My work uses Fourier-analytic ideas to understand geometric configurations, arithmetic structure, signal recovery, and the complexity of functions and data.
The videos page is a curated selection. The complete record of invited talks and lecture series appears in the curriculum vitae.
A central question is how analytic information about a measure or function forces geometric structure. This includes Fourier restriction, oscillatory integrals, generalized Radon transforms, and the Falconer distance problem.
On Falconer’s distance set problem in the plane · Quanta overview
This program studies when partial spectral information determines a signal, how uncertainty principles interact with restriction theory, and how geometric support conditions produce rigidity.
Uncertainty principles on finite abelian groups and sparse signal recovery · Hausdorff Institute lecture series
The Fourier ratio is a simple spectral parameter that measures effective complexity. Current work connects it to approximation, sampling, recovery, localization, learning, and information-theoretic descriptions of signals.
The Fourier ratio and complexity of signals · A unifying measure for recovery, localization, and learning
Many Euclidean configuration problems have finite-field and finite-ring analogues. Fourier analysis, incidence methods, group actions, and arithmetic estimates reveal both common structure and sharp differences between settings.
Group actions and geometric combinatorics in finite fields · The Fuglede conjecture in the finite plane
Recent projects connect harmonic analysis and geometric combinatorics with VC dimension, statistical learning, time-series recovery, forecasting, and collaborative applied research.
Current research programs: