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.

The most famous classical isoperimetric inequality roughly states that among plane figures with the boundary of a given length the largest area is achieved by a disk. It was known already in Ancient Greece, but was proven rigorously only in the 19th century. Since then many generalizations were proven and similar type inequalities discovered in several branches of mathematics, including measure theory, smooth manifolds, and graph theory. In this mini-course, we discuss isoperimetric-type inequalities (such as Log-Sobolev inequality and Gaussian isoperimetric inequality) 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. Several approaches to these topics and some novel variants will be outlined.

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.