Research
Synopses of some current research topics
Can you hear your location on a manifold?
Suppose you find yourself standing at an unknown point in a familiar room with your eyes closed. Can you determine where you are standing by clapping your hands once and listening to the resulting echos and reverberations?
In a paper with Yakun Xi, we ask this question and determine the answer for "most" rooms that take the form of a compact, boundary-less Riemannian manifold. In a follow-up paper with
Eigenfunction concentration on manifolds
The study of the behavior of Laplace eigenfunctions on manifolds is both broad and deep, with connections reaching from quantum physics to number theory. I study eigenfunctions of large eigenvalue–in particular how the geometry of the underlying manifold influences their asymptotics.
I have been particularly excited lately about the prospect of obtaining polynomial improvements to the remainder of the Weyl law if the manifold is the Cartesian product of two or more compact manifolds. See my paper with Alex Iosevich, and a recent paper by Michael Taylor.
Fourier integral operators and their applications
Among the main tools used to study high-frequency eigenfunctions is the theory of Fourier integral operators (FIOs). FIOs are also used to attack problems in geometric measure theory (such as Falconer's distance problem) where the key feature of FIOs are their Sobolev mapping properties.
I have encountered with increasing frequency what could be called "Fourier integral multilinear forms." An example arose in my work on eigenfunction triple products, and I am working to develop general Sobolev continuity estimates for such objects.
VC-dimension and geometry
The Vapnik–Chervonenkis dimension measures the complexity of a family of binary classifiers, and is a core part of learning theory. To illustrate, consider the following game of battleship. Alice secretely draws a triangle in the unit square. Bob then starts throwing darts at the square, and each time, Alice tells Bob if the dart lands inside or outside of the triangle. After throwing some number of darts, Bob must then make a guess at what triangle Alice drew. The VC-dimension of the set of triangles determines how many darts Bob must throw to be able to guess the triangle up to a small error with high confidence. (The VC-dimension of the set of triangles in the plane is 7, by the way.) The higher the VC-dimension, the harder it is to guess. This game can be played in a variety of settings and seems to give rise to fascinating combinatorial and geometric configurations (such as in this paper).
Talks
MAQD (2024) (slides) in memory of Steve Zelditch
Northeastern Analysis Network (2023) (slides)
Colloquium talk at Binghamton University (2023) (slides)
Analysis seminar at Zhejiang University (2022) (slides)
Analysis seminar at University of New Mexico (2022) (see slides for Zhejiang talk)
Harmonic and geometric analysis seminar at Cornell University (2022) (see slides for Zhejiang talk)
Conference on Harmonic Analysis and Symmetric Spaces (2021) (slides)
Analysis and PDE Seminar at Stanford University (2021) (slides)
Analysis Seminar at the University of Rochester (2018) (slides)
Tianyuan Advanced Seminar on Harmonic Analysis, Geometry, and PDEs (2018) (slides)
Mathematics Colloquium at Georgetown University (2017)
Global Harmonic Analysis AMS Special Session (2017)
Publications and preprints
Cospectral vertices, walk-regular planar graphs and the echolocation problem, with Shi-Lei Kong and Yakun Xi. (2024)
Surfaces in which every point sounds the same, with Feng Wang and Yakun Xi. (2023)
Can you hear your location on a manifold? With Yakun Xi. (2023)
The VC-dimension of quadratic residues in finite fields, with Brian McDonald and Anurag Sahay. (2022)
Fractal dimension, approximation and data sets, with many coauthors. CANT Proceedings. (2022)
A two-term asymptotic formula, with Yakun Xi. Communications in Mathematical Physics. https://doi.org/10.1007/s00220-023-04667-z (2023).
Uniform distribution and geometric incidence theory, with Ayla Gafni and Alex Iosevich. To appear in Journal of Mathematical Analysis and Applications. (2022)
The VC-dimension and point configurations in ${\mathbb F_q}^2$, with David Fitzpatrick, Alex Iosevich, and Brian McDonald. Discrete & Computational Geometry (2021)
Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions, with Yakun Xi and Steve Zelditch. Pure and Applied Analysis. (2021)
Triangles and triple products of Laplace eigenfunctions. Journal of Functional Analysis. (2021)
Fourier coefficients of restrictions of eigenfunctions, with Yakun Xi and Steve Zelditch. Science China Mathematics. (2020)
Improved Weyl remainder for products of spheres, with Alex Iosevich. Analysis Mathematica. (2019)
Fourier frames for surface-carried measures, with Alex Iosevich, Chun-Kit Lai, and Bochen Liu. International Mathematics Research Notices. (2020)
Improved generalized period estimates over curves on Riemannian surfaces with nonpositive curvature, with Yakun Xi. Forum Mathematicum. (2020)
Period integrals in nonpositively curved manifolds. Mathematical Research Letters. (2019)
Looping directions and integrals of eigenfunctions over submanifolds. Journal of Geometric Analysis. (2018)
Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature. Journal of Geometric Analysis. (2018)