Topology and Symplectic Geometry Seminar

Fall 2019

September 5 - Kevin Sackel (Stony Brook University / Simons Center for Geometry and Physics)

• Pre-talk: A biased introduction to contact and symplectic manifolds
  • Abstract: This talk serves as an introduction to some basic notions in contact and symplectic geometry. We will describe geometric tubular neighborhood theorems in these settings, as well as the handle decomposition theory for Weinstein manifolds.
• Talk: Getting a handle on contact manifolds
  • Abstract: We discuss the basic handle decomposition theory of convex contact manifolds, which can be thought of as an odd-dimensional analogue of Weinstein manifolds. This theory relates naturally to the study of convex hypersurfaces in contact manifolds and supporting Weinstein open book decompositions, both of which have been very powerful tools in 3-dimensional contact geometry.

September 12 - Umut Varolgunes (Stanford University)

• Pre-talk: Relative Floer theoretic invariants
  • Abstract: I will start by explaining the construction of relative symplectic cohomology, in the c_1(M)=0 case for simplicity, as a graded algebra over the Novikov field. Then, I will list some of its properties including vanishing for stably displaceable subsets and a simple version of the Mayer-Vietoris property. I will also hopefully have some time to discuss the relative version of Lagrangian Floer homology in the tautologically unobstructed case. Part of this is my thesis, and part of it is joint work with D. Tonkonog.
• Talk: Applications of relative invariants in the context of symplectic SC divisors with Liouville complement
  • Abstract: I will start by introducing the elementary notions of SH-visible and SH-full subsets, which are analogous to Entov-Polterovich's heavy and superheavy subsets. Then, I will sketch the proof that in the c_1(M)=0 case, the skeleton of a Liouville domain as appeared in the title is SH-full, and explore some consequences of this (this part is inspired heavily by M. McLean's work). Finally, I will give a speculative discussion about what can happen if c_1(M)=0 is not assumed. This is joint work with D. Tonkonog.

September 26 - Kyle Hayden (Columbia University)

• Pre-talk: Background: Knot traces and 4-manifold embeddings
  • Abstract: A knot trace is a 4-manifold obtained by attaching a single 2-handle to the 4-ball along a knot in its boundary. These simple 4-manifolds arise naturally in many contexts, from the construction of exotic smooth structures on R^4 and small compact 4-manifolds to the recent proof that the Conway knot doesn not bound a smooth disk in the 4-ball. After sketching these applications, we'll discuss "higher genus knot traces", which are obrain by attaching a thickened surface to the 4-ball along a knot. talk about the role knot traces play in Floer homology.
• Talk: Stein structures and spineless 4-manifolds
  • Abstract: We construct a curious family of Stein domains in C^2 that are each homeomorphic but not diffeomorphic to a knot trace a simple 4-manifold obtained by attaching a 2-handle to the 4-ball along a knot in its boundary. This sets up a surprising application of symplectic topology to a classical question about piecewise-linear surfaces in 4-manifolds: Given any integer g≥0, there exists a smooth 4-manifold X that is homotopy equivalent to a closed genus g surface S and contains a topological locally flat spine (i.e. an embedding of S into X that realizes the homotopy equivalence), yet X does not contain a piecewise-linear spine. In particular, for g=0, this gives a fundamentally new solution to Problem 4.25 in Kirbys list, which was recently resolved by Levine and Lidman. This is joint work in progress with Lisa Piccirillo.

October 3 - Semon Rezchikov (Columbia University)

• Pre-talk: Introduction to Floer Homology and Coherent orientations
  • Abstract: I will begin by giving an elementary introduction to Lagrangian Floer homology, and subsequently disucss some of the family index theory involved, with an eye towards the usual construction of coherent orientations in Floer homology.
• Talk: Floer homology via Twisted Loop Spaces
  • Abstract: I will describe a new perspective on signs in (Lagrangian) Floer homology, based on arXiv:1909.01325. The proposal is that one does *not need* coherent orientations of Floer-theroetic moduli spaces if one instead counts curves with coefficients in some appropriate twist of the algebra of chains on the based loop spaces of the Lagrangian boundary conditions. This means that a variant of Lagrangian Floer homology with *integral* coefficients can be defined for any pair of exact Lagrangians, avoiding the Pin conditions required in the standard theory; this answers a question of Witten on Floer homology between "flat Spin^c Lagrangians". As an example, I will describe Lagrangians for which this method improves the self-intersection bound coming from Floer homology.

October 10 - Hang Yuan (Stony Brook University)

• Pre-talk: A biased review of rigid analytic geometry
  • Abstract: We will start by giving an elementary introduction to rigid analytic geometry particularly over the Novikov field. We will also discuss Maurer-Cartan equations appearing in Symplectic Geometry, and use them to construct an analytic subvariety in the non-archimedean torus.
• Talk: Non-archimedean SYZ mirror
  • Abstract: Given a Lagrangian fibration, we will describe a new construction of a mirror Landau-Ginzburg model from the perspective of the family Floer program. The mirror rigid analytic space is locally modeled on the zero locus of weak Maurer-Cartan equations and we show a version of wall-crossing formulas which gives the correct transition maps to glue these loci. We try to improve Abouzaids and Tus previous work by dropping the tautological unobstructedness condition therein. Besides transversality issues, it is also necessary to develop some new Homological Algebra of A infinity algebras.

October 17 - Agustín Moreno (University of Augsburg)

• Pre-talk: Bourgeois contact structures
  • Abstract: In this pre-talk, I will cover some backround on contact structures, on the philosophy of flexibility and rigidity in contact and symplectic topology, on symplectic fillability of contact manifolds, on low vs high dimensions, as well as motivation for studying a broad class of examples constructed by Bourgeois in the early 2000s. This is a preparation for the afternoon talk.
• Talk: Bourgeois contact structures: tightness, fillability and applications
  • Abstract: Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.

October 24 - Abigail Ward (Harvard University)

• Pre-talk: Landau-Ginzburg models in mirror symmetry
  • Abstract: We will give a partial description of how Landau-Ginzburg models appear in the study of symplectic geometry and homological mirror symmetry. We will construct the Fukaya-Seidel category associated to a Landau-Ginzburg model and exhibit homological mirror symmetry results that relate these Fukaya-Seidel categories to the derived categories of projective varieties. The talk will be focused on examples.
• Talk: Homological mirror symmetry for elliptic Hopf surfaces
  • Abstract: We show that homological mirror symmetry is a phenomenon that exists beyond the world of Khler manifolds by exhibiting HMS-type results for a family of complex surfaces which includes the classical Hopf surface (S^1 x S^3). Each surface S we consider can be obtained by performing logarithmic transformations to the product of P^1 with an elliptic curve. We use this fact to associate to each S a mirror "non-algebraic Landau-Ginzburg model" and an associated Fukaya category, and then relate this Fukaya category and the derived category of coherent analytic sheaves on S.

October 31 - Sara Venkatesh (Stanford University)

• Pre-talk: Hamiltonian Floer theory and symplectic cohomology
  • Abstract: Starting with closed symplectic manifolds, we introduce Hamiltonian Floer homology and discuss the dynamical information it encodes. We then translate this story to open symplectic manifolds, on which symplectic cohomology is defined.
• Talk: The quantitative nature of symplectic cohomology
  • Abstract: Symplectic cohomology has been extensively studied on exact symplectic manifolds. How far does our intuition stretch when we move to non-exact cases? We present a tool -- completed symplectic cohomology -- that has interesting properties on non-exact manifolds. We exhibit its behavior on semi-monotone negative line bundles, where mirror symmetry coarsely guides our computations. We end with a discussion on a conjectural dynamical application.

November 7 - Morgan Weiler (Rice University)

• Pre-talk: Introduction to contact dynamics and embedded contact homology
  • Abstract: We will give an introduction to dynamical questions in contact geometry and define embedded contact homology. If there is time, we will also discuss open book decompositions.
• Talk: Mean action of periodic orbits of area-preserving annulus symplectomorphisms
  • Abstract: An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary of the annulus, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

November 21 - Xujia Chen (Stony Brook University)

• Pre-talk: Disk Enumeration and Linking Numbers
  • Abstract: Gromov-Witten invariants count pseudo-holomorphic curves on a symplectic manifold passing through some fixed points and submanifolds. Similarly, open Gromov-Witten invariants are supposed to count disks with boundary on a Lagrangian, but in most cases such counts are not independent of some choices as we would wish. Welschinger defined open invariants on sixfolds in 2012 that count multi-disks weighted by the linking numbers between their boundaries. I will give an introduction to these invariants. I will also explain the difficulties of generalizing them to higher dimensions and how they can be dealt with (which is the conceptual picture of the main talk).
• Talk: A geometric interpretation of Solomon-Tukachinsky's open Gromov-Witten invariants
  • Abstract: Motivated by Fukaya '11, J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms. We present a geometric translation of their construction. This geometric perspective works over arbitrary coefficient rings and the resulting invariants readily reduce to Welschinger's open invariants for symplectic sixfolds, which count multi-disks weighted by the linking numbers between their boundaries. Solomon-Tukachinsky's open WDVV-relations and their proof can be translated similarly.

December 5 - Yingdi Qin (Harvard University)

• Pre-talk: Theta functions and mirror symmetry for the elliptic curve
  • Abstract: I will briefly introduce the Fukaya category of a symplectic manifold and explain homological mirror symmetry in detail for the symplectic 2-torus and elliptic curve.
• Talk: Coisotropic branes on symplectic tori and homological mirror symmetry
  • Abstract: Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. Kapustin and Orlov conjecture that coisotropic branes should be included into the Fukaya category from a physics view point. In this talk, I will construct for linear symplectic tori a version of the Fukaya category including coisotropic branes and show that the usual Fukaya category embeds fully faithfully into it.