicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn selected mathematically relevant methods of continuum mechanics and their applications in physical disciplines. From a range of topics (e.g., chemical/biological reactions, higher moments, Boltzmann equation and electrodynamics, introduction to relativity, self-gravitation of stars) you will deepen individual chapters and be able to understand and apply the underlying mathematical methods.5 ECTSno date this semesterMA5340Self-interacting Random Walks and Statistical PhysicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn methods for the analysis of self-interacting random walks and their connections to statistical physics. In the end you will be able to understand and apply central models (e.g. edge-/vertex-reinforced processes), proofs of localization/delocalization, and connections to supersymmetric sigma models and random Schrödinger operators.3 ECTSno date this semesterMA5941Special Topics in Ordinary Differential Equations: Symmetries and ReductionNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with parameter-dependent ordinary differential equations (analytic or polynomial), their symmetries and special reduction methods. A focus lies on algorithmic/reconstructive methods for determining symmetries and critical parameters as well as applications in simple biochemical reaction networks. In the end you will be able to recognize symmetries, perform reductions and identify critical parameters for singular perturbations.3 ECTSno date this semesterCIT4130011Stability of Nonlinear WavesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn what traveling waves (solutions that propagate with constant shape and speed) are and why their dynamic stability is important. By the end you can sketch the steps of a stability analysis, understand the associated spectral structure, and apply various tools to approximate spectra and infer (in)stability from spectral data.5 ECTSno date this semesterMA5945Statistical Analysis of CopulasNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn to model multivariate dependency structures using Copulas and to understand, estimate, and apply Vine-Copula models (C-, D- and R-Vines). In the end you can select suitable vine models, implement them in R with VineCopula/CDVine, simulate, interpret and assess results.5 ECTSno date this semesterMA5408Statistical Inference for Dynamical SystemsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramIn this module you will learn how to model biological reaction networks with ordinary differential equations and how parameter values of these models can be reliably estimated from experimental data. You will acquire methods for parameter estimation (frequentist and Bayesian) as well as techniques for analyzing uncertainty and practicality of parameters and implement these in MATLAB.6 ECTSno date this semesterMA5612Statistical Inverse ProblemsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn fundamental methods for the statistical treatment of inverse problems. The course covers linear inverse problems with random noise, regularization, convergence rates, adaptive procedures and model selection; at the end you can compare the results and apply them to concrete examples.5 ECTSno date this semesterMA5428Stochastic Models for Tariff Calculation, Loss Reserving and Reinsurance and Their ApplicationsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn stochastic models and their application in property and casualty insurance: fundamentals, rating, reserve and reinsurance. By the end you can model tariffs and reserves and evaluate simple reinsurance contracts as well as practical constraints such as data or model risks.3 ECTSno date this semesterMA5736Structure Preserving Discretisation on Staggered GridsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn principles and methods of structure-preserving discretisations for partial differential equations. Using finite differences on staggered grids and finite element methods, you will learn to design procedures such that fundamental properties of the PDEs (e.g., invariants or Hamiltonian structures) are preserved. In the end you will be able to apply these procedures to classical PDEs such as Maxwell’s or Euler’s equations and assess their advantages and limitations.5 ECTSno date this semesterMA5936Tensor Network MethodsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn modern methods for efficient approximation of high-dimensional functions (large N-tensors). The focus is on Tensor-Train (TT) approximation: its theory, numerical algorithms and applications. In the end you can analyze and simulate simple multidimensional problems with tensor-network methods.5 ECTSno date this semesterCIT413064Theorie der Zellulären AutomatenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the theoretical foundations of cellular automata: topological properties, various classification approaches, and questions of computability and decidability. In the end you will be able to apply the topological viewpoint, understand approaches for classifying automata, and prove (non-)decidability statements for relevant problems.6 ECTSno date this semesterMA5611Theorie der ZufallsmatrizenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn non-asymptotic deviation estimates for random variables and vectors (in particular subgaussian and subexponential as well as isotropic vectors) and apply these concepts to random matrices with independent rows or columns. In the end you can formulate statements about the extreme singular values of such matrices, compare them and apply them to concrete examples as well as applications such as dimensionality reduction and Compressed Sensing.5 ECTSno date this semesterMA5346Time-Frequency AnalysisNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the analysis of signals in time and frequency as well as methods for their approximation using frames. In the end you will know Fourier- and wavelet-like transformations, principles of uncertainty, the basics of frame theory and Gabor frames as well as frame-based reconstruction methods and you will be able to apply these to concrete signal and image analysis tasks.9 ECTSno date this semesterMA5916Topics in Computational BiologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will gain an overview of methods in Computational Biology, such as gene sequence analysis, image analysis, statistical network modeling, and dynamic path modeling. You will learn to assess the strengths and limitations of the different procedures and to select appropriate computational approaches for concrete biological or biomedical questions.6 ECTSno date this semesterMA5607Topics in Dynamical SystemsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with a selected, advanced topic of the theory of dynamical systems (e.g., stochastic systems, multi-scale dynamics, numerical methods, special classes such as one-dimensional maps or infinite-dimensional systems). In the end you will be able to understand, apply, and assess the relevant concepts, methods, or numerical procedures for concrete systems.5 ECTSno date this semesterMA5300Topics in Mathematical Statistical MechanicsNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the mathematical foundations of statistical mechanics and acquire tools to treat thermodynamic limits, Gibbs states and large deviations. You will also examine phase transitions and symmetry breaking using concrete model examples such as the Ising and O(N) models.5 ECTSno date this semesterCIT413037Topics in the Theory of Markov ProcessesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with analytical aspects of Markov processes, in particular Itô diffusions (linear and nonlinear). By the end you will be able to understand Markov semigroups and generators, classify basic existence and uniqueness results for McKean-SDEs/McKean–Vlasov equations, and make statements about long-term behavior as well as phase transitions for nonlinear Fokker–Planck equations.3 ECTSno date this semesterMA5424TopologyNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou learn the fundamentals of set-theoretic topology and algebraic topology. In the first part you analyze topological and metric spaces as well as concepts such as continuity, compactness and connectedness; in the second part you deal with homotopies, fundamental groups and singular homology. At the end of the module you can assess topological properties as well as determine fundamental groups and homology of simple spaces.9 ECTSno date this semesterMA3241TUM Data Innovation LabNo ratings for this module yet.A1.3.1 Related to the Study ProgramIn the TUM Data Innovation Lab you work in your Master's program in small, interdisciplinary teams on real data-driven projects from science or industry. In the end you will be able to process, analyze and visualize data, implement numerical solutions, and present your results both technically and in an understandable way.10 ECTSno date this semesterMA8113Variational Analysis of Thin Elastic BodiesNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn the variational analysis of thin elastic bodies: from fundamentals of linear and nonlinear elasticity, through key inequalities and geometric rigidity, to dimension reduction. In the end you will be able to model variational problems for thin structures and apply the essential convergence and approximation results of the theory.5 ECTSno date this semesterCIT413029Variational Inequalities with Applications in Porous MediaNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou engage with variational inequalities (elliptic, first- and second-order) and their applications in porous media. In the end you will know existence, uniqueness and a priori estimates, master solution methods and be able to apply the procedures in associated software.9 ECTSno date this semesterMA5356Variational Principles for Collective Plasma MotionNo ratings for this module yet.A1.3.2 Additional to the Study ProgramYou will learn how models for collective motions in plasmas can be derived from variational principles. After the course you will be able to apply variational principles in Lagrangian and Eulerian descriptions and derive models such as MHD, Vlasov–Maxwell, drift-kinetics and hybrid variants, as well as understand the corresponding conserved quantities.3 ECTSno date this semesterCIT4130010
1 more module matches, but it is taught in German. Show itAxiomatische Mengentheorie und ihre logischen GrundlagenNo ratings for this module yet.A1.3.2 Additional to the Study ProgramDu lernst die Grundlagen der formalen Logik und der axiomatischen Mengentheorie (Zermelo-Fraenkel). Am Ende kannst du formale Sprachen und Beweise verstehen und Methoden der Modelltheorie anwenden sowie natürliche Zahlen, Ordinal- und Kardinalzahlen innerhalb des Mengenuniversums konstruieren und mit transfiniter Rekursion umgehen.3 ECTSno date this semesterMA5075