The Laplace-Beltrami operator is the natural generalization of the Laplacian from Euclidean space to Riemannian manifolds. The study of the Laplace-Beltrami operator, its eigenvalues and eigenfunctions are of fundamental importance to a number of disciplines, including:
Analytic number theory through the spectral theory of automorphic forms.
Quantum mechanics and semiclassical analysis.
Manifold learning via spectral clustering and embedding.
Stochastic analysis and Brownian motion.
Spectral graph theory via e.g. the Colin de Verdière graph invariant.
My primary research program explores the connection between the behavior of the Laplace-Beltrami eigenfunctions and the geometry of the underlying manifold. This manifests as two types of problems:
Forward problems, such as the Duistermaat-Guillemin theorem, which explore how the geometry of the manifold determines the behavior of the Laplace-Beltrami spectrum.
Inverse problems, such as "Can one hear the shape of a drum?", which explore how the behavior of the Laplace-Beltrami operator determines the geometry of the manifold.
The Weyl law gives sharp-in-general asymptotics for the count of Laplace-Beltrami eigenvalues less than a threshold. It is sharp-in-general because it is sharp for the round sphere. It is not sharp for all manifolds. The Duistermaat-Guillemin Theorem tells us the remainder in the Weyl law can be improved provided the set of covectors in closed orbits of the geodesic flow is measure zero in the cotangent bundle. This result establishes the fundamental relationship between the asymptotic distribution of Laplace-Beltrami eigenvalues and the dynamics of the geodesic flow.
This idea holds for counting functions associated to certain weighted spectral measures. Each has remainder term that can be improved provided a bespoke configuration set of geodesics is suitably sparse. My main research program explores various generalizations, refinements, and applications of this idea.
In a paper with Yakun Xi, we pose the question, "Can one hear where a drum is struck?" This problem pays homage to Kac and his famous question, "Can one hear the shape of the drum?" It is a question that asks about the relationship between the action of the isometry group on points in a manifold and the Laplace-Beltrami eigenfunctions. To state our question more precisely, we ask whether or not it is possible for two points on a compact manifold not in the same orbit of the isometry group to have the same local Weyl counting function.
In our paper, we prove that one can hear where most drums are struck, in some precise generic sense. In spite of this, we only know of a few specific, nontrivial examples for which we can answer this question at all! We know of a handful of examples, such as the planar disk, rectangle, square, flat Klein bottle, and ellipsoids and tori of revolution, where the answer to the question is yes. We know just a couple of topologically disconnected manifolds where the answer is no, e.g. the disjoint union of Milnor's isospectral tori. One part of the current research program is to expand the catalogue of examples, mainly to search for topologically connected examples for which the question is answered no.
April Vertical Integration Workshop (2026) (chalk talk)
Analysis seminar at Zhejiang University (2025) (slides)
Virtual Seminar on Geometry with Symmetries (2025) (slides)
Northeastern Analysis Network (2024) (slides)
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)
Note: Each paper that has appeared in print has a DOI link to the journal article ("Link to journal"). The most up-to-date self-archived copy is also available ("Personal version"). The preprint versions are also linked ("arXiv preprint"), though there is no guarantee that they contain any revisions from peer-review.
High-dimensional signal compression: Lattice point bounds and metric entropy, with Alex Iosevich and Armen Vagharshakyan.
arXiv preprint (2026)
Spectral synthesis on Riemannian manifolds, with Alex Iosevich and Azita Mayeli
arXiv preprint (2026)
Polygons and multi-product of eigenfunctions, with Yakun Xi and Yi Zhang.
arXiv preprint (2026)
A refinement of Vapnik-Chervonenkis' Theorem, with Alex Iosevich and Armen Vagharshakyan.
arXiv preprint (2026)
Kuznecov remainders and generic metrics, with Vadim Kaloshin and Yakun Xi.
arXiv preprint (2025)
Fourier uncertainty principles on Riemannian manifolds, with Alex Iosevich and Azita Mayeli.
arXiv preprint (2024)
Hearing the shape of a drum by knocking around, with Xing Wang and Yakun Xi. Bulletin of the London Mathematical Society.
Link to journal / Personal version (2026)
arXiv preprint (2024)
Multi-linear forms, structure of graphs and Lebesgue spaces, with Alex Iosevich, Eyvindur Palsson, and Yujia Zhai. Mathematische Zeitschrift. (2024)
Link to journal / Personal version (2025)
arXiv preprint (2024)
Cospectral vertices, walk-regular planar graphs and the echolocation problem, with Shi-Lei Kong and Yakun Xi. Advances in Mathematics.
Link to journal / Personal version (2025)
arXiv preprint (2024)
Surfaces in which every point sounds the same, with Feng Wang and Yakun Xi. Proceedings of the AMS.
Link to journal / Personal version (2024)
arXiv preprint (2023)
Can you hear your location on a manifold? With Yakun Xi.
The VC-dimension of quadratic residues in finite fields, with Brian McDonald and Anurag Sahay. Discrete Mathematics. (2022)
Link to journal / Personal version (2025)
arXiv preprint (2022)
Fractal dimension, approximation and data sets, with many coauthors. CANT Proceedings. (2022)
Link to journal / Personal version (2025)
arXiv preprint (2022)
A two term asymptotic formula, with Yakun Xi. Communications in Mathematical Physics.
Link to journal / Personal version (2023)
arXiv preprint (2022)
Uniform distribution and geometric incidence theory, with Ayla Gafni and Alex Iosevich. Journal of Mathematical Analysis and Applications.
Link to journal / Personal version (2023)
arXiv preprint (2022)
The VC-dimension and point configurations in ${\mathbb F_q}^2$, with David Fitzpatrick, Alex Iosevich, and Brian McDonald. Discrete & Computational Geometry.
Link to journal / Personal version (2022)
arXiv preprint (2021)
Geodesic biangles and Fourier coefficients of restrictions of eigenfunctions, with Yakun Xi and Steve Zelditch. Pure and Applied Analysis.
Link to journal / Personal version (2023)
arXiv preprint (2021)
Triangles and triple products of Laplace eigenfunctions. Journal of Functional Analysis.
Link to journal / Personal version (2022)
arXiv preprint (2021)
Fourier coefficients of restrictions of eigenfunctions, with Yakun Xi and Steve Zelditch. Science China Mathematics.
Link to journal / Personal version (2022)
arXiv preprint (2020)
Weyl law improvement for products of spheres, with Alex Iosevich. Analysis Mathematica.
Link to journal / Personal version (2019)
arXiv preprint (2019)
Fourier frames for surface-carried measures, with Alex Iosevich, Chun-Kit Lai, and Bochen Liu. International Mathematics Research Notices.
Link to journal / Personal version (2020)
arXiv preprint (2019)
Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature, with Yakun Xi. Forum Mathematicum.
Link to journal / Personal version (2020)
arXiv preprint (2018)
Period integrals in nonpositively curved manifolds. Mathematical Research Letters.
Link to journal / Personal version (2021)
arXiv preprint (2018)
Looping directions and integrals of eigenfunctions over submanifolds. Journal of Geometric Analysis.
Link to journal / Personal version (2018)
arXiv preprint (2017)
Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature. Journal of Geometric Analysis.
Link to journal / Personal version (2019)
arXiv preprint (2017)
Integrals of eigenfunctions over curves in surfaces of nonpositive curvature.
Note: Unpublished. Result subsumed by "Explicit bounds on..."
Personal version (2017)
arXiv preprint (2017)