Research

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 Besicovitch-Federer 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 Besicovtich-Federer 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 non-vanishing 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 pre-prints:

 

    • Finite trees inside thin subsets of Euclidean space, (with A. Iosevich), (conference proceedings) (pre-print) (2018)

 

 

 

 

 

 

 

 

 

 

 

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).