Semyon Alesker: Valuations and convex geometry
Abstract: Valuations are finitely additive measures on convex bodies. The subject takes its origins in convex and integral geometry. In the last two decades a number of new structures on the space of translation invariant smooth valuations were discovered, notably product, convolution, and Fourier type transform. They will be described in the lectures. A number of new results and conjectures are formulated in terms of these structures. In particular, we will discuss Kotrbaty's conjecture on Hodge-Riemann type bilinear relations on valuations. Known special cases of this conjecture lead to old (e.g. Alexandrov-Fenchel) and new inequalities for mixed volumes of convex bodies, some of them will be discussed in the lectures.
Alina Stancu: Applications of partial differential equations to convex geometry
Abstract: The minicourse will start by introducing a few standard parametrizations of convex bodies and their connections to the curvature measure of the bodies’ boundaries. Particular attention will be devoted to convex bodies with smooth boundaries, the parametrization by the inverse of the Gauss map, thus the support function and the derivation of the Gauss curvature in spherical coordinates. This leads to the first topic as the problem of prescribing curvature measures as Monge-Ampère type equations on the sphere and other such elliptic type partial differential equations. The second topic looks at curvature flows as parabolic partial differential equations with the aim of proving geometric inequalities, or as an alternate approach of proving existence of convex bodies with prescribed curvature type conditions. Each topic will be covered in two 50-minutes lectures for a total of 4 lectures. Only basic knowledge of differential equations is required.
Vladyslav Yaskin: Harmonic Analysis Methods in Geometric Tomography
Abstract: One of the typical questions in geometric tomography is whether one can uniquely determine a convex body using lower-dimensional data, such as areas of sections or projections. We will discuss some problems of this type and show how they can be solved using tools of harmonic analysis.