Additive and Combinatorial Problems in Groups

This project is jointly funded by the Combinatorics program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Algebra and Number Theory program. In this project, the PI aims to study certain problems in groups. A group is a mathematical structure in which we can add and subtract two elements, just like in the integers. Groups are fundamental objects in mathematics and ubiquitous in real life; for example, the numbers on a 12-hour clock form a group of 12 elements. The addressed problems are of combinatorial nature; for example, if we take any subset of a group, then as long as the subset is ''large'' in some sense (without any assumption on its structure), we predict that some pattern will emerge. Graduate students will be trained as part of this proposal. The PI will also be involved in K-12 outreach. The groups studied in this project are: (1) The integers and their polynomial ring analogs, for which techniques from number theory apply, (2) finite groups, including vector spaces over finite fields, for which techniques from algebra apply, and (3) more general amenable groups, for which techniques from Fourier analysis and ergodic theory apply. Problems studied in this proposal include the following: If a subset A of a group G is large in some sense (e.g. A has positive density, or comes from a finite partition of G), then can one find structures in sumsets such as A-A and A+A-A? This question is intimately related to various notions of recurrence in ergodic theory and topological dynamics. If A is a sparse set, e.g. a subset of the primes, then this question will lead to patterns hitherto unexplored in the primes. The PI will also study other additive problems such as additive bases, essential components, and exponential sums using various techniques. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.