Current Research Communities, Collaborations, and Projects:

 In collaboration with Kyle Hambrook at San Jose State, we are investigating the behavior of the Fourier transform of measures supported on sums and products of sets in Euclidean space. The first paper from this joint research program is available on the arXiv.

 In a joint project with Laura Cladek at UCLA, we are working to obtain a variant of the classic Buffon needle problem, in which needles (infinite lines) are replaced with unit circles. The Buffon needle problem can be viewed as a quantitative version of the qualitative BesicovitchFederer projection theorem (which relates the dimension of a subset of the plane to the measure of its orthogonal projections onto lines through the origin) . This is a continuation of my work with K. Simon in which we extend the BesicovtichFederer projection theorem beyond the setting of orthogonal projections. (For more on this topic, see [NVP]). I wrote a short summary of this paper (along with slides) for a summer school organized by Christoph Thiele, which can be found here, (and the slides are here.).

 In collaboration with Simon, we built a research program on investigating the dimension, measure, and interior of large unions of circles; our work has applications to a number of geometric problems, including the study of distance sets (see [1], [2]). A theme throughout our work is the idea that projection and Fourier information on a set can be used to retrieve structural and topological information. In collaboration with A. Iosevich, we move beyond questions about spheres and sum sets to consider sets arising as the support of generalized Radon transform operators satisfying nonvanishing curvature conditions.

 I am working independently on a prescribed or reverse projection problem for transversal families of maps. For more on this extremely interesting problem and its dual formulation for large families of lines and circles, see my research statement.
Published Research and preprints:

 On the Lebesgue measure, Hausdorff dimension, and Fourier dimension of sums and products, (with K. Hambrook) (preprint) (2018)

 Finite trees inside thin subsets of Euclidean space, (with A. Iosevich), (conference proceedings) (preprint) (2018)

 Dimension and Measure of Sums of Planar sets and Curves, (with K. Simon) (preprint) (2017)

 Interior of Sums of Planar sets and Curves, (with K. Simon) (Cambridge Math. Society) (2018)

 Pinned geometric configurations in Euclidean space and Riemannian Manifolds, (with A. Iosevich, and I. UriarteTuero), to appear in Journal d’Analyse Math’ematique

 Finite chains inside thin subsets of Euclidean space , (with M. Bennett, A. Iosevich), Analysis and PDE, (2016).

 Maximal operators: scales, curvature, and the fractal dimension , (with A. Iosevich, E. Sawyer, and I. UriarteTuero), Anal Math (2018).

 Intersections of sets and Fourier analysis , (with S. Eswarathasan and A. Iosevich),Journal d’Analyse Mathematique, (2016).

 The lattice point counting problem on the Heisenberg groups, (with R. Garg, A. Nevo), Annales de l’Institut Fourier, (2015).

 On the MattilaSjolin Theorem for distance sets , (with A. Iosevich and M. Mourgoglou), Annales Academiæ Scientiarum Fennicæ Mathematica, Volume 37, 557–562, (2012).

 Lattice points close to families of surfaces, nonisotropic dilations and regularity of generalized Radon transforms. , (with A. Iosevich), New York Journal of Mathematics, 17, 119, (2011).

 Fourier integral operators, fractal sets, and the regular value theorem , (with S. Eswarathasan and A. Iosevich), Advances in Mathematics, volume 228, pages 23852402 (2011)
Ph.D. thesis:

 Applications of generalized Radon transforms to problems in harmonic analysis, geometric measure, and analytic number theory, Thesis work for Ph.D. in Mathematics, Univeristy of Rochester; Advisor: Alex Iosevich, (2012).