Differential geometry and symplectic topology seminar (Spring 2023)

Abstracts

• Li-Sheng Tseng

  • Often times, manifolds of interests are equipped with a geometrical structure specified by a distinguished closed form. Such manifolds include special holonomy and symplectic manifolds. In this context, I will motivate the usefulness of considering pairs of differential forms that are linked together by a map of the distinguished form. We will show how this can lead to novel functionals and geometric flows. As an application, it leads to new notions of flat connections and Morse theory on symplectic manifolds. This talk is based on joint works with Jiawei Zhou, David Clausen and Xiang Tang.
• Ke Zhu
  • For any compact Riemannian manifold $M$ with smooth metric $g$, using eigenfunctions and eigenvalues of the Laplacian operator, Berard, Besson, and Gallot constructed the heat kernel embedding map $\psi _{t}:x\in M\to \sqrt{2}( 4\pi ) ^{n/4}t^{\frac{n+2}{4}} \{ e^{-\lambda _{j}t/2}\phi _{j}( x ) \} _{j\geq 1}\in \mathcal{l}^2$ and showed it is almost isometric as $t$ goes to $0$, which can be regarded as an asymptotically orthogonal relation in the 1-jet space of $\psi _{t}.$ We further study the higher derivatives of $\psi _{t}$ with respect to an orthonormal basis at $x$ on $M$. As $t$ goes to $0$, it turns out the limiting angles between these derivative vectors in $\mathcal{l}^2$ are universal constants independent of $g$, $x$, and the choice of orthonormal basis. Geometric applications to canonical isometric embedding, mean curvature, and the Riemannian curvature will be given. We will also explore some algebraic structures of the $\infty$-jet space of $\psi _{t}$.
• Shaoyun Bai
  • We show that for any closed symplectic manifold, the number of fixed points of any non-degenerate Hamiltonian diffeomorphism is bounded from below by a version of integral total Betti number, which takes account of torsions of all characteristics. The proof relies on defining integer-valued counts from moduli spaces of J-holomorphic curves, which are only orbispaces in general. This is joint work with Guangbo Xu.
• Chris Wendl
  • As every symplectic topologist learns in Kindergarten, the moduli space of J-holomorphic curves is generically smooth only if one restricts to the open subset of simple curves; the multiple covers are a problem, and they cannot in general be perturbed away. This is a fundamental difficulty in every equivariant Fredholm problem, resulting from the essential incompatibility between transversality and symmetry. Sometimes, however, the symmetry is important and you really don’t want to perturb it away – and it turns out that sometimes, it is possible to achieve transversality anyway, or something almost as nice! In this talk I will present a general method for understanding such issues and illustrate it via two popular examples of equivariant Fredholm problems: (1) closed orbits for a 1-parameter family of contact forms or Hamiltonian structures, and (2) unbranched covers of J-holomorphic curves in a symplectic cobordism with only hyperbolic asymptotic orbits.
• Kenji Fukaya (talk 1)
  • There are two methods to study Lagrangian submanifold of the cotangent bundle. One is using Floer homology. The other is Morse theory of generating functions. In this talk I will explain the relation between them. (Based on a joint work with A. Daemi.) If time permit I will talk also an expected relation to the theory of singular support of sheaves.
• Kenji Fukaya (talk 2)
  • Weinstein proposed to regard the Lagrangian submanifold of the direct product of two symplectic manifolds as a morphism between them. This story is combined with Lagrangian Floer theory by people including Wehrheim and Woowards. In this talk I will explain a way to build a functor from a category of symplectic manifold to a category of A infinity categories. The proof of the main theorem of the first seminar is based on this story. (Two talks are independent however.)
• Franco Vargas Pallete
  • We say that a metric g in a torus cusp is asymptotically cusped if there exists a hyperbolic cusped metric so that its difference with g converges to 0 in C^1 as we go along the cusp. In this talk we will describe minimal surface entropy for asymptotically cusped metrics, which is a geometric invariant established by Calegari-Marques-Neves and defined as the asymptotic counting of compact minimal surfaces as area grows. We will show an entropy rigidity for finite volume hyperbolic 3-manifolds. Namely, if M is a 3-manifold that admits a cusped hyperbolic metric of finite volume, then the hyperbolic metric can be characterized as the metric minimizing minimal surface entropy among negatively curved, asymptotically cusped metrics. This is based on upcoming joint work with Ruojing Jiang.
• Georgios Dimitroglou Rizell
  • We show how Sabloff duality in linearized Legendrian contact homology can be generalised to a relative Calabi-Yau structure of the Legendrian contact homology DGA, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications, including constrains on the dimension of the augmentation variety. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficiens in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.