## Januar 2018

Natural selection begets adaptation through the exponential amplification of initially rare, highly fit lineages. As such, it bears strong ressemblance with topics such as extreme value theory (dominance of extreme events) and coarsening dynamics (dominance of large structures); like them, natural selection mathematical theory is structured by a basic limit theorem. I will present that theorem and discuss its potential application to detect statistical signatures of selection in empirical data.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

In the talk we will present an interacting particle model on the real line which has a connection with the geometry of Wasserstein space. The model is a natural generalization of the coalescing Brownian motions but now particles can sticky-reflected from each other. The fundamental new feature is that particles carry mass which is aggregated as more and more particles occupy the same position and which determines the diffusivity of the individual particle in inverse proportional way. We are going to present the infinite dimensional SDE with discontinuous coefficients which describes the particle model. We will discuss the existence of a weak solution such an equation, using a finite particle approximation. Also the stationary case will be considered and an invariant measure will be constructed. Joint work with Max von Renesse.

Vortrag in der Reihe: Leipziger Gespräche zur Mathematik

Abstract zu dieser Veranstaltung (PDF-Datei)

In this talk we present connections between the true persistent homology of algebraic varieties and their corresponding offset hypersurfaces. By Hardt’s theorem, true persistent homology events occur at algebraic numbers. By studying the offset hypersurface, we characterize when true persistent homology events occur. The degree of the offset hypersurface is bounded by the Euclidean Distance Degree of the starting variety, so this way we show a bound on the degree of the algebraic numbers at which the homological events occur.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Abstract zu dieser Veranstaltung (PDF-Datei)

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

I will introduce the model of spatial random permutations and talk about its connection to quantum physics, in particular to systems of interacting bosons. I will present the main open questions in the theory, and report on the (small) progress that has been made towards understanding some aspects of the model.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

The cubic nonlinear Schrodinger equation (NLS) is energy-critical (s_c = 1) with respect to the scaling symmetry, where s_c is the scaling critical regularity. The initial value problem (IVP) of cubic NLS is scaling invariant in the Sobolev norm H^1 of scaling critical regularity. First this talk introduce the deterministic global well-posedness result of cubic NLS on 4d-torus (T^4) in the critical regime (with H^1 initial data). Second we consider the cubic NLS in the super-critical regime (with H^s data, s=3).

We prove that graphs with non-negative Ollivier-Ricci curvature have at most two ends. To do so, we employ the max flow min cut theorem for constructing Busemann type functions which turn out to be harmonic. As a corollary, we show that graphs with non-negative Ricci curvature and two ends are recurrent.

We prove that the regular $ntimes n$~square grid of points in the integer lattice $mathbb{Z}^{2}$ cannot be recovered from an arbitrary $n^{2}$-element subset of $mathbb{Z}^{2}$ via a mapping with prescribed Lipschitz constant (independent of $n$). This answers negatively a question of Feige. Our resolution of Feige’s question takes place largely in a continuous setting and is based on some new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Firstly we discuss “Lipschitz regular mappings” on Euclidean spaces, with emphasis on their bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly, we build on work of Burago and Kleiner and McMullen on “non-realizable~densities”. We verify the existence, and further prevalence, of strongly non-realisable densities inside spaces of continuous functions. This is joint work with Vojtech Kaluza and Eva Kopecká.

Gaussian heat kernel bounds play a fundamental role in geometric analysis. We present recent results on explicit Gaussian upper bounds for non-compact manifolds depending on locally uniform Ricci curvature integral assumptions. Furthermore, we discuss generalizations of integral curvature bounds in terms of the so-called Kato class. If time allows, topological applications of those heat kernel upper bounds will be given.