Webrelaunch 2020

AG Stochastische Geometrie (Sommersemester 2023)

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, 28.4.2023, 9:45 Uhr

Dominik Pabst (KIT)

Die Betti-Zahlen im stationären Random-Connection-Modell für höherdimensionale Simplizialkomplexe

Abstract: In der Literatur zu Simplizialkomplexen sind die sog. Betti-Zahlen eines der am meist betrachteten Funktionale. Anschaulich gesprochen zählt die $p$-te Betti-Zahl $\beta_p$, $p\in\mathbb{N}$, $p$-dimensionale Löcher eines topologischen Raums, während $\beta_0$ der Anzahl von Zusammenhangskomponenten entspricht. Bei den Betti-Zahlen handelt es sich um hochgradig nichttriviale Funktionale, die beispielsweise auf dem Raum der polykonvexen Mengen nicht additiv sind. Im Vortrag werden zentrale Grenzwertsätze für eine Klasse von Funktionalen, die die Betti-Zahlen enthalten, im stationären Random-Connection-Modell vorgestellt. Dabei werden die verwendeten Eigenschaften der Betti-Zahlen beleuchtet und diskutiert. Der Ansatz beruht auf den Methoden aus <1> für das Random-Connection-Modell als zufälliger Graph.

<1> V.H. Can and K.D. Trinh. Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals. Electronic Journal of Probability (2022).

Freitag, 19.5.2023, 9:45 Uhr (mit Online-Option)

Maxim Glyzhev (KIT)

Spektrale Eigenschaften fraktaler und nicht-fraktaler Trommeln

Freitag, 26.5.2023, 9:45 Uhr

Yogeshwaran Dhandapani (Indian Statistical Institute Bangalore)

Random polyhedral approximation of Strongly C-convex Domains

Abstract: Approximating Euclidean convex domains by polyhedra is a classical topic in convex geometry and random analogues of such questions are well investigated in stochastic geometry. While notions of convexity have been studied in several complex variables (SCV) however study of polyhedral approximations is relatively new. In this talk, we will present some recent progress on random polyhedral approximations in the context of several complex variables. We will discuss differences with optimal polyhedral approximation in SCV as well as contrast the SCV results with those in the Euclidean setting. The talk will be based on a joint work with Siva Athreya and Purvi Gupta.

Dienstag, 13.6.2023, 15:45 Uhr (Termin der AG Stochastik)

Olof Elias (Universität zu Köln)

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

Abstract: In this talk I will present results on percolative properties of (discrete and continuous) excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. I will introduce the models and explain the relation between all of these models. Given time I will also comment on some key ingredients of the proofs of our results.
Based on arXiv:2205.15289

Freitag, 16.6.2023, 9:45 Uhr

Thomas Wannerer (Universität Jena)

The Alesker-Fourier transform of valuations is the Fourier transform

Abstract: Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant continuous valuations that has the same functorial properties as the classical Fourier transform. In this talk, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker-Fourier transform. Two of these properties were previously only conjectured by Alesker.
Based on joint work with Dmitry Faifman.

Freitag, 23.6.2023, 9:45 Uhr

Ilya Molchanov (Universität Bern)

Generalised convexity and related limit theorems

Abstract: The standard convex hull of a subset of Euclidean space is defined as the intersection of all images (under the action of a group of rigid motions) of a half-space containing the given set. We propose a generalisation of this classical notion, that we call a (K,H)-hull, and which is obtained from the above construction by replacing a half-space with some other convex closed subset K of the Euclidean space, and a group of rigid motions by a subset H of the group of invertible affine transformations. The main emphasis is on limit theorems for generalised convex hulls of random samples from K. The talk is based on recent works with Alexander Marynych and Zakhar Kabluchko.

Freitag, 14.7.2023, 9:45 Uhr

Matthew Dickson (LMU München)

Mean-Field Critical Behaviour of Marked Random Connection Models

Abstract: Marked random connection models (RCMs) are random graph models that include both geometric behaviour from assigning vertices a spatial coordinate, and heterogeneity from assigning them `mark' coordinate in some probability space. Vertices are a Poisson point process with a given intensity $\lambda$, and edges then form independently with a probability depending on the spatial and mark coordinates of the two vertices. Under certain conditions, such models are known to exhibit a percolation phase transition: there exists a critical $\lambda_c$ such that for $\lambda<\lambda_c$ clusters are almost surely finite and for $\lambda>\lambda_c$ there is a positive probability that a given cluster is infinite. One may naturally be interested in how properties such as the expected cluster size behave as we approach this critical intensity. Here we first show how a 'triangle condition' ensures that such a critical exponent takes its mean-field value, and then how this triangle condition itself can be viewed as a 'high-dimensional' condition. This is based on joint work with Alejandro Caicedo and Markus Heydenreich.

Dienstag, 18.7.2023, 15:45 Uhr (Raum 2.58, Geb. 20.30)

Ali Khezeli (INRIA Paris)

On the Existence of Balancing Allocations and Factor Point Processes

Abstract: Let Φ and Ψ be ergodic stationary random measures on Rd. The study of balancing allocations and balancing
transport kernels between Φ and Ψ has been of great interest in the last two decades. This was started by
the novel work (Hoffman, Holroyd and Peres, 2005), which provided a generalization of the stable marriage
algorithm and constructed a balancing allocation between the Lebesgue measure and the Poisson point pro-
cess. For the existence of factor allocations in general, it is necessary that Φ and Ψ have the same intensity,
but no simple necessary and sufficient condition is known yet. Recently, (Last, Thorisson, 2023) provided the
sufficient condition that Φ is diffuse (i.e., has no atoms) and there exists a point process as a factor of (Φ, Ψ).
It was conjectured in (Khezeli, 2016) that there always exists a factor point process provided that (Φ, Ψ) has
no nontrivial symmetry a.s. In this talk, we prove this conjecture by using the results of the theory of Borel
equivalence relations. Meanwhile, we will provide a quick introduction into this theory and its applications in
probability theory.

Based on joint work with Samuel Mellick, available at https://arxiv.org/abs/2303.05137