Differential geometry and symplectic topology seminar (Fall 2021)

Abstracts

• Russell Avdek
  • Let S be a convex hypersurface with neighborhood N(S) inside of some contact manifold M. When dim(M)=3, Giroux’s criterion provides a user-friendly way of determining exactly when N(S) is tight in terms of simple closed curves in S. It appears very difficult to state a generalization to the case dim(M)>3 in terms of topological data. Instead, we’ll describe an exact criterion for “algebraic tightness” which is applicable in all dimensions using holomorphic curves. The proof of our result combines obstruction bundle gluing and Kuranishi techniques.
• Jie Min
  • Abstract: Symplectic log Calabi-Yau divisors are the symplectic analogue of anti-canonical divisors in algebraic geometry. We study the rigidity of such divisors. In particular we prove a Torelli type theorem and form an equivalent moduli space of homology configurations which is more suitable for counting. We also discuss their relations to toric actions and almost toric fibrations, reprove a finiteness result and an upper bound for toric actions by Karshon-Kessler-Pinsonnault, and prove a new stability result.
• Demetre Kazaras
  • The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I will use this formula to consider the following question: How flat is an asymptotically flat manifold with very little total mass? We make progress on this problem and confirm special cases of conjectures made by Huisken-Ilmanen and Lee-Sormani. The main results asserts that in the class of asymptotically flat manifolds with non-negative scalar curvature satisfying a uniform lower bound on Ricci curvature, small mass implies Gromov-Hausdorff closeness to flat space.
• Nate Bottman
  • Both this talk and my talk in Monday’s topology seminar are about the symplectic (A-infinity,2)-category Symp, but this talk will focus much more on the symplectic aspects and on giving a ground-up overview of Symp. Symp is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N), and it is the right way to package the functoriality properties of the Fukaya category. I will explain why Symp is a necessary extension of Wehrheim–Woodward’s construction of functors between Fukaya categories from Lagrangian correspondences, and I will explain the blueprint for Symp’s definition. In particular, I will explain what an (A-infinity,2)-category is, and I will describe a new definition of (A-infinity,2)-categories that is enabled by the work I will present in Monday’s topology seminar. Expect lots of pictures!
• Ao Sun
  • We show that after a perturbation on the initial data of mean curvature flow, the perturbed flow can avoid certain non-generic singularities. This contributes to the program of dynamical approach to generic mean curvature flow initiated by Colding and Minicozzi. The key is to prove that a positive perturbation on initial data would drift to the first eigenfunction direction after a long time. This result can be viewed as a global unstable manifold theorem in the most unstable direction for a nonlinear heat equation. This is joint work with Jinxin Xue (Tsinghua University).
• Jacob Rooney
  • An outstanding problem in ECH has been a construction of cobordism maps via a count of holomorphic curves. We give a construction for contact manifolds with no elliptic Reeb orbits up to a certain action and an adaptation for Legendrian surgery cobordisms in Colin-Ghiggini-Honda’s hat version of ECH, where a certain type of elliptic orbit appears. We also explain why the latter map agrees with the cobordism map on the hat version of Heegaard Floer homology and give some applications.
• Mohammad Farajzadeh Tehrani
  • Logarithmic differential forms and other logarithmic structures naturally appear when studying divisors and related structures in algebraic geometry. In this talk, first, I will introduce the logarithmic tangent bundle of a pair (X,D) of a smooth symplectic manifold X and a (smooth or normal crossings) symplectic divisor D (based on joint works with McLean and Zinger). Then, I show how the logarithmic tangent bundle can be used to properly set up and understand the deformation theory of J-holomorphic curves relative to a divisor. As an application, I will talk about the integrality (conjecture) of the count of genus-zero J-holomorphic curves with maximal tangency condition in log Calabi-Yau Symplectic surfaces. Similar techniques can be used to study symplectic cohomology and Fukaya categories relative to a divisor.
• Ke Zhu
  • We study the adiabatic degeneration of Floer trajectories to “disk-flow-disk” configurations and the recovering gluing, where the gradient flow part has positive length. Unlike the standard gluing problem, we study the problem of gluing two objects of different dimensions: 1-dimensional gradient segments and 2-dimensional (perturbed) J-holomorphic maps. Similar configurations also appear in the Morse-Bott approach for symplectic homology and contact homology. As an immediate application, we outline the proof that when a finite number of Hamiltonian deformations of a monotone Lagrangian submanifold collapse simultaneously, the pearl complex moduli spaces by Biran-Cornea are diffeomorphic to the J-holomorohic polygon moduli spaces by Fukaya-Oh-Ono-Ohta, provided the dimension of the moduli spaces is sufficiently small e.g., when the dimension is 0,1 or 2. This is enough to prove that the 𝐴∞-structures appearing in the two pictures are isomorphic to each other. This also provides another proof of the isomorphism property of PSS map which is different from previous work of Oh-Zhu: It bypasses the nodal Floer trajectories by going directly from “disk-flow-disk” configurations to resolved Floer trajectories, without the need of blowing up target.
• Guangbo Xu
  • Gauged linear sigma model (GLSM) is a physical theory introduced by Witten which has played significant roles in the study of symplectic geometry and mirror symmetry. In this talk I will present the progress on the mathematical theory of GLSM in both the closed string and the open string case. In the closed string case, joint with Gang Tian, we have constructed a set of Gromov-Witten type invariants for GLSM spaces satisfying the “geometric phase” condition and proved that they satisfy similar axioms as Gromov-Witten invariants. These invariants are the symplectic counterpart of the quasimap invariants of Ciocan-Fontanie-Kim and the GLSM invariants of Fan-Jarvis-Ruan. In forthcoming works we will also show a relation between GLSM invariants and Gromov-Witten invariants. In the open string case, joint with Chris Woodward, we have constructed an A_\infty morphism between an equivariant Fukaya algebra and a bulk deformed Fukaya algebra of a Lagrangian brane in a GIT quotient. The applications include an easier criterion for weak unobstructedness of Lagrangians and an enumerative interpretation between the Lagrangian Floer disk potentials and the Hori-Vafa type superpotentials.
• Yongjia Zhang
  • I will introduce Bamler’s recent works on the entropy of the Ricci flow. His results show how Nash entropy is involved with local geometry. My coauthors and I have applied his method to various geometric problems and obtained several results. These results include: (1) characterization of a Ricci flow with a closed finite-time singularity model, (2) a local Sobolev inequality on Ricci flow, (3) an optimal volume growth estimate for noncollapsed steady Ricci solitons. The new results presented in this talk are joint works with Richard H. Bamler, Pak-Yeung Chan, and Zilu Ma.
• Josef G Dorfmeister
  • The orbit space of the action of Diff$^+$(M) on $M=\mathbb CP^2\sharp k\overline{\mathbb CP^2}$ will be described. For classes whose orbit includes a reduced class (which encompasses all classes of positive square), this is a well-known result following from the work of numerous authors. For classes whose orbit includes a simplified class, this is less well understood. I will describe the orbit structure in this case and draw some conclusions about the minimal genus of such classes. Switching to orbits which contain a reduced class, I may describe recent results relating to symplectic genus of such classes and restrictions on Lefschetz fibrations arising from an understanding of these classes.
• Andrew Sageman-Furnas
  • Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only a metric and mean curvature function. However, there are exceptions. These may arise in a family, like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid, or as two surfaces called a Bonnet pair.
  • For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. This led them to ask if compact Bonnet pairs exist.
  • In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of special surfaces that are called isothermic. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.
  • The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.