Webrelaunch 2020

AG Stochastische Geometrie (Sommersemester 2022)

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

Freitag, 29.4.2022, 9.45 Uhr

Colin Bretl (KIT)

Real Time Anti-Aliasing Sampling Patterns

Abstract: Computer graphics, as a branch of computer science, is concerned with the generation of images from digital scene descriptions (rendering). This can be understood as the sampling of two-dimensional, real-valued functions, whereby the number of sampling points is technically limited. This is especially the case when a complex scene is to be rendered under real-time conditions (e.g. within 1/60 second).
Typically, equidistant grid points are used as sampling points, whereby an insufficient number of sampling points leads to aliasing artefacts (cf. (Ö20)). Desirable, on the other hand, are sampling points with the so-called blue-noise property: this states that high-frequency and thus non-sampling frequencies should be represented as white noise, while low-frequency details should be preserved. Therefore, different types of sampling points have been investigated in the past, which are modelled as point processes for this purpose (Ö20).
Within computer graphics research, only comparatively simple forms of sampling points with the blue-noise property have been mathematically investigated (H13). The talk therefore focuses on the question which mathematical questions arise in the aforementioned context and which mathematical findings can be applied to it. For this purpose, rendering and the aliasing problem are explained in the talk, requirements for a mathematical model for sampling points are derived from these and point processes are presented as a corresponding model. Finally, some research ideas resulting from the application in computer graphics will be outlined.
The audience is cordially invited to contribute their thoughts and ideas to the lecture.

(H13) Heck, D., Schlömer, T., and Deussen, O.: Blue noise sampling with controlled
ACM Transactions on Graphics (TOG) 32.3 (2013): 1-12.
(Ö20) Öztireli, A.C.: A Comprehensive Theory and Variational Framework for Anti‐aliasing
Sampling Patterns.
Computer Graphics Forum. Vol. 39. No. 4. 2020.

Freitag, 6.5.2022, 9.45 Uhr

Goran Radunovic (University of Zagreb)

An overview of the theory of complex dimensions

Abstract: We will give an overview of the main results of the theory of complex dimensions for subsets of Euclidean spaces. This theory has been developed in a series of papers and in a research monograph ”Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions” coauthored by M. L. Lapidus, G. Radunovic and D. Zubrinic. The theory gives a far-reaching generalization of the one-dimensional theory for fractal strings developed by M. L. Lapidus, M. van Frankenhuijsen and their numerous collaborators. The complex dimensions of a given set generalize the well-known notion of its Minkowski dimension but are defined as the poles (or more general singularities) of the (distance or tube) fractal zeta function associated with the given set. Although the complex dimensions are defined analytically, we will explain how they possess a deep geometric meaning connected to the fractal nature of the given set and the intrinsic oscillations in its geometry. Namely, the complex dimensions of a set appear explicitly in its fractal tube formula which, under appropriate assumptions, gives a (generalized) asymptotic expansion of the volume of its \epsilon-parallel set when \epsilon is close to zero. We will illustrate the theory by interesting examples and also reflect on some of the possible applications.

Freitag, 3.6.2022, 9.45 Uhr

Steffen Winter (KIT)

Aggregation models

Abstract: Diffusion-limited aggregation (DLA) is a stochastic model for cluster growth proposed by Witten and Sander in 1981. Particles started from infinity perform random walks until they hit the cluster for the first time and are attached there. The model is easy to simulate and produces dendrite-like structures mimicking those observed in several physical phenomena, including colloidal aggregation and dielectric breakdown. Despite its popularity in physics and its rather simple definition, not much is known rigorously about this model. One of the few things known is an upper bound on the growth rate of the diameter of the cluster due to Kesten (1987), which can also be interpreted as a lower bound on the fractal dimension of the cluster.
Another popular cluster growth model in physics is ballistic aggregation, which up to now has not attracted much attention in the maths literature. Here particles move along straight lines and similar dendrite-like clusters are generated. In the talk we review both models. Using Kesten's techniques, we obtain a bound for the growth rate of the cluster in the ballistic model, which in this case allows to deduce its fractal dimension.
Based on joint work with Tillmann Bosch.

Dienstag, 14.6.2022, 15.45 Uhr (im Rahmen der AG Stochastik; Raum 2.58)

David Dereudre (Université de Lille)

Fully-connected bond percolation on \mathbb{Z}^d

Abstract: We consider the bond percolation model on the lattice \mathbb{Z}^d with the constraint to be fully connected. Each edge is open with probability p in (0,1), closed with probability 1-p and then the process is conditioned to have a unique open connected component (bounded or unbounded). The model is defined on \mathbb{Z}^d by passing to the limit for a sequence of finite volume models with general boundary conditions. Several questions and problems are investigated: existence, uniqueness, phase transition, DLR equations. Our main result involves the existence of a threshold 0 < p^*(d) < 1 such that any infinite volume model is necessary the vacuum state in subcritical regime (no open edges) and is nontrivial in the supercritical regime (existence of a stationary unbounded connected cluster). Bounds for p^*(d) are given and show that it is drastically smaller than the standard bond percolation threshold in \mathbb{Z}^d. For instance 0.128 < p^*(2) < 0.202 (rigorous bounds) whereas the 2D bond percolation threshold is equal to 1/2.

Montag, 20.6.2022, 15.45 Uhr (Raum 3.069)

Richard Gardner (Western Washington University, USA)

Geometric tomography: An update on open problems, including the hyperplane and Mahler conjectures

Abstract: Geometric tomography aims to retrieve information about a geometric object (such as a convex body, star body, finite set, etc.) from data concerning its intersections with planes or lines and/or projections (i.e., shadows) on planes or lines. The topic offers the researcher many interesting open problems, and we shall describe the current status of several of them.
There is a significant overlap between geometric tomography and analytical convex geometry, and in particular these subjects share two notoriously difficult open problems: The hyperplane conjecture (or slicing problem) of Bourgain, and Mahler's conjecture. Both problems have generated a large literature in which connections with many other open problems from a number of different areas in mathematics have been observed, for example, analysis, probability, information theory, number theory, and computational, Finsler, integral, stochastic, and symplectic geometry. We shall attempt to provide an overview of these connections.
The talk will be aimed at a general audience.

Freitag, 24.6.2022, 9.45 Uhr

Mikhail Chebunin (KIT)

Lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^d$.

Abstract: We consider a random connection model driven by a Poisson process and derive lower bounds for the critical intensities. The talk is based on joint work with Günter Last.

Dienstag, 5.7.2022, 15.45 Uhr (im Rahmen der AG Stochastik; Raum 2.58)

Dominik Pabst (KIT)

Die Euler-Charakteristik im Random Connection Model für höherdimensionale Simplizialkomplexe

Abstract: Graphen finden in vielen Bereichen der Wissenschaft Anwendung, um Strukturen mit Interaktionen zwischen den auftretenden Objekten zu modellieren. In der Stochastik bilden zufällige Graphen deshalb ein großes und anwendungsreiches Forschungsgebiet, wenn Strukturen von zu untersuchenden Objekten zufällig oder schwer vorhersagbar sind. Da in Graphen nur Interaktionen zwischen jeweils zwei Objekten modelliert werden, scheint es natürlich Verallgemeinerungen von zufälligen Graphen zu untersuchen, mit denen auch höhere Interaktionen dargestellt werden können.
Eine Möglichkeit hierfür bilden Simplizialkomplexe, bei denen es sich um vergleichbar einfache topologische Räume handelt, die schon durch ihre kombinatorische Struktur festgelegt sind. Beispielsweise verwendet die topologische Datenanalyse zufällige Simplizialkomplexe, die eine Verallgemeinerung des Gilbert-Graphen bilden.
In diesem Vortrag wird eine Verallgemeinerung des bekannten Random Connection Model von einem zufälligen Graph auf einen zufälligen Simplizialkomplex beliebiger endlicher Dimension vorgestellt und diskutiert. Neben den sogenannten Betti-Zahlen, die topologische Merkmale einer bestimmten Dimension zählen, ist die Euler-Charakteristik eine der am meisten untersuchten Größen bei der Analyse von Simplizialkomplexen. Diese lässt sich sowohl als alternierende Summe der Betti-Zahlen als auch als alternierende Summe der Simplexanzahlen schreiben, wobei vor allem letztere Darstellung sehr hilfreich ist.
Zunächst werden Momente erster und zweiter Ordnung der Verteilung der Euler-Charakteristik im verallgemeinerten Modell betrachtet, um danach die Frage nach der Grenzverteilung in verschiedenen natürlichen Grenzszenarien zu stellen und zu beantworten. Dabei wird es entscheidend sein, die Euler-Charakteristik als ein Funktional eines geeigneten Poissonprozesses zu beschreiben.

Donnerstag, 7.7.2022, 9.45 Uhr (Raum 3.069)

Goran Radunovic (University of Zagreb)

Applications of fractal zeta functions

Abstract: We will illustrate some of the interesting examples of applications of fractal zeta functions and complex dimensions such as obtaining fractal tube formulas and a Minkowski measurability criterion for particular classes of sets and applications to dynamics of parabolic diffeomorphisms.