The workshop will cover advances in harmonic analysis that have been linked to geometric combinatorial problems. Specific examples include the Kakeya problem, the restriction problem, continuous and finite field distance set problems, and continuous and finite field variants of the sum-product theory in additive combinatorics. This workshop will bring together experts in Fourier analysis, geometric measure theory, and combinatorics of finite fields.
The classical Kakeya problem asks how small a set (called a Kakeya set) can be in n-dimensional Euclidean space if it contains a unit line segment in every direction. It is conjectured that for any reasonable definition of fractal dimension, such a set should have dimension n. This is known to be the case in two dimensions, but in three and higher dimensions only weaker partial results are known. The Kakeya problem has been shown (through the work of Fefferman, Bourgain, and others) to be closely related to central problems in Fourier analysis such as Fourier multipliers and restriction estimates, but on the other hand, it involves combinatorics of union of lines that can be viewed as formally analogous to point-line incidence theory. Researchers (notably Wolff) went on to establish and exploit further connections of this type, in the process introducing harmonic analysts to combinatorial geometric problems.
More recently, algebraic methods first introduced in the context of the Kakeya problem (Dvir’s polynomial method in finite fields, and Guth’s endpoint result for the multilinear Kakeya problem based on the polynomial ham sandwich theorem) have also led to dramatic results in incidence geometry. Some geometric aspects of this work inspired the work of Guth and Katz on the joints problem and subsequently on the distance set problem. More recent work of Bourgain. It is possible that some of these ideas could be used in studying the conventional Kakeya problem.
The workshop will also examine questions related to distance sets and sum-product phenomena in continuous and finite field settings. While there has been substantial progress on the corresponding discrete problems, the continuous variants remain less understood. We hope that the new algebraic methods can shed more light on these questions; for instance, a major open problem is whether the sum-product phenomenon can ultimately be understood in an algebraic way. The workshop will be an excellent opportunity for experts who study these problems in different settings to interact and learn from one another.
This workshop will include a poster session; a request for posters will be sent to registered participants in advance of the workshop.
(Massachusetts Institute of Technology)
Alex Iosevich (University of Rochester)
Nets Katz, Chair (California Institute of Technology)
Izabella Laba (University of British Columbia)