Research directions
Problems you can expect to work on
I work on problems at the meeting points of harmonic analysis, combinatorics, number theory, and data science. The directions below overlap substantially; that unity is an important part of the mathematics.
Uncertainty principles and signal recovery
Suppose that a function on a finite abelian group is transmitted through its Fourier transform and some frequencies are lost. Under what hypotheses can the original signal still be recovered exactly? Uncertainty principles connect this question to sparsity, Fourier restriction, spectral synthesis, compressed sensing, and learning theory.
Students working in this circle of ideas include: Pablo Bhowmik, Will Burstein, Shantanu Deodhar, Zhihe Li, Hari Nathan, and Ella Yu.
Geometric measure theory and configurations
How large must a subset of Euclidean space, or of a Riemannian manifold, be before it contains a prescribed geometric configuration—an equilateral triangle, a given angle, or a chain of a given length? Fourier-analytic and combinatorial methods interact here. The central model is the Falconer distance conjecture.
Students and former students in this area include: Shengze Duan, Quy Pham, Donggeun Ryou, Kellen Adkins, Zhihe Li, Brian McDonald, Alex McDonald, Nik Chatzikonstantinou, Bochen Liu, Steven Senger, Belmiro da Silva, Krystal Taylor, and Brianna Vick.
Spectral theory on Riemannian manifolds
I study the distribution of Laplace eigenvalues, remainder terms in Weyl's law, and uncertainty principles on manifolds. One question is whether the classical Weyl remainder can always be improved on product manifolds. Emmett Wyman and I proved such an improvement, with explicit exponents, for products of spheres; the general setting offers many promising problems.
Finite fields and commutative rings
Finite-field configuration problems have a distinctive arithmetic component. We ask how large a subset of a finite vector space must be to determine an equilateral simplex or another prescribed configuration. Related sum-product questions seek lower bounds for the larger of a set's sum set and product set.
Students and former students in this area include: Esen Aksoy, Philipp Birklbauer, Jeremy Chapman, David Covert, Doowon Koh, Alex McDonald, Brendan Murphy, Steven Senger, and Le Anh Vinh.
The Erdős distance problem
How few distinct pairwise distances can a finite subset of Euclidean space determine? Guth and Katz resolved the planar conjecture in 2011, while the higher-dimensional problem remains wide open. Sum-product phenomena and incidence geometry play important roles.
Students and former students in this area include: Steven Senger, Jonathan Passant, and Firdavs Rakhmonov.
Data science and geometric combinatorics
Projects range from applied data science to theoretical questions connecting learning theory with analysis and combinatorics. Topics include sampling guarantees for time-series forecasting and the Vapnik–Chervonenkis dimension of geometric families over finite fields.
Students and former students in this direction include: Brian McDonald, Shashank Chorge, Pablo Bhowmik, Haoyuan Tian, Alex Liu, and Svetlana Pack.
Choosing a direction
These are starting points rather than rigid categories. Many successful projects draw ideas from two or more of them. The best direction depends on a student's interests, background, and mathematical taste, and it often becomes clearer through reading, seminars, and exploratory work.
For names, dissertations, and placements, see the complete list of my current and former Ph.D. students.
