AG Stochastische Geometrie (Sommersemester 2023)
- Dozent*in: Prof. Dr. Daniel Hug, Prof. Dr. Günter Last
- Veranstaltungen: Seminar (0175700)
- Semesterwochenstunden: 2
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).
Termine | ||
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Seminar: | Freitag 9:45-11:15 | 20.30 SR 2.58 |
Lehrende | ||
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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 -te Betti-Zahl
,
,
-dimensionale Löcher eines topologischen Raums, während
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)
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 , 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
such that for
clusters are almost surely finite and for
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