Modules

153 results

Elective Modules153

Computational Plasma PhysicsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA4304Differential GeometryNo ratings for this module yet.Mathematical ModulesYou learn the basics of smooth manifolds, bundles (tangent, vector, and principal bundles) as well as Riemannian metrics, curvature tensors and geodesics. In addition, symmetric spaces and Lie groups are treated. In the end you can transfer metric properties of Euclidean space to 'curved' spaces and derive properties of such spaces using analytical methods.9 ECTSruns this semesterMA3205Discrete Differential GeometryNo ratings for this module yet.Mathematical ModulesYou learn the fundamentals of the differential geometry of curves and surfaces in a discrete framework. The focus is on structurally analogous concepts (e.g. discrete tangents, normals, curvature) and not on numerical approximation. In the end you will be able to treat discrete analogues of smooth curves and surfaces as well as their invariants in a structural and analytical way.6 ECTSruns this semesterMA5205Discrete OptimizationNo ratings for this module yet.Mathematical ModulesYou will learn central concepts and algorithms of linear integer optimization and combinatorial optimization. In the module you will analyze mathematical structures that allow efficient solution methods, and you will be able to model real problems as discrete optimization problems and identify special cases that are efficiently solvable.9 ECTSruns this semesterCIT413041Fundamentals of Mathematical StatisticsNo ratings for this module yet.Mathematical ModulesYou 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
148 more in Elective ModulesFunktionalanalysisNo ratings for this module yet.Mathematical ModulesYou will learn the basics of functional analysis in Banach and Hilbert spaces. In the end you will be able to analyze linear functionals and bounded as well as compact self-adjoint operators, understand duality, and apply concepts such as weak and weak* convergence.9 ECTSruns this semesterMA3001Large DeviationsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5417Mathematische KontinuumsmechanikNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5019Mathematische Methoden zur Unsicherheitsquantifizierung in der HydrologieNo ratings for this module yet.Mathematical ModulesYou will learn methods for quantifying uncertainties in hydrological questions and apply them. In the end you will be able to select suitable UQ methods and apply them to simple to complex hydrological models as well as present the results interdisciplinary.9 ECTSruns this semesterBGU54027Multilinear AlgebraNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT413063Nichtlineare OptimierungNo ratings for this module yet.Mathematical ModulesYou engage with theory and numerical methods of nonlinear optimization. You will learn advanced methods for unconstrained and, in particular, constrained optimization problems (e.g., SQP, barrier and interior-point methods) and you will be able to assess and apply convergence properties of such methods in the end.5 ECTSruns this semesterMA3503Nonconvex Global OptimizationNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT4130019Numerik der DifferentialgleichungenNo ratings for this module yet.Mathematical ModulesYou will learn numerical methods for solving ordinary and simple partial differential equations. Topics include stiff and non-stiff initial value problems, fundamentals of boundary value problems, and methods such as finite differences, finite elements, and spectral methods (focus on 1D, elliptic problems partly in 2D). By the end you will be able to understand, assess and apply basic algorithms on the computer as well as estimate discretization errors.9 ECTSruns this semesterMA3301Probabilistische Techniken und Algorithmen in der DatenanalyseNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA4803Probability TheoryNo ratings for this module yet.Mathematical ModulesYou will learn the measure-theoretic probability theory for sequences of random variables and martingales. By the end you will be able to understand and apply central results such as the law of large numbers, central limit theorems, and fundamental martingale results.9 ECTSruns this semesterMA2409Projektive Geometrie 1No ratings for this module yet.Mathematical ModulesYou 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 semesterMA3203Statistical LearningNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA4802Advanced Finite ElementsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesIn 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 Uncertainty QuantificationNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT4130021An Introduction to the Regularity Theory of Elliptic Partial Differential EquationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5228Approximation AlgorithmsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5360Calculus of VariationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterCIT413058Case Studies Life Science MathematicsNo ratings for this module yet.Mathematical ModulesYou work in groups on concrete problems from the Life Sciences and apply methods of biomathematics and biostatistics (e.g. dynamic systems, stochastic processes). In the end you will be able to mathematically formulate a real biological problem, develop a suitable model, solve it numerically, and present the results professionally.7 ECTSno date this semesterMA5616Case Studies Life Science MathematicsNo ratings for this module yet.Mathematical ModulesYou work in small teams on concrete problems from the Life Sciences. You formulate questions mathematically, develop models (deterministic or stochastic), solve and implement suitable methods, evaluate the results and present them scientifically and for a general audience.10 ECTSno date this semesterCIT413052Case Studies OptimizationNo ratings for this module yet.Mathematical ModulesYou work in small teams on concrete optimization tasks from discrete and non-linear optimization. You model real problems, select and implement suitable solution procedures with modern optimization tools, and present and assess the results both for a disciplinary audience and for a non-scientific audience.10 ECTSno date this semesterCIT413042CombinatoricsNo ratings for this module yet.Mathematical ModulesYou will learn central methods and structures of combinatorics and practice transferring them to concrete problems — with a special focus on applications in cryptography. In the end you will be able to formulate combinatorial models, solve structured counting problems, and present combinatorial arguments clearly.9 ECTSno date this semesterCIT413068Complex Function Theory 2No ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Convexity - Optimal ContainmentNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesIn 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.Mathematical ModulesYou 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 StatisticsNo ratings for this module yet.Mathematical ModulesYou will learn methods of computational statistics for high-dimensional, hierarchical, and latent data structures and how to apply them. The focus is on simulation (univariate and multivariate), Bayesian inference with MCMC (Gibbs, Metropolis-Hastings, Hamiltonian MC), bootstrap procedures and the EM algorithm for missing or latent data. In the end you can theoretically understand the algorithms, implement them in R, and interpret results statistically.5 ECTSno date this semesterMA4402Computer Vision I: Variational MethodsNo ratings for this module yet.Mathematical ModulesYou learn how many tasks of image processing (e.g. denoising, desmearing, segmentation, optical flow, stereo depth estimation, 3D reconstruction) are formulated and solved as variational problems. In the end you know the Euler–Lagrange approach and PDEs, efficient solution methods as well as convex formulations and relaxations and you can implement central concepts in Matlab.8 ECTSno date this semesterIN2246Concentration of MeasureNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT4100004Convex Duality and Applications in Mass Transport and Calculus of VariationsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5910Delay Differential Equations with ApplicationsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5062Differential FormsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterCIT413033Diffusion ProcessesNo ratings for this module yet.Mathematical ModulesYou will learn the mathematical theory of diffusion processes: Gaussian processes, stochastic integrals (Itô, Stratonovich, Klimontovich) and their transformation, Langevin equations to describe diffusions, and the analysis of diffusion operators using semigroup theory. You will also receive an introduction to log-Sobolev inequalities and to stochastic partial differential equations, so that you can formally analyze the models and treat them with semigroup-theoretic methods.9 ECTSno date this semesterCIT413069Direct Methods in the Calculus of VariationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Geometrie: GitterpolytopeNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5215Dynamische SystemeNo ratings for this module yet.Mathematical ModulesYou learn the theory of dynamical systems for ordinary differential equations and mappings in finite-dimensional phase spaces. The module provides tools for analyzing stable and unstable solutions, bifurcations, chaos and ergodic properties; at the end you can analyze geometric and topological properties of trajectories and attractors and transfer the methods to differentiable manifolds and infinite-dimensional spaces.9 ECTSno date this semesterMA3081Elements of the Theory of DistributionsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5319Fine Properties of Sobolev FunctionsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesIn 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.Mathematical ModulesIn 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA4064Functions of Bounded Variations and ApplicationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterCIT413031Generalized Linear ModelsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Methods for Physics of Magnetized PlasmasNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5059Graph TheoryNo ratings for this module yet.Mathematical ModulesYou learn the fundamentals of graph theory: paths and cycles, connectivity, trees, matchings, k-connectivity and Menger's theorem, as well as planar graphs and colorings. In the end you can apply central definitions and theorems, carry out proofs, and transfer the concepts presented to concrete problems and simple applications (including from data analysis).6 ECTSno date this semesterCIT413051Graphical Models in StatisticsNo ratings for this module yet.Mathematical ModulesIn 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.Mathematical ModulesYou 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 semesterMA5954High-Dimensional Partial Differential EquationsNo ratings for this module yet.Mathematical ModulesYou learn how partial differential equations appear in very high dimensions, what theoretical peculiarities and numerical difficulties (e.g., Curse of Dimensionality) arise, and which modern solution approaches exist. In the end you will be able to analyze theoretical properties of solutions and implement and apply various numerical methods (sparse grids, tensor decompositions, neural networks) to applications.5 ECTSno date this semesterCIT413060High-dimensional StatisticsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5921Identification of Artificial Neural Networks: from the Analysis of one Neuron to Deep Neural NetworksNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5929Introduction to Conservation LawsNo ratings for this module yet.Mathematical ModulesYou 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 Regularization and Learning Methods for Inverse ProblemsNo ratings for this module yet.Mathematical ModulesYou will learn the mathematical foundations of inverse problems and regularization as well as modern data-driven solution approaches. In the end you will be able to analyze inverse problems, apply classical regularization methods and classify data-based reconstruction methods.5 ECTSno date this semesterCIT413070Introduction to Stochastic Differential Equations: Theory and NumericsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterCIT413066Isogeometric Analysis: Theory and PracticeNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5938Konvexe Optimierung für Computer VisionNo ratings for this module yet.Mathematical ModulesYou will learn the fundamentals of convex analysis and their application to optimization problems in image processing and computer vision. After the module you will be able to understand, apply and implement common first-order and proximal methods for typical CV tasks (e.g., image reconstruction, segmentation, matrix factorization).6 ECTSno date this semesterIN2330Lattice Boltzmann methodsNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT4130009Low Rank ApproximationNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA4408Maschinelles Lernen für ComputersehenNo ratings for this module yet.Mathematical ModulesYou will learn fundamental machine learning methods that are frequently used in computer vision (e.g., object classification, segmentation, denoising, camera calibration). In the end you will be able to explain the mathematical formulation of central procedures, create simple implementations, and apply them to concrete datasets from the field of computer vision.5 ECTSno date this semesterIN2357Mathematical Data AnalysisNo ratings for this module yet.Mathematical ModulesYou will learn methods for analyzing complex and unstructured data, including regression and classification tasks. The module provides fundamentals such as loss functions, positive definite kernels and reproducing kernel Hilbert spaces as well as regularization strategies (e.g. Tikhonov) and SVM regression. In addition, summation procedures and manifold learning are covered.6 ECTSno date this semesterMA5098Mathematical Foundations of ImagingNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Foundations of Machine LearningNo ratings for this module yet.Mathematical ModulesYou will learn the mathematical foundations of modern methods of machine learning. By the end you can design neural networks and discuss their approximation properties, apply kernel methods in Reproducing Kernel Hilbert Spaces, and assess the statistical efficiency of learning procedures.9 ECTSno date this semesterCIT413048Mathematical Introduction to Quantum Information ProcessingNo ratings for this module yet.Mathematical ModulesYou 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 Modules from other UniversitiesNo ratings for this module yet.Mathematical ModulesYou take mathematics modules that are offered at other universities. In the end you have content and achievements from external mathematics courses recognised and you can have them credited to your study plan.no date this semesterMA8300Mathematische Einführung in die MagnetohydrodynamikNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesThe 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.Mathematical ModulesYou 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 semesterMA5940Methods for Inverse ProblemsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesIn 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5952Modern Methods in Nonlinear OptimizationNo ratings for this module yet.Mathematical ModulesYou will learn selected modern methods of nonlinear optimization, e.g. convex and non-smooth optimization, interior-point methods, semidefinite programming, robustness concepts and duality. In the end you will be able to understand current research articles on the treated topics and you will be prepared to pursue your own research questions in nonlinear optimization.5 ECTSno date this semesterMA4503Moderne Methoden der Nichtlinearen Optimierung (2)No ratings for this module yet.Mathematical ModulesYou 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 semesterMA4505Nichtlineare Dynamik: GrundlagenNo ratings for this module yet.Mathematical ModulesYou will learn the fundamentals of the theory of finite-dimensional continuous and discrete dynamical systems: phase space analysis, stability, bifurcations and initial insights into chaotic dynamics. In the end you will be able to apply these concepts to concrete examples and apply the most important analytical methods.5 ECTSno date this semesterMA3080Nonlinear AnalysisNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Partial Differential EquationsNo ratings for this module yet.Mathematical ModulesYou will learn numerical solution methods for partial differential equations, in particular finite element methods for multi-dimensional elliptic boundary value problems. You will also learn error estimates, adaptive mesh refinement, fast solvers and an introduction to numerical methods for time-dependent problems. In the end you will be able to understand the methods, apply them and use the associated software.9 ECTSno date this semesterMA3303Numerical Methods for Uncertainty QuantificationNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5348Numerics of Dynamical SystemsNo ratings for this module yet.Mathematical ModulesYou deal with numerical methods for dynamic systems: propagation and detection of bifurcations, computation and analysis of invariant sets (e.g. periodic orbits, invariant manifolds, attractors) as well as computational ergodic concepts such as Lyapunov exponents and invariant measures. In the end you will be able to apply these methods, implement them and evaluate their properties.5 ECTSno date this semesterMA3333Operator TheoryNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5012Optimal TransportNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesIn 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5918Particle in Cell Methods for the Vlasov-Maxwell EquationsNo ratings for this module yet.Mathematical ModulesIn this module you will learn numerical Particle-in-Cell methods for the Vlasov–Maxwell equations. You will understand both deterministic and stochastic particle methods, structure-preserving discretizations with the discrete de-Rham complex, and you will be able to formulate and implement in Python geometric algorithms for Vlasov–Maxwell problems.5 ECTSno date this semesterCIT413049Partielle DifferentialgleichungenNo ratings for this module yet.Mathematical ModulesIn this module you will learn fundamental methods for the treatment of partial differential equations (PDEs). You will deal with classical representations of solutions for transport, Laplace, heat and wave equations, with conservation laws, Sobolev spaces, and with weak (variational) solutions of elliptic equations and their properties such as existence, uniqueness and regularity.9 ECTSno date this semesterMA3005PDE2 - Nonlinear Partial Differential EquationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5946Perfect Simulation for Chains of Finite and Infinite OrderNo ratings for this module yet.Mathematical ModulesYou will learn about models of stochastic processes beyond finite Markov chains, in particular chains of infinite order, their properties and how to simulate them exactly (perfect simulation). In the end you will be able to assess existence and uniqueness questions, classify examples and apply perfect-simulation algorithms for such processes.5 ECTSno date this semesterCIT413043Polyhedral CombinatoricsNo ratings for this module yet.Mathematical ModulesYou learn how to approach combinatorial optimization problems through the geometry of polyhedra: representation of polytopes, the connection between geometry and optimization of linear functions, as well as modern algorithms such as branch-and-cut and separation/optimization. In the end you will be able to apply the methods to typical problems (e.g., matching, TSP polytopes) and assess their limits in the context of NP-hardness.6 ECTSno date this semesterMA5225Probability on GraphsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA4406Quantum Dynamics 2No ratings for this module yet.Mathematical ModulesYou 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 Statistical InferenceNo ratings for this module yet.Mathematical ModulesYou will learn mathematical methods for hypothesis testing, parameter estimation, tomography, learning and predictive inference in quantum-statistical experiments. In the end you will be able to formally analyze, describe and design such experiments as well as apply suitable, advanced tools for inference about underlying quantum systems.5 ECTSno date this semesterCIT513009Quantum Tradeoff RelationsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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 semesterMA5306Resurgence in Geometry and PhysicsNo ratings for this module yet.Mathematical ModulesThe module covers methods for the analysis of divergent asymptotic series, their Borel sums, and the associated Stokes data. You will learn techniques of resurgence, Borel summability and applications in exact WKB analysis, cluster algebras, as well as in various areas of geometry and theoretical physics.6 ECTSno date this semesterCIT413050Scheduling: Theory and AlgorithmsNo ratings for this module yet.Mathematical ModulesYou 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 semesterCIT413053Scientific Computing in Circuit SimulationNo ratings for this module yet.Mathematical ModulesYou learn mathematical modeling and numerical methods for the simulation of electronic circuits. The module covers DAEs, sparse handling of matrices and algorithms for parallel computing, so that you understand and can apply basic modeling and solution strategies for industrial circuit simulations at the end.5 ECTSno date this semesterIN2306Selected Chapters from the Mathematical Continuum MechanicsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Inference for Dynamical SystemsNo ratings for this module yet.Mathematical ModulesIn 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.Mathematical ModulesYou 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 AnalysisNo ratings for this module yet.Mathematical ModulesYou will learn the theory and fundamental applications of stochastic analysis. The focus is on Brownian motion (construction and properties), stochastic integrals and the Itô formula, as well as stochastic differential equations and methods such as Girsanov transformation and Donsker's invariance principle. In the end you will be able to formulate central statements and perform simple calculations with Itô integrals and SDEs.9 ECTSno date this semesterMA4405Structure Preserving Discretisation on Staggered GridsNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5936Structure-preserving Finite Elements for Computational ElectromagnetismNo ratings for this module yet.Mathematical ModulesIn this module you will learn finite-element methods that preserve the underlying geometric and functional structure of electromagnetic problems. By the end you will be able to analyse the well-posedness of typical Maxwell model problems and apply stable, structure-preserving FEM approximations on geometrically non-trivial domains.5 ECTSno date this semesterCIT413044Tensor Network MethodsNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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 Dynamical SystemsNo ratings for this module yet.Mathematical ModulesYou 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 Dynamical SystemsNo ratings for this module yet.Mathematical ModulesYou deal with a rotating advanced topic in dynamical systems. At the end you will have in-depth knowledge of the fundamentals, applications, or numerical methods of the respective topic and you can analyze a given system with appropriate concepts or algorithms as well as draw qualitative and quantitative conclusions.5 ECTSno date this semesterCIT415300Topics in Dynamical Systems: Computational DynamicsNo ratings for this module yet.Mathematical ModulesYou study the numerical computation of objects and phenomena of dynamical systems (e.g. fixed/periodic points, invariant manifolds, limit and recurrent sets), with operators such as transfer and Koopman operators and with scalar metrics (e.g. Lyapunov exponents, entropy, dimensions). In the end you can develop, analyze and apply suitable algorithms and make qualitative and quantitative statements about the behavior of given dynamical systems.5 ECTSno date this semesterCIT415303Topics in Dynamical Systems: Network DynamicsNo ratings for this module yet.Mathematical ModulesThe module covers dynamic processes on and from networks. You will learn the basics, models (e.g., Kuramoto-type), random graphs, stochastic descriptions and applications such as epidemics or neural networks as well as related analytical methods. By the end you will be able to analyze given network dynamics with appropriate concepts or algorithms and make qualitative and quantitative statements about the dynamics.5 ECTSno date this semesterCIT415302Topics in Dynamical Systems: Stochastic DynamicsNo ratings for this module yet.Mathematical ModulesYou deal with stochastic or random dynamical systems. You will learn fundamentals, important methods (e.g. Fokker–Planck approach, large deviations, moment methods) as well as phenomena such as noise-induced behavior, synchronization by noise and stochastic bifurcations; at the end you will be able to analyze a given system with appropriate concepts/algorithms and make qualitative as well as quantitative statements about the dynamics.5 ECTSno date this semesterCIT415301Topics in the Theory of Markov ProcessesNo ratings for this module yet.Mathematical ModulesYou 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 semesterMA5424TUM Data Innovation LabNo ratings for this module yet.Mathematical ModulesIn 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 the Ginzburg-Landau Functional: An IntroductionNo ratings for this module yet.Mathematical ModulesYou will learn the variational analytic foundations of Ginzburg–Landau energies. The module introduces Gamma-convergence, relaxation and the treatment of concentration phenomena on objects with co-dimension 1 and 2. By the end you can apply the fundamental concepts and follow the main results in proof lines.5 ECTSno date this semesterCIT413040Variational Analysis of Thin Elastic BodiesNo ratings for this module yet.Mathematical ModulesYou 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.Mathematical ModulesYou 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.Mathematical ModulesYou 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