# Stanford Symplectic Geometry Seminar 2020-2021

Meetings are held on Zoom on Mondays, 4-5pm, with exceptions to the time of day as noted below. The first Monday of every month Stanford's seminar does not happen and in its stead, Northern California Symplectic Geometry seminar takes place (joint between Stanfard, UC Berkeley, UC Davis and UC Santa Cruz). These talks are also listed below for convenience.

*Next*: **Mar 1 - NCSGS **(see http://math.stanford.edu/~ionel/ncsgs.html for details)

**Past Talks****Feb 22 - Benjamin Gammage**(Harvard University)

**Title:**Deformation of Fukaya categories

**Abstract:**Compactification of a symplectic manifold deforms its Fukaya category by counting new holomorphic disks. This deformation may be difficult to compute in general, but it turns out that in the case of a partial compactification by an affine normal-crossings divisor, it is easy to understand. We explain this computation and give some applications to homological mirror symmetry, probably focusing on the case of Berglund-Hübsch Milnor fibers.

**Feb 15 @1PM - Egor Shelukhin**(University of Montreal)

**Title:**Lagrangian configurations and Hamiltonian maps

**Abstract:**We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, and find new instances of Lagrangian Poincare recurrence. In particular, this answers a question of Kapovich-Polterovich from 2006 that appears as Problem 21 in McDuff-Salamon's list of open problems. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.

**Jan 25 @1PM- Sheel Ganatra**(University of Southern California)

**Title:**On the embedding complexity of Liouville manifolds

**Abstract:**Using the formalism of L-infinity structures in linearized contact homology, we introduce new numerical "complexity" invariants of a Liouville manifold which obstruct Liouville embeddings. As an application, we give at many times complete characterizations of Liouville embeddings between normal crossing divisor complements in complex projective space. The main embedding results are deduced explicitly from pseudoholomorphic curves (of genus 0 with many positive ends) without appealing to Hamiltonian or virtual perturbations. This is joint work with Kyler Siegel.

**Nov 16 - Shuhei Maruyama**(Nagoya University)

**Title:**A characteristic class of Hamiltonian fibrations and a group cocycle

**Abstract:**In this talk, I will introduce two cohomology classes. The first one is a third group cohomology class that is analogous to the one introduced by R.S. Ismagilov, M. Losik, and P.W. Michor in 2006. This class is given as a group cocycle of the symplectic diffeomorphism group. The second one is a characteristic class of Hamiltonian fibrations, which is defined by using the Serre spectral sequence. This characteristic class is related to the prequantization of Hamiltonian fibrations. I will introduce the constructions and properties of these classes, and after that, explain how they relate.

**Nov 9 @ 1PM - Yefeng Shen**(University of Oregon)

**Title:**Virasoro constraints in quantum singularity theory.

**Abstract:** In this talk, we introduce Virasoro operators in quantum singularity theories for nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. Using Givental's quantization formula of quadratic Hamiltonians, these operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the genus g generating functions arise in quantum singularity theories are annihilated by these Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and LG/CY correspondence. This talk is based on work joint with Weiqiang He.

**Oct 26 - Konstantin Aleshkin **(Columbia University)

**Title:**Wall-crossing for the mirror quintic.

**Abstract:**It is well-known that Gromov-Witten theory of the quintic threefold is related with the FJRW theory of the Fermat polynomial on the orbifold C^5/Z_5. In particular, Givental I-functions of these theories are related by analytic continuation. This phenomenon is usually called Landau -Ginzurg/Calabi-Yau correspondence. It can be understood in terms of wall-crossing in the stability space of a certain GIT quotient. In the talk I plan to explain how this works for the quintic threefold and how to generalize this to the mirror quintic, which is a subject of my ongoing project with Melissa Liu.

**Oct 19 - Aleksander Doan **(Columbia University)

**Title:**Counting embedded curves in symplectic six-manifolds.

**Abstract:**The number of embedded pseudo-holomorphic curves in a symplectic manifold typically depends on the choice of an almost complex structure on the manifold and so does not lead to a symplectic invariant. However, I will discuss two instances in which such naive counting does define a symplectic invariant, which turns out to be related to the Gopakumar-Vafa conjecture inspired by string theory. The talk is based on joint work with Thomas Walpuski.

**Oct 12 @ 1PM - Gleb Smirnov **(ETH Zurich)

**Title:**Four-dimensional Dehn twists and Gromov-Witten invariants.

**Abstract:**A new proof will be given that Seidel's generalized Dehn twist is not symplectically isotopic to the identity. The argument will stay in the language of counting holomorphic spheres in families and family Seiberg-Witten invariants and will not rely on any Floer-theoretic considerations.

**Oct 5 - NCSGS **(see http://math.stanford.edu/~ionel/ncsgs.html for details)

**Sep 28 - Honghao Gao **(Michigan State University)

**Title:**Exact Lagrangian fillings and clusters

**Abstract:**A general belief is that exact Lagrangian fillings can be distinguished using cluster theory. In this talk, I will present such a framework via Floer theory — given a positive braid Legendrian link, its augmentation variety is a cluster K_2 variety and its admissible fillings induce cluster seeds (joint work with L. Shen and D. Weng). As an application, I will explain how to use Legendrian loops and cluster algebras to construct infinitely many exact Lagrangian fillings for most torus links (joint work with R. Casals, using sheaves in accordance with Shende-Treumann-Williams-Zaslow), and for most positive braid links (joint work with L. Shen and D. Weng, using augmentations). Time permitting, I will also survey other methods to produce infinitely many fillings and compare these approaches.

**Sep 21 @ 1PM - Shira Tanny **(Tel Aviv University)

**Title:**The Poisson bracket invariant: soft and hard approaches.

**Abstract:**In 2006 Entov and Polterovich proved that functions forming a partition of unity with displaceable supports cannot commute with respect to the Poisson bracket. In 2012 Polterovich conjectured a quantitative version of this theorem. I will discuss three interconnected topics: a solution of this conjecture in dimension two (with Lev Buhovsky and Alexander Logunov), a link between this problem and Grothendieck's theorem from functional analysis (with Efim Gluskin), and new results related to the Floer-theoretical approach to this conjecture (with Yaniv Ganor).

Last modified Wed, 24 Feb, 2021 at 21:46