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Mean curvature flow without singularitiesRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Mean curvature flow without singularities is a variant of mean curvature flow, where we only consider smooth solutions. Nevertheless, it is possible to handle singularities. We will present this concept and illustrate it with lots of pictures. Then we will focus on some aspects of that flow like weak interpretations, uniqueness, oszillatory behaviour, variants with Dirichlet boundary conditions or obstacles, asymptotics in space and time, translating solutions and fully nonlinear variants as time permits.

Propagation of gradient flow structures from microscopic to macroscopic modelsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In this talk we will discuss the propagation of gradient flow structures from microscopic models in statistical mechanics such as overdamped particle dynamics or interacting particle systems on lattices to macroscopic partial differential equations. The key insight is that microscopic models can be formulated as linear Markov chains in high-dimensional spaces, e.g. via Liouville equations, for which recent work by Maas, Mielke and others has provided a rather complete picture. The propagation to macroscopic models is then carried out - at least formally - by constructing a metric structure on an associated infinite hierarchy of equations, resembling the BBGKY hierarchy in kinetic theory, and studying mean-field or other limits in this setup.

Riemannian Optimization for Constrained Low Rank Matrix ProblemsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

Low rank matrix factorizations have found many applications in recent years. Often, one is interested in minimizing a function over the manifold of low rank matrices in order to find a data sparse or well structured solution. This can be done using Riemannian optimization, which is the generalization of gradient related algorithms to smooth Riemannian manifolds. In theory, additional constraints on the components, like sparsity or nonnegativity, can be formulated. However, this significantly complicates the procedure and rigorous results are sparse. The talk will give an introduction into Riemannian optimization for constrained low rank problems and illustrate some of the difficulties.

The effect of different velocities on global existence and decay in reaction-diffusion-advection systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of small initial data and even finite time blow-up. In this talk I will show that, if in nonlinearly coupled reaction-diffusion systems components exhibit different group velocities, then quadratic or cubic mix-terms are harmless. Using a careful spatio-temporal analysis one can establish global existence and diffusive Gaussian-like decay for exponentially and polynomially localized initial data. Our approach can be related to the space-time resonances approach as developed by Germain, Masmoudi and Shatah in the dispersive setting. If nonlinear couplings which are not of mix-type are present in the system, the analysis breaks down as the spatial localization imposed by the Gaussian ansatz is too restrictive. For instance, the question whether a `seemingly harmless' Burgers-type nonlinearity can be included in the system is rather subtle.

Approxmation of the Kardar-Parisi-Zhang equation in Weak Disorder in d ≥ 3 (joint work with Francis Comets and Chiranjib Mukherjee)2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

Trying to give a proper definition of the KPZ (Kardar-Parisi-Zhang) equation in dimension d ≥ 3 is a challenging question. A plan to do so, is to first consider the well-defined KPZ equation when the white noise is smoothed in space. Then, the solution of the smoothed equation can be expressed as the free energy of a continuous polymer. Using this representation for d ≥ 3 and small noise intensity, it is possible to study the asymptotic behavior of the solutions in the limit where the smoothing is removed. In particular, we can prove that the stationary solution is a good approximation of the solutions for a wide range of initial conditions. We can further give quantitative estimates of this approximation when the initial profile is flat, which will involve the Gaussian free field in R^d.

Sample Path Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian MotionMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

We study the effect of additive fractional noise with Hurst parameter H > 12 on fast-slow systems. Our strategy is based on sample paths estimates, similar to the sample-paths approach by Berglund and Gentz in the Brownian motion case. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, taylored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood.

Workshop on Replication in Life Insurance/ In cooperation with ERGO GroupBC1 2.02.01 (Parkring 11, 85748 Garching)

Dear colleagues, We are pleased to announce the workshop on “Replication in Life Insurance” hosted by the ERGO Center of Excellence in Insurance at the Chair of Mathematical Finance, Technical University of Munich that will be held on May 10, 2019 in Munich. Our objective is to bring together a small number of practitioners and academics working in the field of life insurance liability replication to create a platform for discussions on improvements of current practice and theory. For organisational purposes, we kindly request interested attendants to confirm their participation by sending an email to the address sekm13@tum.de.

Spatially Inhomogeneous Evolutionary Games and ApplicationsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

We present a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. We conclude with a couple of open problems: the first one is the learning of the game payoff functional from the observation of the dynamics; the second problem is about a novel particle swarm (global) algorithm for global nonconvex optimization based on an evolutionary game. We sketch the motivations for the success of this method and we explain the open problems for a rigorous analysis of convergence.

Rough calculus: pathwise calculus for functionals of irregular pathsB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

Rough calculus: pathwise calculus for functionals of irregular paths Abstract

Hans Foellmer showed that the Ito formula holds pathwise, for functions paths with finite quadratic variation along a sequence of partitions. We build on Foellmer's insight to construct a pathwise calculus for smooth functionals of continuous paths with regularity defined in terms of the p-th variation along a sequence of time partitions for arbitrary large p >0. We construct a pathwise integral, defined as a pointwise limit of compensated Riemann sums and show that it satisfies a change of variable formula and an isometry formula. Results for functions are extended to path-dependent functionals using the concept of vertical derivative of a functional. Finally, we obtain a "signal plus noise" decomposition for regular functionals of paths with strictly increasing p-th variation. Our results apply to sample paths of semi-martingales as well as fractional Brownian motion with arbitrary Hurst parameter H>0.

Based on joint work with: Anna Ananova (Oxford), Henry Chiu (Imperial College London) and Nicholas Perkowski (Humboldt).

https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_summer_term_2019/seminars/oberseminar_finanz_2019/cont/index.html

An optimal investment strategy for future pension plansB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

Since the work of Merton (1969) the construction of optimal investment portfolios in a time-homogeneous market has been well understood. We review the situation when market parameters are allowed to vary stochastically and apply this to the problem of constructing a pension investment scheme which provides a guaranteed minimum sum on retirement at the expense of imposing an upper limit. We limit ourselves to the accumulation phase, during which payments are made into the fund. The aim of the overall project is to devise pension investment strategies whose behaviour is close to optimal but which present investors with easily explained choices.

Indifference Pricing of Pure Endowments via BSDEs under Partial InformationB349 (Theresienstraße 39, 80333 Mathematisches Institut, LMU)

We investigate the pricing problem of a pure endowment contract when the insurer has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative via the solutions of suitable backward stochastic differential equations.

Renormalization of Hamiltonian lattice systems with operator algebrasMI 03.12.020A (Boltzmannstr. 3, 85748 Garching)

We outline an approach to rigorously implement the Wilson-Kadanoff renormalization group for Hamiltonian lattice systems by operator-algebraic methods. We try to convey the main ideas by discussing some working examples such as lattice gauge theory in 1+1 and 1+2 dimensions and discretized (free) scalar fields. Finally, we point out potential connections with tensor networks and the multi-scale entanglement renormalization ansatz.

From stochastic interface models to random walks in time-dependent random environments -- and back?2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

There is a somewhat unexpected connection between random walks in time-dependent random environments and gradient Gibbs measures describing stochastic interfaces in systems arising from statistical mechanics, e.g., the Ginzburg-Landau model, and its dynamics. After reviewing how the space-time covariances of the height of the interface can be expressed in terms of random walks among dynamic random conductances, I will discuss recent progress on the understanding of the behaviour of such random walks. A particular emphasis will be on the results and the methods that has been used to prove invariance principles and local limit theorems for almost every realisation of the environment.

Enforcing discrete maximum principles in discontinuous Galerkin schemes with application to phase-field methodsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Many mathematical CFD models involve transport of conserved quantities that must lie in a certain range to be physically meaningful. The solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds u_min and u_max exist such that u_min <= u(t,x) <= u_max. To enforce such inequality constraints for DG solutions at least for element averages, the numerical fluxes must be defined and constrained in an appropriate manner. We introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve the MP for element averages even if the exact solution of the PDE violates them due to modeling errors or perturbed data. The proposed methodology is based on a combination of flux and slope limiting. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. The (optional) slope limiter adjusts the higher order solution parts to impose local bounds on pointwise values of the high-order DG solution. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. The novel fractional step flux limiting approach is iterative while in each iteration, the MP property is guaranteed and the consistency error is reduced. Practical applicability is demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn-Hilliard equation (parabolic, nonlinear). The flux limiter (similar to slope limiter) is a local/parallelizable postprocessing procedure that can be applied to various types of DG discretizations of a wide range of scalar conservation laws.

Phase-field approximation of Eulerian interfacial energiesRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

We analyze a phase-field approximation of a sharp-interface model for a two-phase elastic material, where the interface is measured in the deformed configuration. To this aim, sets of finite perimeter are characterised in relation to Sobolev deformation mappings. We discuss a functional frame allowing for existence of phase-field minimizers and Gamma-convergence to the sharp-interface limit. This is a joint work with Diego Grandi, Martin Kružík and Ulisse Stefanelli

Festkolloquium in Honor of Professor Claudia KlüppelbergIAS Auditorium (Lichtenbergstraße 2, 85748 Garching)

On May 17th we celebrate Claudia Klüppelberg and her scientific achievements. We do this with a Festkolloquium at the Institute for Advanced Study at TUM. Invited speakers are the former TUM researchers Anita Behme, Vicky Fasen-Hartmann, Alexander Lindner, Gernot Müller, and Robert Stelzer.

For further information please visit http://go.tum.de/345906

TBAB 252 (Theresienstr. 39, 80333 München)

TBA

Unfitted space-time finite element methods for PDEs on evolving geometriesGebäude 33, Raum 1431 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

Many applications of simulation science involve complex and evolving geometries with possibly strong deformations and topology changes during the evolution. In the context of finite element methods most often a “fitted” characterization is used where a parametric description in terms of a computational mesh is available. An alternative approach is based on the idea of separating the computational mesh and the geometry description, resulting in geometrically “unfitted” methods, which allow for a very flexible handling of geometries. On a (typically simple) background mesh a basis discretization is defined. Only afterwards, according to the separately defined geometry this discretization is adapted to the geometrical information. This approach allows to handle complex and possibly time-dependent geometries without the need for complex and time consuming mesh generation or remeshing. In the recent years finite element methods based on this methodology, geometrically unfitted finite element methods, have drawn more and more attention. Despite its advantages unfitted discretizations, often also called cut-cell methods, give rise to new problems. One major issue in the design and realization of higher order unfitted finite element methods is the problem of accurate and stable numerical integration on level set domains. To tackle this problem we combine a high order geometry approximation based on isoparametric mappings with a discontinuous-in-time space-time finite element formulation. In this talk we introduce the method, discuss implementational aspects and a priori error estimates and present numerical results.

An overview of the goodness-of-fit test problem for copulasMI 00.06.11 HS3 (Boltzmannstr. 3, 85748 Garching)

In dependence modeling, a typical problem is related to the selection of "the best", or at least "a convenient" dependence structure, i.e. a copula. We will review the main “omnibus procedures” for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall’s dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values will be discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, will be tackled too.

Aggregation phenomena, secondary instabilities and well-posedness in chemotaxis models of inflammatory diseasesMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Inflammation is the body's response to outside threats. Although it is a protective mechanism, a derangement of the inflammatory response can impair the physiological functions, in fact inflammation is now recognised as the underlying basis of a signicant number of severe and debilitating diseases, like autoimmune diseases. The complex dynamics of the inflammatory process are not yet fully known and an thorough knowledge of these mechanisms could be the key to control the onset and the evolution of autoimmune diseases, such as Multiple Sclerosis (MS). Recently, several mathematical models have been proposed to test biological hypothesis and improve the knowledge of the inflammatory processes ([8], [6], [7]). The subject of this talk is the study of mathematical models aiming to explore the mechanisms of the inflammatory response and the resulting clinical patterns. I shall present: the development and the study of a Reaction-Diffusion-Chemotaxis (RDC) model of acute inflammation [4] (Model I) and the investigation of the pattern-forming and mathematical properties of the RDC model of MS (Model MS) introduced in [7]. In particular, I shall present a study on the onset of primary (Turing and wave) and the secondary (Eckhaus and zigzag) instabilities ([2]). I shall also investigate the conditions which yield the appearance of stationary non constant radially symmetric solutions and, using numerical values of the parameters taken from the experimental literature, I shall show that both models support the formation of stationary patterns that closely reproduce the concentric lesions observed in clinical practice ([4], [1]). Finally, I shall show the qualitative properties (like existence and uniqueness in appropriate function spaces) of the solutions of the models ([3]). Joint work with L. Desvillettes (Universite Paris Diderot, France), M.C. Lombardo (University of Palermo, Italy) and M. Sammartino (University of Palermo, Italy).

Disconnection by Gaussian Free Field level sets and entropic repulsion2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We consider level set percolation for the Gaussian Free Field (GFF) on the Euclidean lattice in dimensions larger or equal to three, in a strongly percolative regime. We study the 'disconnection event' that the level set below a given level disconnects the discrete blow-up of a compact set A from the boundary of an enclosing box. In particular, we give asymptotic large deviation upper and lower bounds on the probability of disconnection that extend earlier results by Sznitman. Moreover, we study the behaviour of local averages of the GFF conditionally on disconnection: If certain critical levels coincide, we show that the GFF experiences a push down to a level that is locally governed by a multiple of the harmonic potential of A, which may be seen as an instance of entropic repulsion. This is a joint work with Alberto Chiarini.

Women in Probability 2019MI 00.07.014 (Boltzmannstr. 3, 85748 Garching)

Sprecherinnen: Luisa Andreis (Weierstraß-Institut), Gioia Carinci (Delft University of Technology), Hanna Döring (Universität Osnabrück), Lisa Hartung (Johannes Gutenberg-Universität Mainz), Cecile Mailler (University of Bath), Eveliina Peltola (University of Geneva), Elena Pulvirenti (Universität Bonn), Ecaterina Sava-Huss (Graz University of Technology)