Dienstag 14 Uhr c.t., Raum 404
TITLE: Minkowski Valuations and the Special Linear Group
ABSTRACT: The systematic study of the space of valuations, i.e. finitely additive maps on convex bodies, that are compatible with some subgroup of linear transformations has its origins in the work of Hadwiger. Through the seminal work of Ludwig, convex-body-valued valuations that intertwine the special linear group have become the focus of recent research. In this talk, I will give a survey on characterizations of these Minkowski valuations.
TITLE: Eine Verallgemeinerung des Gromovschen Satzes über fast flache Mannigfaltigkeiten
26.11.2013 Dr. Florent Balacheff (Universite Lille)
TITLE: Systolic contact geometry
ABSTRACT: Systolic geometry involves a lot of ingredients like algebraic topology, metric geometry or conformal techniques for instance.
In this talk, after briefly recall part of this background, we will explain why contact geometry is a natural setting for the study of isosystolic inequalities and the new perspectives it offers. This is joined work with J.C. Alvarez Paiva and K. Tzanev.
17.12.2013 Sven Jarohs, Frankfurt
Titel: Overdetermined problems involving the fractional Laplacian.
Abstract: In 1971 Serrin proved that if there is a positive solution to the Poissonproblem with constant nonnegative right-hand side and Dirichlet boundary conditions in a domain Omega, such that the outernormal derivative of u along the boundary of Omega is constant, then Omega must be a ball. Several works since then have been devoted to the study of this kind of overdetermined problems. I will show how this result and its generalization can be extended to problems involving the fractional Laplacian. After a short introduction to the fractional Laplacian, I will present a version of Hopf's Lemma for weak continuous solutions for linear problems involving that operator. This especially induces the strong maximum principle. Based on this I will give the main geometric ideas of the proof and reason why the nonlocal structure enables us to a more general result than in the local case. My talk is based on a joint work with Moustapha Fall.
28.01.2014 Dr. Astrid Berg (TU Wien)
Title: Log-Concavity Properties of Minkowski Valuations (joint work with Lukas Parapatits, Franz E. Schuster, Manuel Weberndorfer)
Abstract: The famous Brunn-Minkowski inequality expresses the log-concavity of the volume functional. We use a recent result of Wannerer and Schuster to extend this classical inequality and prove log-concavity properties of intrinsic volumes of rigid motion compatible Minkowski valuations. We also obtain new Orlicz-Brunn-Minkowski inequalities for such valuations. These inequalities extend previous results by Alesker, Bernig, Lutwak, Parapatits and Schuster.