Title: Crofton Measures for O(p,q)-Invariant Valuations
Abstract: We will briefly recall the definitions and some properties of generalized valuations on convex bodies. Then I’ll describe the space of generalized valuations invariant under the action of an indefinite orthogonal group, which is the analogous result to Hadwiger’s classification of valuations invariant under the Euclidean group of motions. I will then focus on how to construct Crofton formulas for these generalized valuations. This is based on a joint ongoing work with Andreas Bernig.
Title: Integral Geometry of the Sphere
Abstract: Although the kinematic formulas for the sphere are in some sense entirely classical, two striking new viewpoints on the subject have arisen recently, both carrying potentially enormous consequences. The first arises from the Alesker theory of valuations on manifolds, and gives rise to a new algebraic structure associated to general Riemannian manifolds. This part of the talk is a progress report on an ongoing project in collaboration with Thomas Wannerer. The second, due to M. McCoy, J. Tropp, et al., uses fundamental ideas from statistics to study convex cones, thought of as cones over convex subsets of the sphere. Through the mechanism of concentration of measure in large dimensions, their work appears to be of central importance to contemporary issues in data compression. This second part of the talk will be essentially a book report on some of their recent papers.
Title: Valuations on Lattice Polytopes
Abstract: Lattice polytopes are convex hulls of finitely many points with integer coordinates in R^n. A function z from a family F of subsets of R^n with values in an abelian group (or more generally, an abelian monoid) is a valuation if
z(P)+z(Q)=z(P\cup Q)+z(P\cap Q)
whenever P, Q, P\cupQ, P\capQ are in F and z(\emptyset)=0. The classification of real-valued invariant valuations on lattice polytopes by Betke & Kneser is classical (and will be recalled). It establishes a characterization of the coefficients of the Ehrhart polynomial.
Building on this, a classification is established of Minkowski valuations on lattice polytopes, that is valuations with values in the abelian semi-group of compact convex sets with Minkowski or vector addition. For valuations that intertwine the special linear group over the integers and are translation invariant, we obtain in the contravariant case that the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined discrete Steiner point.
(Joint work with Kar’oly J.Boeroeczky)
Title: Delocalization of Eigenvectors of Random Matrices
Abstract: A random unit vector is delocalized if it is more or less uniformly distributed on the unit sphere. This intuitive definition can be quantified by saying that with high probability, all coordinates of the vector are small. The conjecture that an eigenvector of a Hermitian random matrix is delocalized has been open for a long time, and was proved recently by Erdos-Schlein-Yau and Tao-Vu using spectral methods. We introduce a different, geometric approach to delocalization, which allows to establish it for general non-Hermitian random matrices with independent entries.
Joint work with Roman Vershynin.
Title: Diametric Completions in Normed Spaces
Abstract: A nonempty bounded subset M of a normed space X is (diametrically) complete if it cannot be enlarged without increasing its diameter. In a Euclidean space, the complete sets are precisely the convex bodies of constant width. A completion of M is a complete set of the same diameter as M that contains M. By a completion mapping we understand a mapping that associates with each nonempty bounded subset M a completion of M. The talk investigates some specific completion mappings. (1) The Maehara completion. (2) The generalized B uckner completion. (3) An application to bodies of constant width. (4) Completions in C(K) spaces.
(Based on joint works with Jose Pedro Moreno and with Imre Barany)
Title: Log-Concavity Properties of Minkowski Valuations
Abstract: In this talk I will discuss log-concavity properties of homogeneous rigid motion compatible Minkowski valuations. These extend the classical Brunn-Minkowski inequalities for intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold.
(Joint work with A. Berg, L. Parapatits, and M. Weberndorfer)
Title: Minkowski Valuations and Generalized Valuations
Abstract: In convex geometry, a valuation is a finitely additive function on the space of convex bodies. As a generalization of measure, valuations have long played a central role in convex and discrete geometry. In this talk, I will focus on valuations taking values again in the space of convex bodies. These valuations were first investigated by Schneider in the 1970s with the goal of describing them completely. Since then a number of descriptions, sometimes even complete characterizations of Minkowski valuations have been established. I will present a new convolution representation of continuous translation-invariant and SO(n)-equivariant Minkowski valuations. The proof is based on new techniques involving translation-invariant generalized valuations which were only recently introduced by Alesker and Faifman.
This is joint work with Franz Schuster.