Galyna Livshyts (Georgia Institute of Technology) will give an online mini-course on inequalities related to convexity and isoperimetry in high dimensions. The course is intended for Ukrainian students, both undergraduate and graduate, as well as everybody who is interested in related topics. The prerequisites are: solid knowledge of undergraduate mathematics courses related to the topic of the course. Lectures will be given in English.

In this mini-course, we discuss isoperimetric-type inequalities related to log-concave measures (and, in particular, convex bodies). We interpret several classical inequalities as concavity principles and employ the powerful idea of linearization to understand further isoperimetric-type questions about functions. We will discuss inequalities such as Prekopa-Leindler inequality, a generalized form of Log-Sobolev inequality, functional Ehrhard inequality, Brascamp-Lieb inequality and Bobkov’s inequality, as well as the phenomenon of the Gaussian isoperimetry. Several approaches to these topics will be discussed, and some novel variants will be outlined as well.

Galyna Livshyts is an associate professor of mathematics at the Georgia Institute of Technology (USA). She graduated from V. N. Karazin Kharkiv National University in 2010 and obtained doctorate degree at Kent State University (USA) in 2015. Galyna worked at several prominent research institutions including Simons Institute for the Theory of Computing and MSRI (Berkeley), Hausdorff Institute for Mathematics (Bonn), ICERM (Providence). Galyna is one of the leading experts in asymptotic geometric analysis and its applications to information theory, probability and convex geometry.