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AG Stochastische Geometrie (Wintersemester 2022/23)

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
Seminar: Freitag 9:45-11:15 20.30 SR 2.58
Lehrende
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, 17.2.2023, 9:45 Uhr

Babette Picker (KIT)

t.b.a.


Freitag, 10.2.2023, 9:45 Uhr

Fabian Mussnig (TU Wien)

Integral geometry on convex functions

Abstract: We introduce a theory of valuations on convex functions of n variables and explain how it relates to valuations on convex bodies in n-dimensional as well as n+1-dimensional space. We will use properties of mixed Monge-Ampère measures to obtain an improved version of the recently established Cauchy-Kubota formulas for convex functions, which generalize their classical counterparts for convex bodies. If time permits, we also discuss a functional version of the additive kinematic formulas.

Based on joint work in progress with Jacopo Ulivelli.


Freitag, 3.2.2023, 9:45 Uhr

Daniel Hug (KIT)

Hyperbolic Poisson flats


Dienstag, 24.1.2023, 15:45 Uhr (Zeit der AG Stochastik), Raum 2.59 (Geb. 20.30)

Günter Last (KIT)

Testing hyperunifomity

Abstract: Hyperuniform structures can be both isotropic like a liquid and homogeneous like a crystal. In that sense, they represent a new state of matter, and they have attracted a quickly growing attention in physics, biology and material science. In mathematical terms, hyperuniformity of a stationary point process in Euclidean space can be described as an anomalous suppression of large-scale density fluctuations. This means that the variance of the number of points in a large ball grows more slowly than its volume. By now many different point processes with these exciting properties have been discovered, among them perturbed lattices, quasi-crystals, dependent thinnings and the Ginibre process.

We devise the first rigorous significance test for hyperuniformity with sensitive results, even for a single sample. Our test is based on the asymptotic behavior of the so-called scattering intensity, which is the squared norm of the empirical Fourier transform of the (localized) point process, suitably normalized. Theoretical results as well as simulations show that this behavior applies to a wide range of stationary point processes. We can then use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.

This talk is based on joint work with Michael Klatt and Norbert Henze.


Freitag, 13.1.2023, 9:45 Uhr

Tobias Hartnick (KIT)

Hyperuniformity and non-hyperuniformity of quasicrystals

Abstract: We introduce a class of stationary jammed hardcore point processes in \R^n known as transverse point processes. Among them are cut-and-project processes, which serve as the main mathematical models of quasicrystals. We then discuss the question of hyperuniformity for such point processes.
We will discuss the spectral formulation of hyperuniformity and its physical meaning in terms of diffraction; we then present various diffraction formulas for transverse point processes and compare them to diffraction formulas for lattices and Poisson processes. This comparison shows that, contrary to popular belief, quasicrystals do not need to be hyperuniform. Whether they are hyperuniform or not does not depend on “geometric” aspects of the quasicrystal, but rather on “number theoretic” aspects. We will give examples of anti-hyperuniform quasicrystals and show that “most” quasicrystals (in a precise sense) are hyperuniform.
Time permitting we will discuss also related results concerning number rigidity and stealth of quasicrystals.


Dienstag, 20.12.2022, 15:45 Uhr (Zeit der AG Stochastik)

Steffen Winter (KIT)

Regularity properties of local parallel volume and surface area

Abstract: We show that at differentiability points r_0>0 of the volume function of a compact set A\subset\mathbb{R}^d, the surface area measures of r-parallel sets of A converge weakly to that of the r_0-parallel set as r \to r_0. We also address the structure of the set of non-differentiability points of the volume function of compact sets. In dimension 1 and 2 we give a complete characterization of the subsets of real numbers that occur as sets of non-differentiability points.


Freitag, 9.12.2022, 9:45 Uhr

Ferenc Fodor (University of Szeged)

Random approximation in generalized convexity

Abstract: We present a collection of results concerning generalizations of random polytope models in which the convex hull is generated by intersections of translates of a fixed convex set. We describe asymptotic formulas for expectations, bounds on variances for various quantities, and central limit theorems via Stein's method.


Freitag, 2.12.2022, 9:45 Uhr

David Willimzig (KIT)

Boxapproximation von Fraktalen


Freitag, 18.11.2022, 9:45 Uhr

Paul Reichert (KIT)

Gleichheitsfälle der Alexandrov-Fenchel-Ungleichung


Freitag, 11.11.2022, 10:00 Uhr

Günter Last (KIT)

Poisson hull estimators