Webrelaunch 2020

AG Stochastische Geometrie (Sommersemester 2015)

Seminar: Freitag 9:45-11:15 SR 2.58
Freitag 9:45-11:15 1C-01
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

Freitag, 17.04.2015

9.45 Uhr Johan DEBAYLE (Ecole Nationale Supérieure des Mines de Saint-Etienne):

Study of Minkowski Valuations for Boolean Models. Application to 2-D Image Analysis of Crystallization Processes

Solution crystallization processes are widely used in the process industry and notably in the pharmaceutical industry as separation and purification operations and are expected to produce solids with desirable properties. More particularly, the size and the shape of crystals are known to have a considerable impact on the final quality of products, such as drugs. Hence, it is of main importance to be able to control in real time the granulometry of the crystals.

From a sequence of 2-D images acquired by an in situ camera (visualizing the particles crystallizing in a reactor), the scientific objective is then to characterize the geometry of crystals. Different issues have to be addressed: the perspective projection of the 3-D crystal shape onto the image plane, the blurred appearance of unfocused particles, the degree of agglomeration or overlapping, and the random variation in size/shape of the observed particles. In this way, different research works on image processing and analysis have been already carried out at the ENSM-SE. The proposed approaches give satisfying results for low solid concentrations and/or low overlapping. At a higher density/overlapping rate, it seems to be more suitable to use other methods such as based on stochastic geometry. This kind of approach is currently investigated through the PhD thesis of Saïd Rahmani at the ENSM-SE.

The presentation will be then be focused on the geometrical characterization and modeling of 2-D images of ammonium oxalate crystals. In a first step, the Minkowski functionals will be investigated to get some useful geometrical characteristics. Indeed, assuming that the binarized images can be represented as Boolean models of rectangles, the Miles formulas (under some assumptions) directly give some mean geometrical characteristics (area, perimeter) of the individual particles. Nevertheless, these features do not give any information about the shape ratio (elongation) or anisotropy of the particles. In this way, other characteristics are required to analytically fit the model to the real data. By using the covariance of dilated models with various structuring elements, the first and second-order moments of some grain geometrical characteristics (area, perimeter, Feret diameter/support function) can be estimated. First quantitative results will be presented to show the performance of the proposed method. Current works in relation to this method are focused on the analysis of the mixed moments (autocorrelation) of the Feret diameters to fully characterize the geometry of the particles.

In a second part, the Minkowski tensors will be investigated. Indeed, the recent work about Minkowski tensor density formulas for Boolean models developed at KIT seems to be very interesting for the analysis of crystal images. These `extended Miles formulas' for Minkowski tensors give additional geometrical information, such as anisotropy of the particles. These formulas will be specifically investigated for Boolean models of rectangles. Perspectives are now focused on the study of Minkowski tensors for dilated Boolean models by hoping that other useful geometrical characteristics will be estimated.

Freitag, 24.04.2015

9.45 Uhr Alexander Koldobsky (MPI-Bonn/University of Missouri):

The slicing problem for sections of proportional dimensions

The slicing problem asks whether there exists an absolute constant such that every symmetric convex body of volume one in every dimension has a central hyperplane section with area greater than this constant. We consider a generalization of this problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies, for duals of bodies with bounded volume ratio, and for k-intersection bodies. We also prove it for arbitrary symmetric convex bodies with a constant dependent only on the dimension, and with an absolute constant when sections are of proportional dimension.

Freitag, 08.05.2015

9.45 Uhr Markus Heydenreich (Ludwig-Maximilians-Universität München):

Random walk on critical percolation clusters

Understanding the geometry of critical percolation clusters is a challenging problem. However, on high-dimensional lattices, lace expansion allows for a rigorous analysis. Critical percolation clusters have an intricate geometry, which is reflected in different volume growth exponents for balls in the intrinsic (graph) and extrinsic (Euclidean) distance. We are considering random walks on critical percolation clusters, and show that they are highly sub-diffusive. More specifically, we identify asymptotics for exit times for random walk on the high-dimensional incipient infinite cluster. Based on joint work with Remco van der Hofstad (Eindhoven) and Tim Hulshof (Vancouver).

Freitag, 15.05.2015

9.45 Uhr Richard Gardner (Western Washington University):

The Orlicz-Brunn-Minkowski theory

The Orlicz-Brunn-Minkowski theory was introduced by Lutwak, Yang, and Zhang in 2010 and represents the latest substantially developed extension of the classical Brunn-Minkowski theory. The talk will be a survey on the current state of the new theory and its significance, with some thoughts about possible future extensions. I will spend some time giving the background, illustrated with a few pictures, for those who are not familiar with the Brunn-Minkowski theory. My own contribution is contained in joint work with Daniel Hug and Wolfgang Weil and with Deping Ye of the University of Newfoundland.

Freitag, 29.05.2015

9.45 Uhr Vlad Yaskin (University of Alberta):

Stability results for sections of convex bodies

$ $ Let $K$ be a convex body in $\mathbb R^n$. The parallel section function of $K$ in the direction $\xi\in S^{n-1}$ is defined by
$$ A_{K,\xi}(t)=\mathrm{vol}_{n-1}(K\cap \{\xi^{\perp}+t\xi\}), \quad t\in \mathbb R. $$
If $K$ is origin-symmetric (i.e. $K=-K$), then Brunn's theorem implies
$$ A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t) $$
for all $\xi\in S^{n-1}$.

The converse statement was proved by Makai, Martini and \'Odor. Namely, if $A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all 
$\xi\in S^{n-1}$, then $K$ is origin-symmetric.

We provide a stability version of this result. If $A_{K,\xi}(0)$ is close to $\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all $\xi\in S^{n-1}$, then $K$ is close to $-K$.

Joint work with Matthew Stephen.$

Freitag 12.06.2015

9.45 Uhr Daniel Hug:

Random polytopes in halfspheres

Freitag 26.06.2015

9.45 Uhr Ilya Molchanov (Universität Bern):

The semigroup of metric measure spaces and its infinitely divisible probability measures

The family of metric measure spaces can be endowed with the semigroup operation being the Cartesian product. The aim of this talk is to arrive at the generalisation of the fundamental theorem of arithmetic for metric measure spaces that provides a unique decomposition of a general space into prime factors. These results are complementary to several partial results available for metric spaces (like de Rham theorem on decomposition of manifolds). Finally, the infinitely divisible and stable laws on the semigroup of metric measure spaces are characterised.

This is based on joint work with S.N. Evans (Berkeley).

Freitag 03.07.2015

9.45 Uhr Dennis Müller:

Normal-Approximation der Euler-Charakteristik Gaußscher Exkursionsmengen

Freitag 10.07.2015

9.45 Uhr Uta Freiberg (Universität Stuttgart):

Spectral asymptotics on random Sierpinski gaskets

Self similar fractals are often used in modeling porous media. Hence, defining a Laplacian and a Brownian motion on such sets describes transport through such materials. However, the assumption of strict self similarity could be too restricting. So, we present several models of random fractals which could be used instead. After recalling the classical approaches of random homogenous and recursive random fractals, we show how to interpolate between these two model classes with the help of so called V-variable fractals. This concept (developed by Barnsley, Hutchinson & Stenflo) allows the definition of new families of random fractals, hereby the parameter V describes the degree of `variability' of the realizations. We discuss how the degree of variability influences the geometric, analytic and stochastic properties of these sets. - These results have been obtained with Ben Hambly (University of Oxford) and John Hutchinson (ANU Canberra).

Freitag 17.07.2015

9.45 Uhr Lukas Parapatits (ETH Zürich):

Centro-Affine Tensor Valuations

$ $ The tensor valued map
   K \mapsto \int_K x^{\odot p} dx
is a natural generalization of volume and moment vector.
It is a continuous $\mathrm{SL}(n)$-covariant valuation
for each non-negative integer $p$.
A new example with the same properties is
   K \mapsto \int_{S^{n-1}} u^{\odot p} dS_p(K^*,u) ,
where $S_p(K^*, \cdot)$ denotes the $L_p$ surface area measure of the polar body of $K$.
We show that these operators are essentially the only
measurable $\mathrm{SL}(n)$-covariant tensor valued valuations
on polytopes with the origin as interior point. 

\noindent This is joint work with Christoph Haberl.