Friday, October 25, 2013 (4-5pm):
Title: The theory of valuations and what it can do for you!
Speaker: Franz Schuster (Vienna University of Technology)
Abstract: A function $\phi$ defined on convex (or more general) compact sets in $\mathbf{R}^n$ and taking values in an Abelian semigroup is called a valuation if
$$\phi(K) + \phi(L) = \phi(K \cup L) + \phi(K \cap L)$$
whenever $K \cup L$ is convex. The theory of valuations on convex sets is a classical part of (convex) geometry with traditionally strong relations to integral geometry. However, in the last 15 years there have been dynamic developments that have led to enormous progress both conceptual and technical. Even the notion of valuation itself has evolved in different directions (e.g. \emph{finitely additive} smooth functionals on smooth manifolds or operators on function spaces) and the ties of valuation theory to other areas of both pure and applied mathematics have become much more diverse.
The purpose of this talk is to give a (by no means complete) survey on the recent developments in the theory of valuations and the new connections to other branches of mathematics, like differential geometry, harmonic analysis, and the theory of isoperimetric and analytic inequalities. To be more specific, we discuss the recent breakthrough in the structure theory of invariant valuations which in turn was the starting point for what is now called \emph{algebraic integral geometry}, we explain recent \emph{classifications of affine invariant notions of surface area} by Ludwig, Reitzner and Haberl, Parapatits, and present \emph{new geometric inequalities} for convex body valued valuations which strengthen several classical isoperimetric and analytic inequalities. The latter results also build a bridge to harmonic analysis as these valuations are closely related to Radon, cosine, and other convolution transforms.