Dmytro Bilyk (University of Minnesota) will give an online mini-course on geometric discrepancy theory. The course is intended for undergraduate or graduate students with some basic knowledge of analysis, as well as for anyone interested in related topics. The problems in this field are often easy to state, though many of them remain challenging or even open. Lectures will be given in English.

The main goal of discrepancy theory is to understand how well continuous objects can be approximated by discrete point configurations in different geometric contexts. It goes back to the fundamental work of Weyl on the uniform distribution of sequences modulo one (i.e. on the unit circle) and a conjecture of van der Corput that the discrepancy of any such sequence cannot stay bounded as the number of points goes to infinity (irregularities of distribution). We shall discuss the origins and applications of the theory, various constructions of well distributed (low-discrepancy) point sets (lattices, van der Corput digit-reversing set, jittered sampling), bounds for various geometric discrepancies (axis-parallel boxes, spherical caps etc) with a particular emphasis on the methods of analysis, as well as connections to some other areas (energy minimization, combinatorial discrepancy, etc).

Dmytro Bilyk is a Professor in the Department of Mathematics at the University of Minnesota. He obtained his Master’s degree from Kharkiv National University and earned his PhD in Mathematics from the University of Missouri–Columbia (USA). Prof. Bilyk’s research focuses on uniform distribution, discrepancy theory, harmonic and functional analysis, number theory, discrete geometry, and geometric measure theory.

Teaching Assistant: Maryna Manskova (Graz University of Technology)
Grader: Anna Goncharuk (V. N. Karazin Kharkiv National University)