L4

In this talk I will discuss different types of coproducts on cohomological Hall algebras associated to 3-Calabi-Yau categories, as recently defined by Kinjo-Park-Safronov and Descombes. In general, these are actually a more exotic structure we call a vertex coproduct. I will explain the important special case of a quiver with potential. In this case, these coproducts are related to deformed Drinfeld coproducts on Yangians. I will also discuss how vertex coproducts are related to factorization type coproducts, recently defined in work of Davison-Hennecart-Kinjo-Schiffmann-Vasserot. This is joint work with Jindal and Latyntsev and work in progress with Latyntsev and Descombes.

In their recent ICM address, Dan Halpern-Leistner and Jarod Alper posed the question: does the stack of canonically polarized curves admit a theta-stratification whose semistable locus is the moduli space of stable curves? In other, perhaps more familiar, language: is there some notion for curves that understands instability the way Harder-Narasimhan type does for vector bundles/coherent sheaves?

I will explain some joint work with Dave Swinarski that solves the analogous problem for certain rational curves, and then sketch out an argument that I think should solve the problem in general.

Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.

Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation we study the deformations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.

Hypertoric varieties are conical symplectic singularities equipped with a hamiltonian action of a torus of maximal possible dimension. They behave analogously to toric varieties in many ways, with the hamiltonian torus replacing the dense torus. Examples include: (crepant partial resolutions of) rational surface singularities of type A, the cotangent bundle of projective space, and Nakajima quiver varieties with the `all 1s' dimension vector.

Arbo and Proudfoot conjectured a combinatorial classification of hypertoric varieties by zonotopal tilings. In this talk I will discuss a proof of the conjecture.

We study the random motion of interfaces in stochastic Allen–Cahn equations. In one dimension, we show that, on a suitable long time scale, the solution stays close to the family of standing wave solutions, while the interface location converges to a diffusion. Our main novelty is the construction of functional correctors to cancel potential divergences. We also establish an analogous result for more singular noise leading to distribution-valued solutions. In this setting, we show that two divergent quantities arising from renormalization and Itô’s formula cancel each other.

In higher dimensions, we construct stochastic mean curvature flow driven by noise that is white in time and smooth in space, and identify it as the sharp interface limit of the stochastic Allen–Cahn equation with the same noise slightly smoothed in time. To construct this flow, we use a transformation developed by Buckdahn, Keller, Ma and Zhang (2020), which reduces the stochastic equation to a classical PDE with random coefficients.

This talk is based on joint works with Weijun Xu and Wenhao Zhao.