Tuesday

• Speaker: Joyce, Dominic
• Time: 9:00-10:00
• Abstract:

  In https://arxiv.org/abs/2111.04694 I developed a theory of enumerative invariants in homology counting semistable objects in abelian categories, with wall-crossing formulae under change of stability condition. In joint work with Reginald Anderson https://arxiv.org/abs/2604.05664, I apply this to prove the rationality conjecture for generating functions of Pandharipande-Thomas invariants of smooth projective complex 3-folds $X$, for superpositive curve classes $\beta$, that is, curve classes $\beta$ all of whose effective summands $\gamma$ have $c_1(X) \cdot \gamma > 0$. If $X$ is a Fano 3-fold then every curve class is superpositive. The proof uses a wall-crossing formulae relating Pandharipande-Thomas virtual classes to Donaldson-Thomas type invariants counting 1-dimensional sheaves in superpositive curve classes $\gamma$ on $X$.

• Speaker: Lee, Tang-Kai
• Time: 10:30-11:30
• Abstract:

  Singularity formation plays an essential role in the analysis and applications of geometric flows. In mean curvature flow, singularities are modeled by self-shrinking solutions, or shrinkers. Many shrinkers have been constructed, but it has remained unclear whether they can actually be realized as singularities of compact mean curvature flows. We will see that every currently known smoothly embedded shrinker whose asymptotic behavior is understood does arise in this way, with potential applications to the construction of fattening examples in all dimensions. This is based on joint work with Jingwen Chen, Ao Sun, and Xinrui Zhao.

• Speaker: Fredrickson, Laura
• Time: 1:15-2:15
• Abstract:

  Gravitational instantons are defined as non-compact, non-flat, complete hyperkähler 4 manifolds with L^2 curvature decay. They have been recently classified, and all arise as bubbling limits of K3 surfaces. The Modularization Conjecture posits that any gravitational instanton arises as the moduli space of gauge-theoretic equations.

  In this talk, I'll focus on a special kind of gravitational instanton: ALG D_4 gravitational instantons. These can be conjecturally realized as moduli spaces of Hitchin's equations, a system of gauge-theoretic equations on a Riemann surface that is recognized as a central object in mathematics. In joint work (arXiv:2603.17020) and ongoing work with Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss, we prove the Modularization Conjecture in this case.

• Speaker: Parker, Gregory
• Time: 2:30-3:30

Singularities of minimal submanifolds are often characterized by their tangent cones. Foundational work of Simon and others showed that in the simplest case --- isolated conical singularities --- these tangent cones are unique, and later work of Caffarelli-Hardt-Simon revealed there is a rich deformation theory of the corresponding submanifolds. Shortly after, L. Simon posed the question of whether analogous constructions could be carried out for cylindrical tangent cones, where the singular set has positive dimension. In this talk, I will describe a general construction of infinite-dimensional families of minimal submanifolds, in arbitrary codimension, with cylindrical tangent cones along singular submanifolds of these. These examples do not rely on extra structure such as symmetry or the presence of a calibration, and produce submanifolds whose singular set may be prescribed from a finite-codimension set of analytic submanifolds of the ambient space. The construction uses techniques from the analysis of singular elliptic edge operators and Nash-Moser theory. This talk is based on joint work with Rafe Mazzeo.

• Speaker: Oliveira, Gonçalo
• Time: 4:00-5:00

I will explain how the construction of closed geodesics for certain Ricci-flat Kähler metrics on K3 surfaces is related to an open problem in electrostatics posed by Maxwell in 1873. Then, I will use recent joint work with Christopher Fillmore and Herbert Edelsbrunner to tackle both of these problems, culminating in the construction of several closed geodesics for these metrics on K3 surfaces.