Research

Noncommutative analysis & spectral theory

How does the spectrum of an operator control analysis when the classical Fourier transform is unavailable? My work addresses this question on non-abelian groups, quantum groups, and von Neumann algebras.

Foundations

With collaborators, I have developed Fourier multiplier estimates, smooth dense subalgebras on compact quantum groups, and methods that turn the growth of spectral projections into L^p–L^q bounds. These results connect harmonic analysis, operator algebras, and spectral geometry.

Regularity and evolution equations

My current programme seeks sharper multiplier conditions and dispersive estimates in affiliated von Neumann algebras. Applications include heat and Klein–Gordon equations, with spectral growth determining the relevant decay and regularity estimates.

A global pseudo-differential calculus

I propose to combine compatible Fourier structures, quantum Riemannian geometry, and connection-based symbolic calculus. The goals are a calculus on von Neumann algebras, L^p bounds, lower-bound inequalities, and applications to nonlinear evolution equations. Quantum tori and quantum Euclidean spaces provide concrete starting points.

Further spectral directions

• Spectral asymmetry and projections associated with Dirac operators.
• Hankel and Toeplitz operators with group symmetry.
• Dirac eigenvalue problems on homogeneous spaces and their quantum deformations.

These directions summarise my pure mathematics and spectral theory research statements. Proposed extensions are ongoing research goals.

Spectral geometry of learning

Neural-network training takes place on loss landscapes with highly degenerate minima. My programme explores whether an evolving Dirac-type operator can connect the geometry of these landscapes with optimisation, generalisation, and changes in learned representations.

Learning coefficients and spectral geometry

I propose to relate the local geometry studied in singular learning theory to the low-lying spectrum and heat-trace asymptotics of Witten Laplacians. The first targets are simple singular landscapes and small networks where theoretical predictions can be checked.

Training dynamics and spectral transitions

I aim to study training as a stochastic evolution of operators, testing whether changes in spectral gaps and low eigenvalues track transitions such as grokking. This work would connect parameter-space geometry with the formation of learned circuits.

Algorithms and experiments

• Investigate spectral diagnostics and directions for moving between basins in the loss landscape.
• Compare spectral changes with circuit formation on modular-arithmetic and sparse-parity tasks.
• Study attention through interacting-particle dynamics and the spectra of linearised mean-field operators.

This is a research programme: the proposed links between spectral quantities and learning behaviour remain questions to establish through proofs and experiments.

Earlier research note

Compressing long contexts with log-signatures — an exploratory proposal from March 2026.