The Deutsche Bahn Group as an employer for mathematicians is shortly presented. General fields of application and some specific mathematically related projects and tasks are considered.
While mathematics plays an increasing role in business application, little mathematics has been embedded in standard business software. Beyond that mathematicians are highly welcome in businesses due to increasing complexity.
I worked the last 4 years and a half years for TNG Technology Consulting. In the talk I will describe how I got there. Furthermore I explain, how it is like to work in IT consulting based on the example of the projects I worked on, the task I performed and the people I met.
We discuss a formulation of elasticity in which compatibility and equilibrium equations are separated from the material law relating stresses and strains, resulting in a model defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity and for some examples in nonlinear elasticity.
We develop a corresponding concept of relaxation, which turns out to be fundamentally different from the classical relaxation of energy functions. This talk is based on joint work with Stefan Müller and Michael Ortiz.
Chiral skyrmions are topological solitons occurring in magnets without inversion symmetry. In this talk I shall explain the analytic structure and geometric realization of chiral interactions, responsible for the occurrence of new magnetic phases in the spirit of Ginzburg-Landau, and for the stabilization of excited states in form of isolated magnetic skyrmions. If time permits I shall also discuss dynamic aspects such as the loss of symmetry and emergent spin-orbit coupling in the framework of the governing Landau-Lifshitz equation.
We will discuss several ways in which local and non local diffusion processes interact with each other: Through trasmission across a fixed surface, in a common domain like double membranes or a bid ask process or for adjacent phase transition interactions. We will go through some detain in this last case.
Recently great progress has been achieved by Alessio Figalli et al. doing a fine analysis of the singular set in the obstacle problem. Much less is known about the behavior of the regular set close to singularities. The structure of this set may be very complicated, for example it may contain infinitely many connected components approaching the singularity, funnel-type singularities etc.
A characterization of global solutions, that is solutions in the entire space, would help but this problem (related to conjectures in potential analysis) is open since the 1980s. In this talk we will discuss progess in the characterization of global solutions.
In this talk I will discuss a striking dichotomy which occurs in the mathematical analysis of microstructures in shape-memory alloys: On the one hand, some models for these materials display a very rigid structure with only very specific microstructures, if one assumes that surface energies are penalised. On the other hand, without this penalisation, for the same models a plethora of very ``wild'' solutions exists. Motivated by this observation, we seek to further understand and analyse the underlying mechanisms. Constructing explicit solutions for a model problem, we show that adding only little regularity to the model does not suffice to exclude the wild solutions. We illustrate these constructions by presenting numerical simulations of them. The talk is based on joint work with J. M. Taylor, Ch. Zillinger and B. Zwicknagl.
I will discuss the recent non-uniqueness result with Vlad Vicol on the non-uniqueness of weak solutions to the Navier-Stokes equations, as well as the follow up paper by myself, Maria Colombo and Vicol. I hope to phrase the results within the context of a broader program of resolving a number of important open problems in the field.
Many important biological processes evolve on different time scales and therefore consist of slow and fast components. Differential equations involving variables evolving on widely different time scales yield rich dynamics and notoriously hard mathematical questions. Geometric methods and dynamical systems theory play important roles in the study of such so-called slow-fast systems. During the last decades geometric singular perturbation theory (GSPT) and the blow-up method have become powerful tools for analyzing low-dimensional slow-fast systems in standard form and have been successfully used in many areas of mathematical biology. However, GSPT of mathematical models arising in molecular cell biology is much less established. The main reason for this seems to be that the corresponding models typically do not have an obvious slow-fast structure of the standard form. Nevertheless, many of these models exhibit some form of hidden slow-fast dynamics, which can be utilized in the analysis.
In this talk I will survey recent advances in GSPT beyond the standard form in the context of prototypical examples from cell biology. I will show that geometric methods based on the blow-up method provide a systematic approach to problems of this type.
In this talk I present a proof of the Monge problem of optimal transport which only relies on a non-branching property of geodesics and lower volume distortion assumptions. It is based on the proof of the Lp-Monge-Kantorovich problem, p>1, and applies to a wide range of geometries, in particular, to smooth Riemannian and Finsler manifolds as well as Lorentzian manifolds. Whereas previous proofs rely on the existence of dual solutions, the present proof for finite target measures is based on a simple geometric idea and its extension to arbitrary target measures is based on measure-theoretic arguments.
Many ordinary and partial differential equations can be written as a gradient flow, which means that there is an energy functional that drives the evolution of the the solutions by flowing down in the energy landscape. The gradient is defined in terms of a dissipation structure, which in the simplest case is a Riemannian metric. We discuss classical and nontrivial examples in reaction-diffusion systems or friction mechanics. We will emphasize that having a gradient structure for a given differential equation means that we add additional physical information.
Considering a family of gradient systems depending on a small parameter, it is natural to ask for the limiting (also called effective) gradient system if the parameter tends to 0. This can be achieved on the basis of De Giorgi's Energy-Dissipation Principle (EDP). We discuss the new notion of "EDP convergence" and show by examples that this theory is flexible enough to allow for situations where starting from a linear kinetic relation (or quadratic dissipation potentials) we arrive at physically relevant, nonlinear effective kinetic relations. The connections between macroscopic kinetic kinetic relations and microscopic non-equilibrium steady states will be discussed.
In this talk I will describe the construction we set up with M. Morini (U. Parma), M. Novaga (U. Pisa), M. Ponsiglione (U. Roma I) to build and show uniqueness of some types of mean curvature flows driven by an anisotropic, nonsmooth perimeter. In particular a striking result is the convergence of the scheme designed by S. Luckhaus and T. Sturzenhecker to a unique limit in almost all situations (generically, including for nonsmooth distances or surface tensions).
In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets. As a consequence, classical solution concepts for such PDEs are naturally limited to short-time existence results or particular initial configurations like perturbations of a steady state. At the same time, the transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions. In the absence of a comparison principle, the relation between weak solution concepts and strong solution concepts for interface evolution problems has remained a mostly open question. We establish weak-strong uniqueness principles for two interface evolution problems, namely for planar multiphase mean curvature flow and for the evolution of the free boundary between two viscous fluids: As long as a classical solution to these evolution problems exists, it is also the unique BV solution respectively varifold solution.
I will present new results about qualitative properties of the solutions of stochastic Hamilton-Jacobi equations. These
Include domain of dependence, intermittent regularization, long time behavior and regularity properties.
It is well known that elastic effects can cause surface instability. In this talk, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the "step-bunching" phenomena in epitaxial growth on vicinal surfaces. The surface steps are subject to pairwise long range interactions which takes the form of a general Lennard--Jones (LJ) type potential. It is characterized by two exponents $m$ and $n$ describing the singular and decaying behaviors of the potential for $x\\ll 1$ and $x\\gg 1$. (We henceforth call these potentials generalized LJ $(m,n)$ potentials.) We provide a systematic study of the asymptotic properties of the step configurations and the value of the minimum energy for different regimes of $m$ and $n$. Both linear stability and nonlinear minimization problems are considered. Our results show that there is a phase transition between the bunching and non-bunching regimes. Some of our statements also hold for critical points of the energy, not just minimizers. This work extends the technique and results of Luo-Xiang-Yip 2016 which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). We emphasize on discrete models but continuum description will also be discussed. This is joint work with Tao Luo and Yang Xiang.
I will present both the curvature flow and the elastic flow of a network of curves in the plane, discussing existence, uniqueness, singularity formation and asymptotic behavior of the evolution.
In this talk I will present some recent results on the analysis of the free boundary for the thin obstacle problem, in connection with the theory of stationary varifolds which are parametrized as graphs of multiple valued functions. In particular, I will focus on the global structure of the free boundary and on the finiteness of the local measure. All the results are in collaboration with M. Focardi (Uni. Firenze).
The exchange-driven growth model describes a process in which pairs of clusters interact through the exchange of single monomers. The rate of exchange is given by an interaction kernel which depends on the size of the two interacting clusters. The model was recently obtained as the mean-field limit of stochastic particle systems (zero-range process).
Well-posedness of the model is discussed under suitable growth bounds on the kernel and arbitrary initial data. The central part of the talk considers the longtime behavior under a detailed balance condition on the kernel. The total mass density, determined by the initial data, acts as an order parameter, in which the system shows a phase transition in the following sense: There is a critical mass density characterized by the rate kernel. In the subcritical regime, there exists a unique equilibrium state, and the solution converges in a strong sense (conservation of mass). In the supercritical regime, the solution converges only in a weak sense, where excess mass gets lost due to the formation of larger and larger clusters.
In this regard, the model behaves similarly to the Becker-Döring equation. The main ingredient for the longtime behavior is the free energy acting as Lyapunov function for the evolution. It is also the driving functional for a gradient flow structure of the system under the assumption of detailed balance.
The talk closes with an outlook on work in progress together with Bob Pego and Juan Velazquez on related models without detailed balance showing stable oscillations as the longtime behavior.
In 1995, Luckhaus and Sturzenhecker proved that the implicit time discretization for mean curvature flow converges to a distributional solution provided that the time-integrated perimeters of the approximations converge to those of the limit. In this talk I will show that in the case of strictly mean convex initial conditions, this condition is indeed verifiable. The proof establishes, by compensated compactness techniques, the strict convergence of the arrival time functions. This is joint work with Guido De Philippis.
In this talk, we consider boundary value problems on domains with non smooth boundaries. We approach this problem by transferring it to non-compact manifolds with a nice geometry -- the bounded geometry. This gives a more general framework how to handle Dirichlet (or Dirichlet-Neumann mixed) boundary value problems for domains with a larger class of singularities on the boundary and gives a nice geometric interpretation.
This is joint work with Bernd Ammann (Regensburg) and Victor Nistor (Metz).