Research

Much of my research to date has focused on the Fourier restriction problem for certain hypersurfaces with degenerate curvature, such as hyperboloids and cones.  More broadly, I'm interested in geometric problems in harmonic analysis, as well as problems at the intersection of harmonic analysis, geometric measure theory, and combinatorics.

Publications and preprints:

1. Two-point patterns determined by curves (with M. Pramanik), Math. Ann. (2025), 45 pages.  published version, preprint

   • Does every sufficiently complex fractal contain two points whose difference lies on a prescribed curve?  We answer this question affirmatively and show that the more 'curved' the curve is, the easier it is to find such a pair of points.

2. Keller properties for integer tilings (with I. Łaba), Electron. J. Combin., 31 (2024), no. 4, 26 pages.  published version, preprint 

   • Which finite sets can be used to tile the integer line by translation?  We draw connections between this unsolved problem in the integers and Keller's conjecture on cube tilings in Euclidean space.

3. Local extension estimates for the hyperbolic hyperboloid in three dimensions, J. Fourier Anal. Appl., 28 (2022), 32 pages.  published version, preprint 

   • Progress on the Fourier restriction problem for the one-sheet hyperboloid in three dimensions using polynomial partitioning.

4. Global restriction estimates for elliptic hyperboloids, Math. Z., 301 (2022), no. 2, 2111-2128.  published version, preprint

   • Progress on the Fourier restriction problem for two-sheet hyperboloids using a combination of elliptic and conic techniques.

5. Restriction inequalities for the hyperbolic hyperboloid (with D. Oliveira e Silva and B. Stovall), J. Math. Pures Appl., 149 (2021), 186-215.  published version, preprint 

   • Progress on the Fourier restriction problem for the one-sheet hyperboloid in three dimensions using a combination of elliptic and conic techniques.

6. Fourier restriction to a hyperbolic cone, J. Funct. Anal., 279 (2020), no. 3, 16 pages.  published version, preprint 

   • New 'endline' Fourier restriction estimates for a hyperbolic cone in four dimensions, resulting in a complete solution of the restriction problem for this surface.