# Graduiertenkolleg

## Energie, Entropie und Dissipative Dynamik

# Joint Analysis Seminar

Das Joint Analysis Seminar findet dienstags von 10:30 - 11:30 Uhr im Hörsaal „1950|008 SeMath” (Pontdriesch 14/16) statt.

## Bevorstehende Vorträge

Zurzeit sind keine zukünftigen Termine bekannt.

17.07.2018, 10:30 Uhr

**Simon Foucart** (Texas A&M University):**Semidefinite programming in approximation theory: two examples**

10.07.2018, 10:30 Uhr

**Dario Trevisan** (Universität Pisa):**A PDE approach to random matching problems**

05.06.2018, 10:30 Uhr

**Paolo Bonicatto** (Universität Basel):**Untangling of trajectories for non-smooth vector fields and Bressan’s compactness conjecture**

Given d≥1, T>0 and a vector field **b**: [0,T] × ℝ^{d} → ℝ^{d},
we study the problem of uniqueness of weak solutions to the associated transport equation
∂_{t}u + **b**·∇u = 0 where u: [0,T] × ℝ^{d} → ℝ
is an unknown scalar function. In the classical setting, the method of characteristics is available
and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field
**b**. However, when we drop regularity assumptions on the velocity field, uniqueness is in
general lost.
In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian
representation. This tool allows to represent a suitable class of vector fields as superposition of
trajectories: we will then give local conditions to ensure that this representation induces a partition
of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a
family of 1-dimensional equations. We will finally show that if **b** is locally of class BV
in the space variable, the decomposition satisfies this local structural assumption: this yields in
particular the renormalization property for nearly incompressible BV vector fields and thus gives
a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.

24.04.2018, 10:30 Uhr

**Enno Lenzmann** (Universität Basel):**Energy-critical half-wave maps: solitons and Lax pair structure**

The eneryg-critical half-wave maps equation arises as a classical continuum limit of Calogero-Moser and Haldane-Shastry type spin systems in one space dimension. In this talk, I will discuss some essential features such as the complete classification of traveling solitary waves with finite energy, by using a close connection to minimal surfaces with free and non-free boundary conditions. Furthermore, I will present a recently found Lax pair structure and we explain its potential applications to the dynamics of the half-wave maps equation. Finally, I will mention some open problems. This talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

20.02.2018, 10:30 Uhr

**Filip Rindler** (University of Warwick):**Rademacher’s Theorem, Cheeger’s conjecture and PDEs for measures**

23.01.2018, 10:30 Uhr

**Herbert Koch** (University of Bonn):**The renormalized nonlinear wave equation in 2d with additive white noise**

By the classical uniformization theorem, every smooth Riemann surface is conformally diffeomorphic to a surface of constant curvature. What happens if the smooth Riemannian metric is replaced by a non-smooth distance? Does the so obtained metric surface still admit parametrizations with good geometric and analytic properties? Such questions have been widely studied in the field of Analysis on metric spaces and are important for example in view of applications to Geometric Group Theory. I will explain how one can use recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. In particular, we obtain a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Joint work with Alexander Lytchak.

19.12.2017, 10:30 Uhr

**Daniel Cremers** (TU München):**Convex relaxation methods for computer vision**

Numerous problems in computer vision and image analysis can be solved by optimization methods. Yet, many classical approaches correspond to non-convex energies giving rise to suboptimal solutions and often strong dependency on appropriate initialization. In my presentation, I will show how problems like stereo or multiple view reconstruction and 3D shape matching can be tackled by means of convex relaxation methods which allow to efficiently compute globally optimal or near-optimal solutions. The arising large-scale convex problems can be solved by means of provably convergent primal-dual algorithms. They are easily parallelized on GPUs providing high-quality solutions in acceptable runtimes.

21.11.2017, 10:30 Uhr

**Stefan Wenger** (Université de Fribourg):**Finding good parametrizations for metric surfaces**

11.07.2017, 10:30 Uhr

**Jean V. Bellissard** (Westfälische Wilhelms-Universität, Münster, retired from the Georgia Institute of Technology):**Special Joint Analysis Seminar (Mathematics & Physics):
A Toy Model for Viscosity**

A short review of the temperature behavior of liquids viscosity will be provided. The concept of anankeon and as a new degree of freedom will be described and its relation with elastic degrees of freedom discussed. Then a simplistic solvable model, based on a Stochastic Markov dynamics, will be proposed and the solution explained and discussed. One consequence is the prediction that in a certain subclass of liquids, there is a bifurcation leading to a new time scale, the Maxwell time, which is liable to explains the exponential increase of the viscosity near the liquid-solid transition. A comparison with numerical simulations using molecular dynamics will be discussed.

13.06.2017, 10:30 Uhr

**Helmut Abels** (Universität Regensburg):**Diffuse Interface Models for Two-Phase Flows of Incompressible Fluids and Their Sharp Interface Limits**

We discuss so-called "diffuse interface models" for the flow of two viscous incompressible Newtonian fluids in a bounded domain. Such models were introduced to describe the flow when singularities in the interface, which separates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. We discuss analytic results on well-posedness and qualitative behavior for some of these models and relation to classical sharp interface models, when the small parameter related to the thickness of the interface tends to zero.

23.05.2017, 10:30 Uhr

**Etienne Sandier** (Université Paris 12):**Lower bound for energy growth of locally minimizing solutions of -Δu=u(1-|u| ^{2}) for u: ℝ^{3}→ℝ^{2}**

09.05.2017, 10:30 Uhr

**Guido de Philippis** (SISSA, Trieste):**On the structure of measures satisfying a PDE constraint**

After a general introduction concerning the study of the interplay between PDE constraint and concentration/oscillation, I will present a general structure theorem for the singular part of Radon measure satisfying a PDE constraint. I will then present some applications.

02.05.2017, 10:30 Uhr

**David Gross** (Universität zu Köln):**Low rank matrix recovery, the Clifford group, and some quantum mechanics**

07.02.2017, 14:15 Uhr

**Giuseppe Savare** (University of Pavia):**Singular perturbation of gradient flows and rate-independent evolution problems**

We will present some new results concerning rate-independent limits of singularly perturbed gradient flows. A particularly important case arises when the total variation of the approximating curves is not uniformly bounded and one has to recover a limit by combining Kuratowski convergence of the graphs with a variational description of the energy dissipation. This approach leads to a new notion of dissipative solution to a family of stationary problems parametrized by the time variable. (In collaboration with Virginia Agostiniani and Riccarda Rossi)

06.12.2016, 14:15 Uhr

**John M. Ball** (University of Oxford):**The mathematics of liquid crystals**

15.11.2016, 14:15 Uhr

**Qiji J. Zhu** (Western Michigan University):**Variational methods in the presence of symmetry**

25.10.2016, 14:15 Uhr

**Gero Friesecke** (TU München):**Inferring atomic structure from X-ray diffraction patterns**

05.07.2016, 10:45 Uhr

**Andrea Mondino** (ETH Zürich):**Non-smooth spaces with Ricci curvature lower bounds**

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can a give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, etc.), permitting to give further insights in the theory. In the seminar I will give an overview of the topic. In the talk, after a brief introduction to the topic I will present some recent results.

07.06.2016, 10:45 Uhr

**Robert L. Jerrard** (University of Toronto):**Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii
equation**

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous 1858 work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this talk, we discuss a rigorous derivation of the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation, which describes an ideal quantum fluid. This is joint work with Didier Smets.

24.05.2016, 10:45 Uhr

**Eric A. Carlen** (Rutgers University):**The structure of near minimizers of a non-local free energy functional**

The Gates-Pentrose-Lebowitz free energy functional is a non-local analog of the phenomenological van der Waals free energy function. Unlike the local van der Waals functional, it arises in the continuum limit of lattice gas models. As in many statistical mechanical models in which free energy functionals are large deviations functional, one is interested not only in the absolute minimizers, but also the near minimizers. In classical work on the near minimizers of the van der Waals functional, the co-area formula and the quantitative isoperimetric inequality play a crucial role. We present joint work with Maggi on a non-local quantitative isoperimetric inequality, and discuss its application to the description of near minimizers of the Gates-Pentrose-Lebowitz free energy functional in work with Carvalho, Esposito, Marra and Lebowitz.

26.04.2016, 10:45 Uhr

**Dorin Bucur** (Université de Savoie):**Optimal shapes and isoperimetric inequalities for spectral functionals**

In this talk I will discuss isoperimetric inequalities involving the spectrum of the Laplace operator (of Faber-Krahn or Saint-Venant type) seen from the perspective of shape optimization. I will focus on problems involving the spectrum of the Robin-Laplacian and the Steklov problem, and show techniques developed around the Mumford-Shah functional in image segmentation theory can be used to prove such inequalities.

26.01.2016, 14:15 Uhr

**Alexander Lytchak** (University of Cologne):**Classical Plateau Problem in metric spaces**

The question of Plateau concerns the existences soap films: objects of minimial area spanning a given curve in the Euclidean spaces. The most classical answer to this question has been provided by Rado and Douglas who have proved the existence of parametrized discs of minimal area spanning an arbitrary Jordan curve. This result was generalized by Morrey to Riemannian manifolds. In the talk I will discuss a solution of the Plateau problem in arbitrary metric spaces, regularity of solutions and some applications to isoperimetric problems.

15.12.2015, 14:15 Uhr

**Laure Saint-Raymond** (Pierre et Marie Curie University (Paris VI)):**The Stokes-Fourier equations as scaling limits of interacting system
of particles**

In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium. This control of the collision process can be obtained in fluctuation regimes. In a recent work with Thierry Bodineau and Isabelle Gallagher, we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.

24.11.2015, 14:15 Uhr

**Tim Sullivan** (University of Warwick):**Brittleness and robustness of Bayesian inference**

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors and others) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint work with Houman Owhadi and Clint Scovel (Caltech).

10.11.2015, 14:15 Uhr

**Eduard Feireisl** (Academy of Sciences of the Czech Republic, Prague):**On solvability of certain problems in fluid mechanics involving inviscid
fluids**

We present a general method of "construction" of solutions to systems of partial differential equations describing the motion of inviscid fluids. The method is based on a variable coefficients version of the oscillatory lemma proved in the context of the incompressible Euler system by C. De Lellis and L. Székelyhidi. Several specific examples of fluid systems including the Korteweg fluid models, quantum fluids, the Euler-Fourier system, and the Savage-Hutter model will be presented. We also discuss suitable admissibility criteria to ensure well-posedness of these problems.

02.06.2015, 14:00 Uhr

**Benjamin Schlein** (University of Zurich):**Hartree-Fock Dynamics for Weakly Interacting Fermions**

Fermions are quantum particles described by wave functions that are antisymmetric with respect to permutations. According to first principle quantum mechanics, the evolution of fermonic systems is described by the many body Schroedinger equation. We are interested, in particular, in the mean field regime, which is characterized by a large number of weak collisions among the particles. For fermionic systems, the mean field regime is naturally linked to a semiclassical limit. Asymptotically, the many body evolution can be described by the classical Vlasov equation. A better approximation, however, is given by the Hartree-Fock (or the Hartree) equation. In this talk, we will show precise bounds on the convergence of the Schroedinger dynamics towards the Hartree-Fock dynamics, for initial data close to Slater determinants with the appropriate semiclassical structure.

12.05.2015, 14:00 Uhr

**Karlheinz Gröchenig** (University of Vienna):**Deformation of Gabor Systems**

We prove a very general deformation result for Gabor frames and Gabor Riesz sequences. The new notion of deformation is non-linear and includes in particular the standard jitter error and linear deformations of phase space. As a consequence the main result covers the known perturbation and deformation results for Gabor frames. As part of the theory we derive new characterizations of Gabor frames and Gabor Riesz sequences with windows in the Feichtinger algebra. These are in the style of Beurling's characterization of sets of sampling for bandlimited functions and extend significantly the characterization of Gabor frames &ldqo;without inequalities&rdqo; from lattices to non-uniform sets.

05.05.2015, 14:00 Uhr

**Aldo Pratelli** (University of Erlangen-Nürnberg):**On the Approximation of Sobolev Homeomorphisms**

The problem of approximating a Sobolev homeomorphism with diffeomorphisms or piecewise affine homeomorphisms is studied since the early 1980's, when it was proposed by Ball and Evans in the framework of the nonlinear elasticity. Despite the simplicity of the question, the problem is hard, because the standard approximation methods fail to preserve the injectivity of the maps. We will describe the motivation and the history of this problem, and then the positive (still partial) solutions of the last years. We will also discuss the open problems (joint works with Mora-Corral, Daneri, Hencl, Radici).

21.04.2015, 14:00 Uhr

**Nicola Gigli** (University Pierre et Marie Curie (Paris VI)):**Nonsmooth Differential Geometry**

In this talk I will discuss in which sense general metric measure spaces possess a first order differential structure and how on spaces with Ricci curvature bounded from below a second order one emerges. In this latter framework, the notions of Hessian, covariant/exterior derivative and Ricci curvature are all well defined.

20.01.2015, 16:00 Uhr

**Laurent Demanet** (MIT):**1930s analysis for 2010s signal processing: recent progress on the super-resolution question**

06.01.2015, 16:00 Uhr

**Felix Otto** (MPI for Mathematics in the Sciences):**A regularity theory for elliptic equations with random coefficients**

18.11.2014, 16:00 Uhr

**Martin Rumpf** (University of Bonn):**Time Discrete Geodesics on Image Manifolds**

28.10.2014, 16:00 Uhr

**Camillo DeLellis** (University of Zurich):**Dissipative continuous solutions of the incompressible Euler equations with Onsager-critical spatial regularity**

13.10.2014, 16:00 Uhr

**Martin Hairer** (Warwick University):**Weak universality of the KPZ equation**

15.07.2014, 16:00 Uhr

**Zdzislaw Brzezniak** (University of York):**Strong and weak solutions to stochastic Landau-Lifshitz equations**

I will speak about the of weak (and the existence and uniqueness
of strong solutions) to the stochastic Landau-Lifshitz equations for
multi (one)-dimensional spatial domains with. I will also describe
the corresponding Large Deviations principle and it's applications
to magnetization switching in a ferromagnetic wire.

The talk is based on a joint works with B. Goldys, Liang Li and T. Jegaraj.

01.07.2014, 16:00 Uhr

**Felix Schulze** (University College London):**Networks of curves evolving by curvature in the plane**

The network flow is the evolution of a regular network of embedded curves under curve shortening flow in the plane, where it is allowed that at triple points three curves meet under a 120 degree condition. A network is called non-regular if at multiple points more than three embedded curves can meet, without any angle condition but with distinct unit tangents. Studying the singularity formation under the flow of regular networks one expects that at the first singular time a non-regular network forms. In this talk we will present recent work together with Tom Ilmanen and Andre Neves, showing that starting from any non-regular initial network there exists a flow of regular networks.

17.06.2014, 16:00 Uhr

**Massimo Fornasier** (TU München):**Consistency of probability measure quantization by means of power repulsion-attraction potentials**

In this talk we present the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually Gamma-converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient flows. To model situations where the given probability is affected by noise, we additionally consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of Gamma-convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.

06.05.2014, 16:00 Uhr

**Mauro Maggioni** (Duke University):**Multiscale geometric methods for statistical learning and data in high-dimensions**

We discuss a family of ideas, algorithms, and results for analyzing various new and classical problems in the analysis of high-dimensional data sets. These methods rely on the idea of performing suitable multiscale geometric decompositions of the data, and exploiting such a decomposition to perform a variety of tasks in signal processing and statistical learning. In particular, we discuss the problem of dictionary learning, where one is interested in constructing, given a training set of signals, a set of vectors (dictionary) such that the signals admit a sparse representation in terms of the dictionary vectors. We discuss a multiscale geometric construction of such dictionaries, its computational cost and online versions, and finite sample guarantees on its quality. We then generalize part of this construction to other tasks, such as learning an estimator for the probability measure generating the data, again with fast algorithms with finite sample guarantees, and for learning certain types of stochastic dynamical system in high-dimensions.

04.02.2014, 16:45 Uhr

**Alexander Mielke** (Weierstrass Institute for Applied Analysis and Stochastics (WIAS)):**On gradient structures for reaction-diffusion systems**

04.02.2014, 15:00 Uhr

**Barbara Niethammer** (University of Bonn):**Phase transitions in a nonlocal Fokker-Planck equation with multiple time scales**

07.01.2014, 15:00 Uhr

**Thierry Gallay** (University of Grenoble):**Distribution of energy and convergence to equilibria in extended dissipative systems**

We study the local energy dissipation in gradient-like nonlinear partial differential equations on unbounded domains. Our basic assumption, which happens to be satisfied in many classical examples, is a pointwise upper bound on the energy flux in terms of the energy dissipation rate. Under this hypothesis, we derive a simple and general bound on the integrated energy flux which implies that, in low space dimensions, our ``extended dissipative system'' has a gradient-like dynamics in a suitable averaged sense. In particular, we can estimate the time spent by any trajectory outside a neighborhood of the set of equilibria. As an application, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equation in an infinite cylinder. This talk is based on a collaboration with S. Slijepcevic (Zagreb, Croatia).

10.12.2013, 15:00 Uhr

**Didier Bresch** (University of Savoie):**Mathematical topics around shallow-water type equations**

16.07.2013, 15:45 Uhr

**Lia Bronsard** (McMaster University):**Vortices for a 2 component Ginzburg Landau system**

We study vortices in a Ginzburg-Landau model for a pair of complex-valued order parameters. Multi-component functionals have been introduced in the context of unconventional p-wave superconductors and spinor Bose-Einstein condensates to include spin coupling effects. As in the classical Ginzburg-Landau model, minimizers will exhibit quantized vortices in response to boundary conditions or applied fields. However, we show that the interaction between the two components allows for vortices with a more exotic core structure. Our results are based on a combination of variational and PDE methods, blowing up around the vortex core and studying the resulting system and its local minimizers. This is joint work with Stan Alama and Petru Mironescu.

18.06.2013, 15:45 Uhr

**Michael Struwe** (ETH):**The gradient flow for a supercritical elliptic variational problem**

04.06.2013, 15:45 Uhr

**Sergio Conti** (Universität Bonn):**Derivation of a line-tension model for dislocations in the plane**

Dislocations are topological defects in crystals which generate long-range elastic stresses. For dislocations in the plane the elastic interactions can be represented via a singular kernel behaving as the H^{1/2} norm of the slip. We obtain a sharp-interface limit within the framework of Gamma convergence in the limit of small elastic spacing. One key ingredient is a proof of the fact that the presence of infinitely many equivalent length scales gives strong restrictions on the geometry of the microstructure. In particular we show that the micrustructure must be one-dimensional on most length scales, and that only few are available for the relaxation. This talk is based on joint work with Adriana Garroni, Annalisa Massaccesi and Stefan Müller.

28.05.2013, 15:45 Uhr

**Mark Peletier** (University of Eindhoven):**Energy-driven pattern formation via competing long- and short-range interactions**

I will discuss patterns in block copolymer melts. This is a model system that is mathematically tractable, physically meaningful (and experimentally accessible) and representative for a large class of energy-driven pattern-forming systems. Such systems show a remarkable variety of different patterns, of which only a small fraction is well understood.

In this talk I will focus on a variational model for this system, in a parameter regime in which the system forms regular patterns of small spheroid blobs, called particles. The energy for these structures is dominated by a single-particle term, which penalizes each particle independently. This term drives the system towards particles of a well-defined size. At the next level the interaction between the particles is given by a Coulomb interaction potential, giving rise to approximately periodic arrangements.

07.05.2013, 15:45 Uhr

**Giuseppe Buttazzo** (University of Pisa):**Optimal potentials for Schrödinger operators**

We consider the Schrödinger operator −Δ+V(x) on H^{1}_{0}(O), where O is a given domain of ℝ^{d}. Our goal is to study some optimization problems where an optimal potential V≥0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

# Oberseminar für spezielle Probleme der Analysis

Das Oberseminar findet in Raum 1010|114 (Hauptgebäude) statt.

## Bevorstehende Vorträge

Zurzeit sind keine zukünftigen Termine bekannt.

## Vergangene Vorträge

In den vergangenen sechs Monaten haben keine Veranstaltungen stattgefunden.

# Ferien-Seminar im Kloster Steinfeld

Das Ferienseminar findet in der Regel Ende August in Kloster Steinfeld in der Eifel statt.

Von Montag bis Freitag werden Vorträge über interessante Themen der Analysis gehalten.

Die Auswahl der Themen und die Verteilung der Vorträge richtet sich nach dem Vorwissen der Teilnehmer.

Mindestvoraussetzung ist der bestandene Zyklus Analysis I - III.

Bitte besuchen Sie die die Seite des Seminars für weiterführende Informationen.