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My primary interests are in pure analysis, especially functional analysis, harmonic analysis, spectral theory, and operator theory, with frequent connections to linear algebra and abstract algebra. I am most attracted to problems where a rigid analytic or algebraic structure forces something nontrivial to happen: linear independence, spectral concentration, operator convergence, statistical complexity, or strong restrictions on the possible behavior of a system. I am increasingly interested in places where these analytic ideas meet learning theory and sequence models.
I tend to prefer analytic and operator-theoretic arguments to geometric intuition. A recurring theme in the problems I like is that an object may look complicated at first, but after choosing the right transform, basis, operator, or invariant, the problem becomes a question about a surprisingly rigid piece of linear or functional analysis.
Time–frequency analysis is the subject I keep returning to. I am especially interested in finite Gabor systems and questions about when collections of time–frequency shifts can or cannot be linearly dependent. The Heil–Ramanathan–Topiwala problem was what originally pulled me toward this area.
The recent construction of a Schwartz function admitting a finite linearly dependent family of time–frequency shifts changes the unrestricted HRT question substantially. What interests me now is the structure around that failure: which hypotheses still force independence, which configurations remain rigid, and how operator-theoretic formulations such as Weyl polynomials, Zak transforms, group-algebra methods, and spectral constraints distinguish positive regimes from counterexamples.
More generally, I am interested in invariant subspaces, zero-divisor phenomena, Rosenblum-type operators, and problems where algebraic relations strongly constrain the spectrum or kernel of an operator.
Another major interest of mine is understanding how regularity and spectral information control the complexity of a function. My recent work with the Fourier Ratio studies this question through spectral expansions: how eigenvalue growth, Sobolev or energy regularity, and the geometry of the underlying space constrain how spread out the Fourier coefficients can be.
I am particularly interested in the interaction between Weyl laws, compact-resolvent operators, spectral dimension, resistance forms, and sampling theory. On resistance spaces, the reciprocal spectrum can be identified with average resistance distance, connecting an operator-theoretic quantity directly to geometry and then to random reconstruction. The same ideas specialize to graph Laplacians and electrical networks, where energy becomes dissipated power and resistance distance becomes effective resistance.
This is the sort of connection I enjoy most: a spectral identity begins as a statement in functional analysis and eventually controls approximation, sampling, inverse problems, and the stability of physical measurements.
I am also interested in analytic number theory, especially families of automorphic L-functions and the statistics of their zeros near the central point. My work in this area has involved low-lying zeros of holomorphic cuspidal newforms, centered moments, n-level densities, test functions, and the comparison between arithmetic families and Random Matrix Theory.
What I find appealing here is the same kind of rigidity that appears elsewhere in my work: global arithmetic information is encoded in analytic objects, and carefully chosen transforms and averages reveal statistical structure that is otherwise difficult to see. I am especially interested in questions that connect zero statistics, Fourier analysis, spectral distributions, and arithmetic invariants.
I am also interested in arithmetic functions such as the Möbius and Liouville functions, especially when their apparent randomness can be studied through exponential sums and Fourier complexity. Questions of the form “how structured must an arithmetic function be if its Fourier transform is unusually concentrated?” are particularly appealing to me because they turn classical number-theoretic behavior into an analytic rigidity problem.
A direction I want to understand much better is the interface between harmonic analysis and statistical learning theory. In particular, I am interested in using Fourier-analytic quantities as complexity measures for hypothesis classes, and in understanding when spectral or arithmetic structure forces large Vapnik–Chervonenkis dimension, large sample complexity, or statistical hardness of learning.
The connection between arithmetic functions and learning theory is especially interesting to me. For the Möbius function, for example, lower bounds on a Fourier Ratio can be converted into lower bounds on the VC dimension of an associated hypothesis class, and then into linear sample complexity lower bounds. I like this because it translates a statement about exponential sums into a precise statement about learnability. More broadly, I am interested in the distinction between functions that are easy to describe algebraically and functions that are nevertheless hard to learn statistically from partial data.
I am interested in approximation theory when it is phrased in operator-theoretic language. Korovkin-type theorems are a good example: convergence of a large class of linear operators can be forced by their behavior on a very small collection of test functions. I am interested in abstract versions of these ideas, statistical and summability notions of convergence, positive and non-positive operators, and the extent to which a finite set of structural tests can control convergence in larger function spaces.
This direction fits naturally with my broader interest in functional analysis because the main object is not an approximation formula by itself, but the operator acting on a function space and the structural conditions that determine its limiting behavior.
My applied interests tend to be problems in machine learning that can be reduced to a clean mathematical model. I am interested in memory mechanisms, recurrent state models, attention, and sequence architectures when their behavior can be described through linear operators, kernels, stability estimates, approximation bounds, asymptotic scaling laws, or explicit computational complexity.
Self-attention is particularly interesting to me because it replaces an explicitly sequential recurrence with a global interaction rule between positions. I am interested in the mathematics of query-key compatibility, multi-head attention, positional information, long-range dependence, and the tradeoff between expressivity and computational cost. The fact that self-attention can connect distant positions with a constant number of sequential operations makes it a natural object for studying how architectural structure changes the effective path length through which information must propagate.
I am also interested in architectural minimality: which pieces of a large sequence model are actually necessary, which are redundant, and what is lost when one simplifies the system. My work on Transformer-based time-series forecasting comes from this instinct. Rather than assuming a full encoder–decoder architecture is necessary, I prefer to isolate its components and ask which pieces are actually doing the mathematical or statistical work. In that setting, decoder-only models frequently matched or outperformed full encoder–decoder models while using a simpler architecture.
For memory systems, I have been studying a two-timescale model in which a temporary eligibility state retains a decaying trace of recent writes before a later capture decision moves information into persistent memory. Once the capture sequence is fixed, the mechanism becomes a causal linear operator, which makes delayed relevance, stability, memory capacity, and computational scaling accessible to exact analysis. I am interested in developing similar exact analyses for attention and other sequence mechanisms whenever the architecture admits a tractable operator or kernel description.
Across all of these areas, the subjects themselves are less important to me than the kind of question being asked. I like problems involving:
I am happy to move between fields when the underlying structure is interesting. My long-term goal is to build enough analysis, algebra, spectral theory, and operator theory that I can recognize the same rigid mechanisms when they appear in very different-looking problems.
LAST UPDATED: September 13, 2026
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