Research

My research lies at the intersection of number theory, spectral theory, dynamics, and mathematical physics. I am particularly interested in the relationship between deterministic structure and statistical behavior: when do systems governed by precise rules exhibit patterns associated with randomness, and when do geometry or arithmetic prevent this?

Spectral theory and geometry

The spectrum of a differential operator connects analysis with the geometry of the underlying space. I am especially interested in the Laplacian on hyperbolic and locally symmetric spaces, where geometric and arithmetic structures interact.

How does geometry constrain the spectrum? How many exceptional eigenvalues can a space have, and how do these vary in families? How strongly can eigenfunctions concentrate? I am interested in these questions both for individual spaces and for families in which the geometry becomes increasingly complicated.

Quantum chaos

Quantum chaos concerns the relationship between classical motion and the behavior of quantum states and energy levels. A central theme of my research is how the distinction between integrable and chaotic classical dynamics is reflected in spectral statistics.

When should neighboring energy levels behave like independent random points, and when should they exhibit the correlations of random matrix theory? How uniformly do high-energy eigenfunctions spread through a space? What role do symmetries, and especially arithmetic symmetries, play in these questions? I am interested in understanding both the statistical predictions and the mechanisms behind them.

Number theory and dynamics

Simple arithmetic rules can generate sequences with remarkably intricate distributions. I study questions about equidistribution, the spacing of points, and correlations between them, particularly for sequences considered modulo one and for orbits of group actions.

When does a deterministic sequence resemble a random one at small scales? What distinguishes uniform distribution from genuinely Poissonian local statistics? How do Diophantine properties influence these patterns? A related interest is counting: how accurately can we count points in an orbit or a geometric region, and what can spectral information tell us about the error?

Statistical mechanics and transport

I am interested in how macroscopic laws emerge from microscopic motion, especially in models of particles moving among scatterers. These systems provide a setting in which deterministic dynamics, geometric constraints, and randomness meet.

When does a particle’s motion become diffusive? On what time scales does it lose memory of its past? How do repeated encounters, long-range correlations, and the arrangement of scatterers affect transport? More broadly, I would like to understand when a complicated microscopic evolution can be described by a simpler stochastic process or a macroscopic differential equation.