## Januar 2018

Principal component analysis is a widely-used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes. This is joint with with Leon Zhang and Xu Zhang.

The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. The talk will not assume any prior knowledge of oriented matroids.

Abstract: In this talk we introduce the concept of singular Finsler

foliation, which generalizes the concepts of Finsler actions, Finsler

submersions and (regular) Finsler foliations. We show that if $F$ is a

singular Finsler foliation with closed leaves on a Randers manifold

$(M,Z)$

with Zermelo data $(h,W),$ then $F$ is a singular Riemannian foliation on

the Riemannian manifold $(M,h)$.

We also present a slice theorem that locally relates singular Finsler

foliations on Finsler manifolds with singular Finsler foliations on

Minkowski spaces. This talk is based on a joint work with Prof. Miguel

Angel Javaloyes (Murcia-Spain) and Dr. Benigno O. Alves (IME USP-Brazil).

Given a set of predictor variables and a response variable, how much information do the predictors have about the response, and how is this information distributed between unique, complementary, and shared components? Recent work has proposed to quantify the unique component of the decomposition as the minimum value of the conditional mutual information over a constrained set of information channels. We present an efficient iterative divergence minimization algorithm to solve this optimization problem with convergence guarantees, and we evaluate its performance against other techniques.

Joint work with Johannes Rauh and Guido Montúfar (https://arxiv.org/abs/1709.07487)

In general dimensions, weak solutions of the Euler Lagrange equation of the area functional are naturally constructed in the class of integral varifolds. As the regularity theory of area-stationary integral varifolds is still substantially incomplete, it is important to enlarge the related mathematical machinery, both to handle the possible singularities and to advance regularity theory.

In this regard, the talk will survey the implementation of the concept of weakly differentiable functions on integral varifolds of locally bounded first variation of area. The present theory of Sobolev spaces on metric measure spaces turns out to ill-adapted for this purpose. Instead, a coherent theory can be constructed by defining a non-linear class of functions taking into account the extrinsic geometry of the varifold. The inclusion of non-vanishing first variation therein is not only theoretically preferable but is also important for the needs of problems from applied analysis.

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