Tag - Symplectic geometry

Zijun Zhou: Virtual Coulomb branch and quantum K-theory

21st March, 2023

In this talk, I will discuss a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of
quasimaps into the holomorphic symplectic quotient T *N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under the variation of GIT.

Morgan Weiler: Dynamics of Seifert Surfaces of Torus Knots Via ECH

20th March, 2023

Embedded contact homology (ECH) is a diffeomorphism invariant of three-manifolds due to Hutchings, defined using a contact form. This very diffeomorphism invariance makes it quite useful when studying contact dynamics, because it is possible to apply calculations using simpler contact forms to situations involving more complex ones. We will outline how a knot filtration on ECH was used in a 2015 paper of Hutchings to identify low mean action periodic orbits of disk maps, as well as several more recent generalizations due to other authors. We will then explain the correspondence between the existence of an action function and the construction of a contact three-manifold (via a mapping torus) when starting with a specific surface symplectomorphism. Finally, we will mention how the ECH computations change by analyzing the case of T(2,3) and its genus one Seifert surface. Based on work in progress with Jo Nelson. Note: The material discussed in this talk will differ from that of Jo Nelson's February 28 talk. However, enough background will be given to make this talk self-contained.

Yuhan Sun: Heaviness and Relative Symplectic Cohomology

17th March, 2023

For a compact subset K of a closed symplectic manifold, Entov-Polterovich introduced the notion of (super)heaviness, which reveals surprising symplectic rigidity. When K is a Lagrangian submanifold, there is a well-established criterion for its heaviness, by using closed-open maps. We will discuss an equivalence between the heaviness and the non-vanishing of the relative symplectic cohomology, for a general compact set K.

Marco Mazzucchelli: Surfaces of Section, Anosov Reeb Flows, and the C2-Stability Conjecture for Geodesic Flows

3rd March, 2023

In this talk, based on joint work with Gonzalo Contreras, I will briefly sketch the proof of the existence of global surfaces of section for the Reeb flows of closed 3-manifolds satisfying a condition à la Kupka-Smale: non-degeneracy of the closed Reeb orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed Reeb orbits. I will then present an application of this theorem to hyperbolic Reeb dynamics: a Reeb flow on a closed 3-manifold is Anosov if and only if the closure of the subspace of closed Reeb orbits is hyperbolic and the Kupka-Smale transversality condition holds. This result implies the validity of the C2 stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C2 structurally stable within the class of Riemannian geodesic flows must be Anosov.

Jo Nelson: From Embedded Contact Homology to Surface Dynamics

27th February, 2023

I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of S3 along T(2,q) torus knots to deduce information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages which are freely isotopic to rotation by 1/(2q) along the boundary. I will explain the interplay between the topology of the open book, its presentation as an orbi-bundle, and our computation of the knot filtered ECH chain complex. I will describe how knot filtered ECH realizes the relationship between the action and linking of Reeb orbits and its application to the study of the Calabi invariant and periodic orbits of symplectomorphisms of the pages.

Noah Porcelli: Floer Theory and Framed Cobordisms Between Exact Lagrangian Submanifolds

24th February, 2023

Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more- we show it can detect the framed bordism class of certain Lagrangians and in particular recover a result of Abouzaid-Alvarez-Gavela-Courte-Kragh about smooth structures on Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large's recent construction of a stable-homotopical enrichment of Lagrangian Floer theory.

Jo Nelson: Floer Theories and Reeb Dynamics for Contact Manifolds

13th February, 2023

Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a nondegeneracy condition called maximal non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will then explain how to construct and compute Floer-theoretic contact invariants. These are a sort of infinite-dimensional version of Morse theory wherein the chain complexes are generated by closed Reeb orbits and the differential counts certain J-holomorphic curves. This talk will feature numerous graphics and anecdotes.

Kai Hugtenburg: Open Gromov-Witten Invariants from the Fukaya Category

10th February, 2023

Enumerative mirror symmetry is a correspondence between closed Gromov-Witten invariants of a space X, and period integrals of a family Y. One of the predictions of Homological Mirror Symmetry is that the closed Gromov-Witten invariants can be obtained from the Fukaya category. For Calabi-Yau varieties this has been demonstrated by Ganatra-Perutz-Sheridan. Recently, enumerative mirror symmetry has been extended, by including open Gromov-Witten invariants and extended period integrals. It is natural to expect that open Gromov-Witten invariants can be obtained from the Fukaya category. In this talk I will outline a construction which will demonstrate this for certain open Gromov-Witten invariants.