Weekly Bulletin

Can commercially available LLMs solve research-level mathematics problems? Can mathematicians use these models to support their own research? The current answer to both questions seems to be a cautious “yes,” which grows less cautious over time. In this talk, I will give an overview of the mathematical reasoning abilities of current state-of-the-art models, with a special focus on agentic systems. I will also highlight best practices, limitations, and open questions surrounding the use of AI in mathematical research.

Diffusion and flow-matching models have become the default tools for generative modeling, but in scientific and structured domains their value depends on one ability: to generate not just plausible samples, but samples that satisfy prescribed constraints and guidance signals — known physics, structure, and target properties. Such requirements come in several distinct types, and a generative model has to be designed in a specific way to accommodate each. This talk frames a line of work around that question — how do we build diffusion and flow models that respect different kinds of constraints? — and presents one design per type. The first type is conditioning on target properties: a score-based diffusion with co-evolving processes that exchange information through loop guidance, steering generation toward desired attributes. The second is structural: a continuous-time flow constrained by a causal dependency graph, learned jointly with the dynamics of interacting time series. The third is partial physics: when the governing equations are known only in part, a grey-box flow-matching model embeds the available physics while a structured latent absorbs the unknown parameters and stochasticity. The fourth is hard physical constraints — conservation laws, boundary conditions — enforced at sampling time by projecting generation onto the constraint manifold.

I will first review the Virasoro constraints for Gromov–Witten theory and the curve case proved by Okounkov and Pandharipande. I will then discuss ongoing work on extending these constraints to orbifold curves using degeneration techniques.