My research lies at the interface of harmonic analysis, geometric measure theory, centered on understanding how the geometric and combinatorial structure of a set constrains the configurations of distances, areas, and other geometric quantities it can generate. Many such questions, from continuous settings in Euclidean spaces to their discrete and finite field analogues, can be reformulated as problems about controlling these configurations, which frequently leads to the study of decay and boundedness properties of associated operators using tools from Fourier and harmonic analysis.
A central question driving my research is: how large does a set in Euclidean space need to be to guarantee that it determines a rich collection of geometric patterns, such as distances, triangles, or general k-point configurations?
My primary tools are Fourier integral operators and microlocal analysis, which I use to obtain sharp dimensional thresholds for configuration set problems, generalizing the Falconer distance conjecture to a broad class of k-point configurations.
Asymmetric Lp improving and maximal estimates for the triangle averaging operator (with A. Iosevich, E.A. Palsson), arXiv:2610.42061, (2026), submitted.
Pinned configuration sets and slicing theorems via Fourier integral operators (with A. Greenleaf, E.A. Palsson, G. Psaromiligkos), arXiv:2610.32088, (2026), submitted.
On Hausdorff dimensions of k-point configuration sets and Elekes-Ronyai type theorems, arXiv:2603.03567, (2026), submitted.
Discretized sum-product type problems: Energy variants and Applications (with T. Pham, and C-Y. Shen) (2022), submitted.
Packing sets in Euclidean space by affine transformations (with A. Iosevich, P. Mattila, E. Palsson, T. Pham, S. Senger, and C-Y. Shen), arXiv:2405.03087, accepted for publication in the Bull. Lond. Math. Soc. (2026).
On the Falconer type functions and the distance set problem, Math. Ann. (2026).
Lp -integrability of functions with Fourier supports on fractal sets on the moment curve, (with S. Duan and D. Ryou), J. Funct. Anal. (2025).
Pinned simplices and connections to product of sets on paraboloids (with A. Iosevich, T. Pham, and C-Y. Shen), Indiana Univ. Math. J. (2025).
Structural theorems on the distance sets over finite fields (with D. Koh and T. Pham), Forum Math. (2023).
On the k-resultant modulus set problem on varieties over finite fields, Int. J. Number Theory (2023).
Bound for volumes of sub-level sets of polynomials and applications to singular integrals (with Loi Ta Le) (2020).