Webrelaunch 2020

AG Stochastische Geometrie (Wintersemester 2023/24)

Studierende und Gäste sind jederzeit herzlich willkommen. Wenn nicht explizit anders unten angegeben, finden alle Vorträge in Präsenz im Raum 2.058 statt. Für die Aufnahme in den E-Mail-Verteiler für die Einladungen kontaktieren Sie bitte Steffen Winter (steffen.winter@kit.edu).

Seminar: Freitag 9:45-11:15 20.30 SR 2.58
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

Friday, 27.10.2023, 9:45 h

Olav Kallenberg (Auburn University)

Point processes from early history to some modern marvels

Abstract: With mathematical fashions swinging back and forth, some people may have got the idea that point processes constitute a part of probability theory that was completely explored ages ago, and whose results are too elementary to be of much interest. My aim is to dispel this misconception by presenting some quite deep and powerful results, which are also new enough that you won't find them in any standard textbook. My talk will begin with an elementary historical discussion, emphasizing the fundamental role of the binomial processes for both theory and applications.

Friday, 3.11.2023, 14 - 18:30 h

Colloquium and Celebration of Günter Last's 65th birthday

See here for the Programme of the colloquium

Miniworkshop on Percolation and related areas

Monday, 6.11.2023, 10:00 h, room 3.60

Mathew Penrose (University of Bath)

Random Euclidean coverage and connectivity problems

Abstract: pdf

Monday, 6.11.2023, 11:00 h, room 3.60

Hermann Thorisson (University of Iceland)

Shift-Coupling and Maximality

Tuesday, 7.11.2023, 15:45 h, room 2.59 (time of the Stochastics Seminar)

Takis Konstantopoulos (University of Liverpool)

Barak-Erdös Graphs and last passage percolation

Abstract: I will present new and older work on directed random graphs focusing mostly on behavior of longest path. Their growth rate is a function of the parameters of the graph (connectivity probability, weights on edges). This function (that can be called last passage percolation constant) has interesting properties. For example, if we assign weight 1 to every existing edge and weight x to every nonexisting one then, as a function of x, we obtain a convex function of x whose derivative fails to exist when x is a nonpositive rational number or x = 2, 3, ... or x=1/2, 1/3,... Depending on the order structure of the set of vertices, we can obtain functional central limit theorems that can range from Brownian motions to Brownian percolation processes whose distribution is related to the largest eigenvalue of a GUE random matrix. There are also relations with branching processes and the PWIT (Poisson weighted infinite tree) that appear in a sparse regime.
The talk will be based on work that has been done with various collaborators over the years: D Denisov, S Foss, B Mallein, A Pyatkin, S Ramassamy.

Friday, 8.12.2023, 9:45 h

Luca Lotz (KIT)

Stabiles Matching von Punktprozessen und Hyperuniformität

Abstract: In diesem Vortrag präsentiere ich die Konstruktion hyperuniformer Verdünnungen von geeigneten Punktprozessen mithilfe des Stabilen Matchings. Das Stabile Matching von zwei Punktmengen erfolgt durch das paarweise Verbinden von Punkten, wobei nah beieinander liegende Paare präferiert werden. Damit kann ausgehend von zwei stationären, ergodischen Punktprozessen unterschiedlicher Intensität eine Verdünnung des Prozesses mit höherer Intensität erzeugt werden. Diese ist unter Regularitäts- und Mixing-Voraussetzungen genau dann hyperuniform, wenn der Prozess mit geringerer Intensität hyperuniform ist.

Friday, 15.12.2023, 9:45 h

Mikhail Chebunin (KIT)

Uniqueness of the infinite cluster in the random connection model

Friday, 12.1.2024, 9:45 h

Tillmann Bühler (KIT)

Johnson-Mehl Percolation in the Hyperbolic Plane

Friday, 26.1.2024, 9:45 h

Matthias Neumann (Ulm University)

Stochastic 3D nanostructure modeling for virtual testing of electrode materials

Abstract: Based on 3D image data, a parametric stochastic model is developed for the simulation of differently manufactured nanoporous particles, which are used as active material in battery electrodes. Functional properties of these particles, like effective diffusivity, depend on the 3D morphology of the nanopores, which are influenced by the underlying production parameters. In order to study relationships between the pore morphology and effective properties influencing the performance of the electrode, data-driven stochastic 3D modeling is a powerful tool. It enables for the generation of a large range of virtual, but realistic nanostructures on the computer, which overcomes limitations of time consuming 3D imaging. For the virtual structures, morphological descriptors as well as effective properties can be determined, in order to statistically quantify relationships between them. We call this virtual materials testing. The stochastic 3D nanostructure model is based on tools from stochastic geometry. The solid phase of aggregate particles is modeled by an excursion set of a certain class of chi-square random fields. Model fitting is performed using analytical relationships between the covariance function of the chi-square field and two-point coverage probabilities, where the latter can be directly estimated from 3D image data. After having fitted the model parameters to image data, model validation is performed by comparing morphological descriptors (not used for model fitting) of simulated and experimental image data. This work is based on joint work with Phillip Gräfensteiner and Volker Schmidt.

Friday, 2.2.2024, 9:45 h

Evgeny Spodarev (Ulm University)

Prediction of random functions via excursion sets

Abstract: We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted L1-distance. Using equivalent forms of this metric and the specific choice of excursion levels, we formulate the prediction problem as a minimization of a certain target functional which involves the excursion metric. Existence of the solution and weak consistency of the predictor are discussed. An application to the extrapolation of stationary heavy-tailed random functions illustrates the use of the aforementioned theory. Numerical experiments with the prediction of Gaussian, \alpha- and \max-stable random functions show the practical merits of the approach.
Joint work with A. Das and V. Makogin

[1] A. Das, V. Makogin, and E. Spodarev. Extrapolation of stationary random fields via level sets. Theory of Probability and Mathematical Statistics, 106:85–103, 2022.
[2] V. Makogin and E. Spodarev. Prediction of random variables by excursion metric projections. Preprint, arXiv:2207.00447v2, September 2022. https://doi.org/10.48550/arXiv.2207.00447.

Friday, 16.2.2024, 9:45 h

Daniel Bonnema (KIT)

The average and expected distortion of Voronoi scapes

Friday, 23.2.2024, 9:45 h

Claudia Redenbach (TU Kaiserslautern-Landau)

Image segmentation by neural networks trained on synthetic data

Abstract: Neural networks are commonly used for image segmentation. Training a network requires a suitable amount of training data, that is, images along with the desired segmentation result. Manual annotation of images is common practice, but time consuming and error prone.
We suggest to use synthetic image data. For their simulation, virtual micro structures are generated from stochastic geometry models. Then, a model for the imaging process is applied to simulate realistic images of the synthetic structures. Binary images of the model realizations yield a ground truth for the segmentation. The resulting pairs are then used to train the neural network. We present two examples of application: segmentation of cracks in µCT images of concrete and of FIB-SEM images of porous structures.