Research

Geometric deep learning

This is most of my own work. I study identifiability, expressivity, and the algebraic geometry of equivariant architectures, and in particular the trade-off between equivariance that is imposed by construction and equivariance that is learned from data. In a recent preprint, I show that no single polynomially parameterized equivariant architecture can represent all equivariant functions representable by an unconstrained multi-head self-attention layer. So, the first question I ask about an architecture is which maps it can represent at all, and only then how well it trains.

Machine learning for science

I work on two things here. The first is symmetry and geometric structure in models for molecular and materials data. The second is machine learning of exchange-correlation functionals for density functional theory, which connects directly to the physics I work on full time.

Foundations of density functional theory

I work full time as a researcher in mathematical physics, on what a density functional theory can represent once its constraints are fixed. My recent work there is on N-representability and on the generalized BEC force for strongly correlated bosons. The mathematics is different from the mathematics of equivariant networks, and the question I ask in both cases is the same one: given a set of constraints, which objects can a parameterized family represent? For that reason, I do not treat the two as separate careers.

Where I want to go next

I am looking for a PhD position in geometric deep learning, in a group that treats the mathematics as part of the work. Groups working on machine learning for the physical sciences interest me for the same reason.