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AG Stochastische Geometrie (Wintersemester 2016/17)

Studierende und Gäste sind jederzeit willkommen. Wenn nicht explizit anders unten angegeben, finden alle Vorträge im Seminarraum 2.58 im Mathebau (Gebäude 20.30) statt.

Seminar: Freitag 9:45-11:15 SR 2.58
Seminarleitung Prof. Dr. Günter Last
Sprechstunde: nach Vereinbarung
Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30)
Email: guenter.last@kit.edu
Seminarleitung Prof. Dr. Daniel Hug
Sprechstunde: Nach Vereinbarung.
Zimmer 2.051 Kollegiengebäude Mathematik (20.30)
Email: daniel.hug@kit.edu

Freitag, 21.10.2016

9.45 Uhr Daniel Hug

Integral geometry of tensorial curvature measures

(based on joint work with Jan Weis)

Freitag, 28.10.2016

9.45 Uhr Jakub Večeřa (Charles University, Prag)

Part I: Estimation of parameters in a planar segment process with a biological application

(joint work in progress with Viktor Benes, Benjamin Eltzner, Carina Wollnik, Florian
Rehfeldt, Veronika Kralova and Stephan Huckemann)

Abstract: We deal with modeling of segment systems in the plane by means of random Poisson processes. The model presented involves the length distribution and distances from the centre of the cell. Estimation of parameters of the models is suggested based on Takacz-Fiksel method. The method is tested first using simulated data. Further the real data from fluorescence imaging of stress fibres in mesenchymal human stem cells are evaluated. In order to classify three groups of cells further characteristics are involved like entropy of the direction distribution and frequency of intersections. Finally the most chaotic group of cells is fitted to the second segment process model with quite satisfactory results.

Part II: Asymptotic distribution of Gibbsian statistics with increasing window

(joint work in progress with Günter Last)

Abstract: We deal with Gibbs processes defined by Papangelou conditional intensity in form of exponential U-statistic. First we start with example of segment process with intensity involving the number of intersections and calculate asymptotic expectation and variance of the number of segments.

Freitag, 11.11.2016

9.45 Uhr Felix Herold

Konzentrationsungleichungen für Poissonprozesse

Abstract: Wir beschäftigen uns mit der Entwicklung von Konzentrationsungleichungen für Poissonprozesse. Hierfür wird die von S. Bachmann und G. Peccati entwickelte kombinierte Entropie-Ungleichung verwendet. Im Anschluss werden die gewonnenen Ergebnisse auf zufällige geometrische Graphen angewendet um Aussagen über die Verteilung relevanter Größen zu treffen. Der Vortrag basiert auf der Masterarbeit des Vortragenden.

Freitag, 18.11.2016

9.45 Uhr Wolfgang Weil

Convex bodies and Grassmann measures

Abstract: In recent years flag measures have been studied as local descriptors of convex bodies. Here, we consider certain image measures on the Grassmann manifold and give corresponding uniqueness results. We also discuss the role of Grassmann measures in the description of even valuations.

Freitag, 25.11.2016

9.45 Uhr Günter Last

Einbettungen Brownscher Exkursionen

Freitag, 2.12.2016

9.45 Uhr Moritz Otto

Extremale Eigenschaften Poissonscher Mosaike

Abstract: Es wird ein allgemeines Vorgehen präsentiert, um das Extremverhalten verschiedener Charakteristika der Zellen in stationären Poissonschen Mosaiken zu untersuchen, die eine starke Mischungseigenschaft erfüllen. Das Verfahren beruht auf einer Anwendung der Chen-Stein-Methode zur Poissonapproximation in einem geeigneten Abhängigkeitsgraphen (vgl. [1]). In Anwendungsbeispielen wird etwa das asymptotische Verhalten des minimalen Umkreisradius der Zellen im Delaunay-Mosaik in einem endlichen Beobachtungsfenster studiert. Der Vortrag basiert auf der Arbeit [2].

  1. R. Arratia, L. Goldstein, L. Gordon, Poisson approximation and the Chen-Stein method, Statistical Science (1990): 403-424.
  2. N. Chenavier, A general study of extremes of stationary tessellations with examples, Stochastic Processes and their Applications 124.9 (2014): 2917-2953.

Dienstag, 6.12.2016 (im Rahmen der AG Stochastik, Raum 2.059)

15.45 Uhr Gerd Schröder-Turk (Murdoch University Perth, Australia)

Hyperuniformisation by Lloyd's algorithm for Centroidal Voronoi diagrams (?)

Mittwoch, 7.12. - Freitag, 9.12.2016

GPSRS Workshop, Hohenwart Forum, Pforzheim

Dienstag, 13.12.2016 (im Rahmen der AG Stochastik, Raum 2.059)

15.45 Uhr Mathew Penrose (University of Bath, UK)

Optimal cuts of random geometric graphs

Freitag, 16.12.2016

9.45 Uhr Steffen Winter

Lokalisierung von Minkowski-Inhalten

Abstract: In den letzten Jahren wurde gezeigt, dass die Minkowski-Inhalte beschränkter Mengen im R^n als Grenzwerte der Oberflächeninhalte der zugehörigen Parallelmengen charakterisiert werden können. Im Vortrag werden wir Lokalisierungen solcher Resultate diskutieren. Sie beruhen wesentlich auf der Lokalisierung und Verfeinerung eines Satzes von Stacho, der den Oberflächeninhalt von Parallelmengen in Beziehung setzt zur Ableitung ihres Volumens.

Dienstag, 10.1.2017 (im Rahmen der AG Stochastik, Raum 2.059)

15.45 Uhr Pierre Calka (Université de Rouen, France)

The typical Poisson-Voronoi cell around an isolated nucleus

Freitag, 20.1.2017

9.45 Uhr Michael Klatt

Characterization of Maximally Random Jammed Sphere Packings

Abstract: Packings of hard, impenetrable spheres are useful models of granular media, low-temperature states of matter, suspensions and biological systems. What is the structure of the most disordered among all mechanical stable packings? A unique property of this maximally random jammed (MRJ) state is that despite the local disorder, similar to a liquid, there is a hidden long-range order that anomalously suppresses density fluctuations on large length scales, more like in a crystalline solid. We describe both the local and global structure of such disordered sphere packings using a variety of different structural characteristics. For example, we introduced Voronoi correlation functions to characterize the structure of sphere packings across length scales. By comparing the structure of MRJ packings to common models of disordered materials, our shape analysis helps to distinguish, despite seemingly similar features in all of those systems, their distinctly different structure. Moreover, these structural characteristics are related to a host of different effective physical behavior, for example, flow or diffusion in these systems as well as their elastic moduli or electromagnetic properties. We thus link problems from material science, chemistry, physics, mathematics and biology.

Freitag, 27.1.2017

9.45 Uhr Daniel Hug

Exkursionen in planaren zufälligen Mosaiken

Freitag, 3.2.2017

9.45 Uhr Thomas Richthammer (Universität Paderborn)

Non-rigidity of 2D Gibbisan point processes

Abstract: Many 2D Gibbsian point processes are believed to show some sort of crystalline phase: If the temperature is sufficiently low (or the density sufficiently high) the points arrange themselves into a regular pattern, which is characterized by long-range correlations. So far there is no rigorous proof of this phenomenon. We show that the expected regular pattern can not be too rigid: In a system of size n, positions of points near the center of the system fluctuate by at least a constant times (log n)^(1/2). Our result holds for fairly general interaction potentials (including all interesting examples of interacting particle systems we know of) and arbitrary values of temperature and particle density. (Joint work with Michael Fiedler.)

Dienstag, 14.2.2017 (im Rahmen der AG Stochastik, Raum 2.059)

15.45 Uhr Sebastian Kapfer (Universität Erlangen)

Geometric properties of extreme Poisson-Voronoi cells

Abstract: Many exact results exist on the geometric properties of Poisson-Voronoi cells, such as the mean volume, surface area and integral mean curvature of the typical cell. No exact formula is known, however, for the full distribution of these quantities, which empirically resembles a generalized Gamma distribution. The talk will present ongoing work on how to use the Wang-Landau Monte Carlo method, originally devised for problems in statistical mechanics, to numerically explore the tails of the distribution of the Minkowski functionals. The method can be generalized to other properties of Poisson-Voronoi cells, and other point processes.