Izabella Laba's Home Page

Research interests

Publications

Teaching

Professional information

CV

Links: people, places, resources

Conferences and programs

Miscellaneous links

  
Izabella Laba - Research interests


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. I've tried to mention some of the main motivating questions in each area as well as the problems that I have actually worked on. In order to keep this reasonably brief, I had to omit most of the history and context. 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 distances

 Harmonic analysis offers an effective and quantitative approach to many beautiful questions in geometric measure theory. One such question, inspired in part by 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. Kelan Zhai and I have extended this result to more general sets obeying a "tiling condition"; further work is in progress.

Another family of problems concerns distance sets. The prototype question is due to Falconer: if a set E in Rn 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 sets

 Together with Malabika Pramanik, I'm developing a program of transferring results and methods of additive combinatorics (see below) 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. 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.

Another result 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 Lp(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 Lp functions.

Additive combinatorics

 The harmonic-analytic side of this area 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 and their structure, with Mariah Hamel and Sergei Konyagin.

Kakeya sets, restriction conjecture, and related questions

 A Besicovitch set is a subset of Rn 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.

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).

Translational tilings and spectral sets

There 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 Rn by translations if and only if L2(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. I also have a somewhat outdated web page on the subject.

Incidence geometry

 This 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 Cn (up to logarithmic factors) distinct distances between them. The closely related unit distance conjecture, that such a configuration can contain at most n 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: December 2010.