Research

Shadows Can Be Deceiving          While a single projection may not tell much about a set,  the more angles we consider, the more we may know about it. (image courtesy of J. Zahl)

Direct link to my CV and list of research talks

Current Research Communities, Collaborations, and Projects:

    • In a joint project with Laura Cladek at UCLA and Blair Davey at CUNY, we recently obtained 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.  Our first paper on this topic has appeared on the arXiv and will appear in the Indiana University Math Journal. (For more on this topic, see [NVP]; I wrote a short summary of this topic, along with slides, for a summer school organized by C. Thiele).
    • 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 and will appear in the Proceedings of the AMS.
    • In collaboration with Karoly Simon at Budapest University, we are building 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 with Tyler Bongers on obtaining geometric bounds for the Favard curve length.  The Favard length measures the size of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. Additionally, we are interested in prescribed or reverse projection problems for transversal families of maps.
    • In a joint work with my PhD student, we are investigating lattice point counting problems in the intersection of Fourier analysis and number theory.

Published Research and pre-prints:

 

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, University of Rochester; Advisor: Alex Iosevich, (2012).