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There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still open in most cases. My talk will be about global well-posedness for the 2D stochastic Abelian-Higgs model, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant (i.e. geometric) approach, which consists of two parts: (1) introducing covariant stochastic objects (2) controlling nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

Grothendieck’s Hilbert scheme is a compact parameter space for subschemes of a projective scheme X. It is one of the basic moduli spaces in algebraic geometry, in the sense that it is the starting point for the construction of many others. One simple question about the Hilbert scheme is the following: as X undergoes a nice degeneration, what is the right way to degenerate the Hilbert scheme of X along with it? One possible answer, proposed in the PhD thesis of Kennedy-Hunt, comes from an object called the logarithmic Hilbert scheme. I will give an introduction to this circle of these ideas, explain the basic geometric properties of the logarithmic Hilbert scheme, and sketch connections with certain moduli spaces of higher dimensional varieties. The talk reports on work of Kennedy-Hunt, joint work with Kennedy-Hunt, and joint work with Maulik.