I approach machine learning from pure mathematics — asking which analytic, geometric, and probabilistic structures make learning possible, and how they become concrete algorithms for generation and inference. Six threads organize the work, with optimal transport, geometry, and analysis increasingly at the centre.
The throughline: analysis, geometry, and probability as the grammar behind modern models — not decoration, but a generative source of ideas.
I believe abstract mathematics is where the good ideas in learning come from. Flows and differential equations, transport and geometry, diffusion and stochastic calculus — I want to keep pulling these threads toward a PhD in pure mathematics and generative models.
The geometry of moving one distribution onto another — Wasserstein geometry, couplings, and transport as the backbone of modern generative flows.
Optimal transport gives distributions a geometry. I'm especially interested in OT couplings that straighten generative paths — conditional flow matching with minibatch OT — and in the analytic side: Wasserstein gradient flows and their link to diffusion.
Learning to sample complex distributions — diffusion, flow matching, normalizing flows, and latent-variable models.
How do we turn noise into structure? I work across diffusion, flow matching, and exact-likelihood flows, with a bias toward the probabilistic and dynamical principles beneath the architectures.
Generation as integrating a stochastic differential equation — and the search for processes better matched to data.
The score-based view recasts generation as a reverse-time SDE. I'm interested in which stochastic processes — and which geometries on their paths — are analytically and topologically better suited to real, structured data.
Curvature, manifolds, and the shape of latent space — continuous-depth models as homeomorphisms that deform geometry.
Data and latent spaces have shape. I study manifold structure, Neural ODEs as continuous deformations preserving topology, and how curvature and geometry inform where and how models can learn.
Function spaces and operators — the analytic setting where scores, densities, and flows actually live.
Densities, score functions, and velocity fields are elements of function spaces; training is optimization over them. I want to bring the tools of analysis — operators, spaces, and their geometry — to bear on why these methods work.