Differential geometry and symplectic topology seminar (Fall 2022)

Abstracts

• Ruobing Zhang
  • This talk focuses on the recent resolution of the following three well-known conjectures in the field.
    • Any volume collapsed limit of unit-diameter K3 metrics is isometrically classified as: the quotient of a flat 3D torus by an involution, a singular special Kähler metric on the topological 2-sphere, or the unit interval.
    • Any complete non-compact hyperkähler 4-manifold with quadratically integrable curvature, must have a classified model end.
    • Any gravitational instanton can be compactified to an open dense subset of a compact algebraic surface.
• Kristen Hendricks
  • Abstract: We discuss the history of applying localization results in equivariant Lagrangian Floer homology to Heegaard Floer homology to obtain rank inequalities for various versions of the Heegaard Floer homology of branched double covers. We then use recent refinements on the symplectic side to establish an inequality between the dimensions of the Heegaard Floer homologies of any double cover between three-manifolds branched along a nulhomologous link. We discuss the relationship with the L-space conjecture and (time permitting) mention some other topological applications. This is joint work with R. Lipshitz and T. Lidman.
• Vinicius G. B. Ramos
  • The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectomorphic to balls. This is joint work with Y. Ostrover and D. Sepe.
• Daniel Stern
  • I’ll discuss recent progress on the existence theory for harmonic maps on higher-dimensional manifolds, giving harmonic maps of optimal regularity from manifolds of dimension n>2 to every non-aspherical closed manifold containing no stable minimal two-spheres. As an application, we’ll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (for n<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of Laplace eigenvalues on surfaces. Based on joint work with Mikhail Karpukhin.
• Ao Sun
  • We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University).
• Nick Edelen
  • We prove a strong maximum principle for minimizers of the one-phase Alt-Caffarelli functional. We use this to construct a Hardt-Simon-type foliation associated to any 1-homogenous global minimizer.
• Beomjun Choi
  • We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.
  • For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.
• Matthew Stoffregen
  • We’ll give a brief description of what exactly (Seiberg-Witten) monopole Floer spectra are, and then present an exact triangle associated to surgeries for these Floer spectra. As an application, we calculate the spectra associated to some plumbed three-manifolds, extending the limited array of calculations done for Floer spectra. This is joint work with Irving Dai and Hirofumi Sasahira.
• Siyuan Lu
  • In this talk, I will first introduce the background and motivation for the study of Lagrangian phase equation. I will then discuss my recent work on the regularity of Lagrangian phase equation. In the second part, I will discuss some open problems relating to Lagrangian phase equation.