KIT - Fakultät für Mathematik - AG Stochastische Geometrie (Sommersemester 2017)

Termine
Seminar:	Freitag 9:45-11:15	SR 2.58

Lehrende
Seminarleitung	Prof. Dr. Daniel Hug
	Sprechstunde: Nach Vereinbarung.
	Zimmer 2.051 Kollegiengebäude Mathematik (20.30)
	Email: daniel.hug@kit.edu
Seminarleitung	Prof. Dr. Günter Last
	Sprechstunde: nach Vereinbarung.
	Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30)
	Email: guenter.last@kit.edu

Studierende und Gäste sind jederzeit willkommen. Wenn nicht explizit anders unten angegeben, finden alle Vorträge im Seminarraum 2.58 im Mathebau (Gebäude 20.30) statt.

Freitag, 5.5.2017

9.45 Uhr Benjamin Reichenwallner (Universität Salzburg)

Monotonicity of moments of volumes of random convex hulls

Abstract: The convex hull of $d+1$ points randomly chosen from a $d$ -dimensional convex body $K$ according to the uniform distribution in $K$ forms a simplex with probability $1$ . We denote its volume by $V_K$ .

In 2006, Mark Meckes raised the question whether $K \subseteq L$ would imply $\mathbb{E} V_{K} \leq \mathbb{E} V_{L}$ . Luis Rademacher gave a surprising answer, only leaving the expected volume of a random tetrahedron in dimension three as an open task.

We use analytic methods to give a counterexample to this monotonicity of inclusion in dimension three. Moreover, we discuss similar results on the monotonicity of higher moments of $V_K$ as well as of moments of the (n-1)-dimensional volume of the convex hull of $n<d+1$ random points in a $d$ -dimensional convex body.

Dienstag, 9.5.2017 (SR 2.29, Mathebau; im Rahmen der AG Stochastik)

15.45 Uhr Richard Gardner (Western Washington University)

Symmetrization

Abstract: The idea of symmetrization---taking a subset of Euclidean space (for example) and replacing it by one which preserves some quantitative aspect of the set but which is symmetric in some sense---is both prevalent and important in mathematics. The most famous example is Steiner symmetrization, introduced by Jakob Steiner around 1838 in his attempt to prove the isoperimetric inequality (the inequality which essentially explains why soap bubbles are spheres rather than some other shape). Steiner symmetrization is still a very widely used tool in geometry, but it and other types of symmetrization are of vital significance in analysis, PDE's, and mathematical physics as well.
The talk focuses on symmetrization processes that associate to a given set one that is symmetric with respect to a subspace. In the first phase of an ongoing joint project with Gabriele Bianchi and Paolo Gronchi of the University of Florence, we consider various properties of an arbitrary symmetrization, the relations between these properties, and which properties characterize Steiner symmetrization. Several other well-known symmetrizations, such as Minkowski symmetrization and central symmetrization, will also be discussed. After summarizing these results, we shall discuss the second phase, in which we attempt to understand the convergence of iterated symmetrals.

Freitag, 12.5.2017

9.45 Uhr Nina Gantert (TU München)

Large deviations for random projections of some convex sets or Cramer's theorem is atypical

Abstract: We give large deviation results for random projections of some convex sets. They quantify the well-known statement that two independently drawn vectors whose law is uniform on a high-dimensional sphere, are nearly orthogonal. We explain how this geometric point of view generalizes the classical Cramer’s Theorem which turns out to be atypical in our setup.

The talk is based on joint work with Steven Soojin Kim and Kavita Ramanan, Brown University.

Freitag, 2.6.2017

9.45 Uhr Günter Last

Wann ist die asymptotische Varianz geometrischer Funktionale des Booleschen Modells positiv?

Freitag, 9.6.2017

9.45 Uhr Franz Nestmann

Cluster counting in the random connection model

Abstract: The classical random connection model (RCM) can be obtained by a Poisson process $\eta$ on $\mathbb{R}^d$ and a connection function $\varphi:[0,\infty)\to[0,1]$ . Each pair of points $(x,y) \in \eta^2$ , with $x\ne y$ , is connected via an edge with probability $\varphi(|x_1-x_2|)$ , independently of all other points and connections.

This model can be extended by adding a random and independent mark in $[0,\infty)$ to each Poisson point. In this marked model, the probability for an edge between two Poisson points also depends on the marks of the two points. This marked model includes the classical Gilbert model. In the Gilbert model the points are marked with random radii and two points are connected if the balls around them intersect.

In this talk we are interested in the number of clusters of the classical and the marked RCM that are isomorphic to a connected finite graph and are located in an observation window. We will use the Stein-Malliavin method to derive bounds for the normal approximation of this random number in the Wasserstein and the Kolmogorov distance.

This talk is based on joint work with Günter Last (Karlsruhe) and Matthias Schulte (Bern).

Dienstag, 13.6.2017 (im Rahmen der AG Stochasik)

15.45 Uhr Jan Weis

Integral Geometric Formulae for Tensorial Curvature Measures

Abstract: The tensorial curvature measures are the natural tensor-valued generalizations of the curvature measures of convex bodies in Euclidean space. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. The talk provides two complete sets of integral geometric formulae, so called kinematic formulae and Crofton formulae, for such (generalized) tensorial curvature measures. These formulae treat the intersection of a convex body with a second geometric object (in the kinematic formulae this is another convex body; in the Crofton formulae this is an affine subspace) which is uniformly moved by a proper rigid motion. The proofs, which are also sketched in this talk, proceed in a more direct way than the classical proofs of the corresponding integral formulae for curvature measures.

Dienstag, 27.6.2017 (im Rahmen der AG Stochasik)

15.45 Uhr Fabian Schaller (Uni Erlangen)

Die Struktur zufälliger granularer Packungen

Abstract: In granular packings and other particulate systems with short-range interactions, the local structure has crucial influence on the macroscopic physical properties such as stability or response to external forces. Packings of monodisperse spherical particles are a common simple model for granular matter and packing problems, but particles in nature and industries are rarely spherical. This talk focuses on packings of ellipsoidal particles, a system which offers the possibility to study the influence of particle shape on packing properties. Results of a large scale experimental study of jammed packings of oblate ellipsoids as well as results of simulations of frictional and frictionless particles with and without gravity are presented. The structure of the packings is analyzed by Set Voronoi diagrams, an extension of the conventional Voronoi diagram to aspherical particles, and the average number of contacts between the particles. Furthermore, a common model for the Voronoi Volume distribution of spheres is extended towards aspherical particles.
Finally, the effect of particle polydispersity to the packing structure is addressed. In the broader context of the physics of particulate systems, our analysis emphasizes the need for simple toy models to understand packing properties of complex shaped particles.

Freitag, 30.6.2017

9.45 Uhr Raman Sanyal (Uni Frankfurt)

A combinatorial theory of valuations on polytopes

Abstract: The confluence of Minkowski addition and volume gives rise to the vast theory of mixed volumes and intrinsic volumes. Mixed volumes and the related geometric inequalities have applications in many areas including algebra and combinatorics. A celebrated result of Hadwiger asserts that the intrinsic volumes constitute a basis for the continuous and rigid-motion invariant valuations on convex bodies and, moreover, this basis is distinguished by nonnegativity- and monotonicity properties. This is a quite complete and mathematically appealing situation for general convex bodies.

In the discrete setting (i.e. lattice polytopes), discrete volume (i.e. counting lattice points) takes the place of the volume. But whereas the structure of the space of lattice-invariant valuations on lattice polytopes parallels the continuous situation above quite remarkably, our understanding of nonnegative and monotone valuations is simply unsatisfactory. Even worse, the nonnegativity and monotonicity of the mixed volumes is genuinely lost for the discrete mixed volumes.

In this talk I will advocate a combinatorial theory of nonnegative, monotone, and mixed valuations. In the continuous case, this underlines the prominent role played by volume and recovers the classical mixed volumes. For lattice polytopes, we obtain a Hadwiger-type theorem and we show that our notion of combinatorial mixed valuation preserves nonnegativity and monotonicity for many valuations including the discrete volume. I will emphasize the strong ties to the combinatorics of subdivisions of (lattice) polytopes and, if time permits, I will remark on applications to recent results of DiRocco, Haase, and Nill on the (motivic) arithmetic genus. The talk is based on joint work with Katharina Jochemko.

Freitag, 7.7.2017

9.45 Uhr Steffen Winter

Does the expected Euler characteristic provide a lower bound for the percolation threshold of Mandelbrot percolation?

Freitag, 21.7.2017

9.45 Uhr Jun Luo (Chongqing University, China)

Topological properties of self-similar fractal squares

Abstract: In this talk we will present some basic topological properties of a class of self-similar sets arising from a unit square. First we give a complete classification of their topology; Secondly we discuss the Lipschitz equivalence between them. The main approach we used is to introduce the so called Gromov hyperbolic graph on the symbolic space of the self-similar set and study its hyperbolic boundary properties. Further works on more general self-similar sets and even self-affine sets will be mentioned finally.

Freitag, 28.7.2017

9.45 Uhr Anna Gusakova (Uni Bielefeld)

Integral geometry formulas for ellipsoids

Abstract: For any non-degenerate ellipsoid $\mathcal{E}\subset\mathbb{R}^d$ we prove that

$\[ \frac{1}{|\mathcal{E}|^{k+1}}\int\limits_{\mathcal{E}^{k+1}}{|\mathrm{conv}( x_0,\ldots, x_k)|^p\dd x_0\ldots \dd x_k}=\alpha_{d,k,p}\int_{G_{d,k}}|P_L\mathcal{E}|^p\nu_{d,k}(dL), \]$

$P_L$ denotes the orthogonal projection to the $k$ -dimensional linear
subspace $L$ uniformly distributed in the linear Grassmannian
$G_{d,k}$ . We rewrite this relation in terms of random Gaussian
matrices. We also show that

$\int\limits_{\mathcal{E}^{k+1}}{|\mathrm{conv}( x_0,\ldots, x_k)|^p\dd x_0\ldots \dd x_k}=\beta_{d,k,p}\int_{A_{d,k}}|\mathcal{E}\cap E|^{p+d+1}\mu_{d,k}(dE),$
where $A_{d,k}$ is the affine Grassmannian endowed with the
standard motion invariant measure $\mu_{d,k}$ . In the case $p=0$ this
formula reduces to an affine version of the integral formula of
Furstenberg and Tzkoni.

Based on a joint work with F. Götze and D. Zaporozhets.