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Izabella Laba - Research interests Below are short descriptions of some of my current and past research interests. Clicking on the links will take you to separate web pages (where available), featuring a more detailed description of each area, its history, recent results, and further references. Kakeya sets, restriction conjecture, and related questions Translational tilings and spectral sets Distance sets Kakeya sets, restriction conjecture, and related questions A Besicovitch set, or Kakeya set is a subset of Rn which contains a unit line segment in each direction. The conjecture (proved for n=2 but open otherwise) is that such sets must have dimension n. The current best partial results have involved a variety of methods from geometry, combinatorics, and additive number theory; but most experts believe that we are still far from a complete solution. This geometrical question turns out to be a crucial part of several major open problems in harmonic analysis. The restriction conjecture (in one formulation) concerns the properties of Fourier transforms of measures supported on hypersurfaces. The Bochner-Riesz conjecture addresses the Lp summability of Fourier series. Also closely related is the local smoothing conjecture on the regularity of solutions to the wave equation. All of these conjectures remain open, though significant advances were made in recent years, involving both Kakeya-type geometric input and sophisticated Fourier-analytic arguments. This is a large and active area of research, where each of the main questions leads to a plethora of intriguing roads to explore. See my main Kakeya page for more information. Translational tilings and spectral sets Fourier analysis can be unexpectedly helpful in Euclidean geometry - the study of translational tilings being just one example. Fuglede's conjecture 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, but remains open in dimensions 1 and 2. To me, its appeal and entertainment value lie in its quite unexpected connections to many different areas of mathematics, including computational harmonic analysis, metric geometry, number theory, combinatorics and algebra. Distance sets Given a set E in Rn, what can we say about the size of the set D(E) of all possible distances between pairs of points in E, depending on the size of E itself? For discrete sets, this is a notorious unsolved combinatorial problem due to Erdos; there is also a continuous version, due to Falconer, where the best results so far have been obtained by Fourier-analytic methods. These are very natural questions, but my interest in them is also motivated by the recently discovered -- and probably not yet fully explored -- connections to the Kakeya-restriction group of problems mentioned above. My recent work has concerned variants of this problem for non-Euclidean distance functions. Last updated: August 2005.