Discrepancy theory studies various forms of the following question: How well can uniform distribution be approximated by a discrete set? This project is devoted to exploring fundamental problems and conjectures of discrepancy theory through the prism of harmonic and functional analysis. The strategic goals of the project are two-fold: to infuse discrepancy theory with new methods stemming from analysis, as well as to enrich the field of analysis with new problems and ideas. The program includes a wide range of problems and topics: understanding the precise behavior of the irregularity of distributions in various function spaces, producing new constructions of various low-discrepancy sets, investigating the correlation between discrepancy estimates and geometry, as well as numerous connections of the field to other areas of mathematics, such as probability and approximation theory.
The ideas and questions of discrepancy theory are strongly interlaced with other areas of mathematics: combinatorics, geometry, number theory, probability, approximation theory. Moreover, this field is closely connected to computational mathematics, namely, the methods of numerical integration, and thus, it has direct applications in science, engineering, finance etc. The scientific output of this project will expand and deepen the understanding of the underlying phenomena of discrepancy theory and its relations to other branches of mathematics, as well as yield important applications to science and technology. The results will be broadly disseminated through publications in high-level journals, presentations in conferences, seminars, and colloquia, active collaborations with researchers around the world, a number of educational and mentoring activities.
