I've spent the last few years building things: diffusion models, flows, graph networks, inference machinery. I loved it. But if I'm honest about what actually pulls me forward, it isn't the engineering — it's the mathematics underneath.
So I'm making the turn explicit. My centre of gravity now is pure mathematics, and I'm treating generative modeling and deep learning as the place where that mathematics becomes concrete and testable.
Why now
Three questions kept recurring in my work, and none of them are really about neural networks:
- When a diffusion model generates, it integrates a **reverse-time stochastic
differential equation**. What geometry lives on those paths?
- When a flow transports noise to data, it's solving a transport problem. What
does optimal transport tell us about the right way to do that?
- When a Neural ODE deforms a latent space, it's acting as a homeomorphism.
Which topological and differential-geometric invariants does it preserve?
Each of these is a mathematics question wearing a machine-learning costume.
I don't want to use the mathematics as after-the-fact justification. I want it to be the source of the next idea.
Where I'm heading
Over the next few months I'm focusing my PhD applications on pure mathematics and generative models. The directions I most want to work in:
- Optimal transport — the geometry of moving one distribution onto another.
- Differential geometry — curvature, manifolds, and the shape of latent space.
- Functional analysis — the spaces where scores, densities, and velocity
fields actually live.
And the thread tying them together: the likely connections between stochastic differential equations, geometry, and analysis.
This blog is where I'll think out loud about all of it — sometimes a clean derivation, sometimes a half-formed intuition. If any of it resonates, reach out.