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research interest, n. - A research area where I have dabbled extensively enough to write one or more papers and that I try to keep up with. This list focuses on questions I have actually worked on. While I have provided some context and history, this is in no way intended to be a complete overview of any area of mathematics. If you're interested in any of the questions below or my contributions to them, I have most of my research and expository papers available here.
Geometric measure theory: projections and distancesHarmonic analysis offers an effective and quantitative approach to many beautiful questions in geometric measure theory. One such question, of interest in both ergodic theory and complex analysis (theory of "analytic capacity"), is as follows. Let E be a self-similar planar set of Hausdorff dimension 1, so that E is a union of L disjoint copies of itself, each rescaled by a factor of 1/L. (There are many ways to construct such sets, for instance by scaling the Sierpinski triangle construction a little bit differently.) What can we say about the linear projections of E? By an old theorem of Besicovitch, almost every projection of E has length 0; the hard question is to prove quantitative estimates (in terms of n) on the average length of projections of the n-th iteration of the set. The harmonic-analytic approach introduced by Nazarov, Peres and Volberg (2008) for the "4-corner set" made it possible, for the first time, to prove power-type bounds for this problem. My work, first with Kelan Zhai and then with Matthew Bond and Alexander Volberg, has extended this to more general classes of sets by introducing number-theoretic methods. In particular, we have made connections to tilings of integers (a subject I'd worked on in a different context), vanishing sums of roots of unity (a classic question in combinatorial/algebraic number theory), and some aspects of additive combinatorics. In a joint paper with Bond and Joshua Zahl, we also prove quantitative bounds for the "visibility problem", where instead of averages of linear projections we consider radial projections from a single point.Another family of problems concerns distance sets. The prototype question is due to Falconer: if a set E in R^{n} has Hausdorff dimension s, what can we say about the dimension of the set D(E) of all possible distances between pairs of points in E? The best results so far, due to Wolff and Erdogan, have been obtained by Fourier-analytic methods related to the Kakeya-restriction group of problems (see below). Together with Alex Iosevich and Sergei Konyagin, I have worked on variants of this problem for non-Euclidean distance functions. Geometry of sparse setsI'm developing a program of transferring results and methods of additive combinatorics to the continuous setting of fractal sets in Euclidean spaces. For example, Roth's theorem in additive number theory states that a "dense" set of positive integers (containing a positive proportion of N, in a sense that can be made precise) must contain 3-term arithmetic progressions. By the transference arguments of Green and Tao, the same is true for sets (such as the primes) which are not dense but are sufficiently randomly distributed. Malabika Pramanik and I have proved a continuous analogue of the latter result: if a fractal set on the line has Hausdorff dimension close enough to 1, and if it satisfies a Fourier-analytic "randomness" condition, it must contain non-trivial 3-term arithmetic progressions. (The analogous problem for longer progressions remains open.) In a follow-up paper with Pramanik and our graduate student Vincent Chan, we proved a multidimensional analogue of this results, namely that "random" (in a Fourier-analytic sense) sets in n dimensions that have Hausdorff dimension close enough to n must contain certain patterns prescribed by systems of matrices.Another joint result with Pramanik is a differentiation theorem for sparse sets, answering an old question of Aversa and Preiss: there is a fractal set E on the line, of dimension 1 but Lebesgue measure 0, such that E differentiates all functions f in L^{p}(R), p>1. More precisely, consider the behaviour of the averages of f (with respect to an appropriate singular measure) over the sets x + tE as t approaches 0. We prove that E can be constructed so that these averages converge to f(x) for almost every x. Differentiation theorems for zero measure sets had been known previously in higher dimensions, for example the spherical maximal theorem of Stein and Bourgain, but not in dimension 1. Many related questions remain open, such as the best range of p for which a set of given dimension s can differentiate L^{p} functions. Additive combinatoricsThe harmonic-analytic side of this new and exciting area of research often focuses on finding good quantitative answers to questions in additive number theory. For example, what is the largest possible size of a subset A of {1,2,...,N} that does not contain a 3-term arithmetic progression? (A recent result of Sanders says that A can have size at most N/log(N), up to log(log(N)) factors.) How does the size of the sumset A+A depend on the "structure" of A? What else can we say about sumsets, for example what types of patterns (long arithmetic progressions, square differences, etc.) must they contain? I have worked on quantitative results concerning sumsets, especially finding long arithmetic progressions in sumsets and related results concerning "almost periodicity". This includes my papers with Mariah Hamel, Sergei Konyagin, and with Ernie Croot and Olof Sisask.Kakeya sets, restriction estimates, and related questionsA Besicovitch set is a subset of R^{n} which contains a unit line segment in each direction. It is known (due to Besicovitch) that such sets may have n-dimensional measure zero. Can they be even smaller, in the sense that their Hausdorff dimension is strictly less than n? The conjecture is that this is not possible; this has been proved for n=2 but remains open in higher dimensions. Attempts to resolve the question have involved a variety of methods from geometry, combinatorics, additive number theory and more. For historical reasons, Besicovitch sets are also often called "Kakeya sets". (See here for more information.)Ever since their invention, Kakeya sets of measure zero have been used as archetypal "bad sets" in harmonic analysis, providing examples of Fourier-analytic operators behaving as badly as they can. Kakeya sets of Hausdorff dimension strictly less than n would be a harmonic analyst's absolute nightmare, disproving several longstanding major conjectures including restriction, Bochner-Riesz and local smoothing conjectures. In the converse direction, partial results on Kakeya sets (such as lower bounds on their dimension) can be used to prove partial results on these conjectures, for example restriction estimates in certain ranges of exponents. My work in this area has concerned lower bounds on the dimension of Kakeya sets (joint work with Nets Katz and Terry Tao) and local smoothing inequalities (joint with Tom Wolff and Malabika Pramanik). While the classical setting for restriction theorems concerns nice smooth surfaces such as spheres or cones, there is a more recent line of work extending such estimates to other, less intuitively geometric objects. Mockenhaupt (2000) proved that it is possible to have restriction estimates for certain types of fractal sets, including fractal subsets of the line. Bourgain (1990s) proved the first restriction-type estimates for integer sets, a result that was re-worked by Green in 2003 (and later by Green and Tao) and played a significant role in additive combinatorics. This work has also changed the way we think about "traditional" restriction results: for example, the old gospel had it that the sphere admits restriction estimates because of its smoothness and curvature, whereas the more modern point of view is to focus on issues of "randomness" (in an appropriate sense) and additive structure. In a paper with my graduate student Kyle Hambrook, we proved that the range of exponents in Mockenhaupt's theorem for fractals in 1 dimension is sharp, by constructing a 1-dimensional analogue of the geometric "Knapp example" used for a similar purpose in higher dimension. Follow-up work is in progress. Translational tilings and spectral setsThere is a number of results relating translational tilings of Euclidean spaces to Fourier analysis. In particular, a conjecture due to Fuglede (1974) states that a set E tiles R^{n} by translations if and only if L^{2}(E) admits an orthogonal basis consisting of exponential functions; this has now been disproved in dimensions 3 and higher, in both directions (due to Tao, Kolountzakis, Matolcsi, Farkas and Mora), but remains open in dimensions 1 and 2. The appeal and entertainment value of this question lie in its somewhat unexpected connections to many different areas of mathematics, from wavelets to number theory, combinatorics and algebra. My work in this area includes joint papers with Mihalis Kolountzakis, Sergei Konyagin and Yang Wang.Incidence geometryThis set of questions concerns counting incidences between points and geometric objects, such as lines, curves or surfaces, in Euclidean spaces. For instance, Erdos's distance set conjecture - now a theorem of Guth and Katz - asserts that for any configuration of n points in the plane, there must be at least C_{p}n^{p} distinct distances between themi for any p<1. The closely related unit distance conjecture, that for any p>1 such a configuration can contain at most O(n^{p}) pairs x, x' such that |x-x'|=1, remains open and can be stated in terms of a bound on the number of incidences between points and circles in the plane. Many other questions of this type, especially in higher dimensions, are not at all well understood. I have worked on several variants of incidence and distance set problems - this includes my papers with Alex Iosevich, Hadi Jorati, Sergei Konyagin, and Jozsef Solymosi.Last updated: August 2013. |