# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### Lagrangian cell complexes and Markov numbers

Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of (2 pi / p). We call the resulting cell complex a 'p-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels. In particular, we will see that a p-pinwheel admits a Lagrangian embedding in CP^2 if and only if p is a Markov number. Time permitting, I will discuss nondisplaceability results, which are a purely symplectic analogue of the Hacking-Prokhorov classification of Q-Gorenstein degenerations of CP^2.

##### Cost of splitting Lagrangians

Assume that the derived Fukaya category of a symplectic manifold admits a collection of triangular generators. By definition, this means that any other Lagrangian submanifold which is an object of this category can be decomposed in terms of exact triangles involving the generators. The purpose of the talk is to explain why such a decomposition requires a certain non-trivial amount of ''energy''. The notion of energy that appears here is an extension of Hofer's energy. A first part of the argument consists in bounding from below the shadow of multiple ended Lagrangian cobordisms and will be discussed in some detail in the talk. I will only be sketch a second part that remains, for technical reasons, somewhat speculative at this time.

##### Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof. Also it is useful to see some properties of it such as its relation to the (target space) group action or compatibility with forgetful map.

##### Projective Dehn twist via Lagrangian cobordism

In this talk, I would like to explain my joint work with Weiwei Wu about understanding projective Dehn twist using Lagrangian cobordism.

##### Monotone Lagrangians in cotangent bundles

We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles. This is joint work with Mohammed Abouzaid.

##### Length and Width of Lagrangian Cobordisms

In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant. This is joint work with Lisa Traynor.

##### From Lusternik-Schnirelmann theory to Conley conjecture

In this talk I will discuss a recent result showing that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. Based on joint work with Viktor Ginzburg.

##### Towards a theory of singular symplectic varieties

Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field. I will describe a new perspective on this question and motivate its appropriateness in the case of normal crossings singularities. It leads to multifold versions of symplectic sum and cut constructions expected by Gromov and notions of one-parameter families of degenerations of symplectic manifolds and logarithmic tangent bundles in the spirit of the Gross-Siebert program. In our category, the standard triple point condition of algebraic geometry is the only obstruction to the smoothability of NC singularities.

##### Lagrangian Whitney sphere links

Let n>1. Given two maps of an n-dimensional sphere into Euclidean 2n-space with disjoint images, there is a Z/2 valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on n, that this vanishes when the components are immersed Lagrangian spheres each with exactly one double point of high Maslov index. This is joint work with Tobias Ekholm.

##### The gauged symplectic sigma-model

I will recall the construction of the space of states in a gauged topological A-model. Conjecturally, this gives the quantum cohomology of Fano symplectic quotients: in the toric case, this is Batyrev's presentation of quantum cohomology of toric varieties. Time permitting, I will discuss the role of "Coulomb branches" in gauge theory in relation to equivariant quantum and symplectic cohomology.

##### Rectification and the Floer complex: Quantizing Lagrangians in T^N

We shall give a construction of the quantized sheaf of a Lagrangian submanifold in T^N and explain a number of features and applications.

##### C^0 Hamiltonian dynamics and a counterexample to the Arnold conjecture

After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a C^0 counterexample to the Arnold conjecture in dimensions four and higher. This is joint work with Lev Buhovsky and Vincent Humiliere.

##### Contact manifolds with flexible fillings

In this talk, I will prove that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, I will show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Using similar methods, I will construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and using positive symplectic homology.

##### Positive loops of loose Legendrians and applications

In this talk, I will give a simple and geometrical proof of the following theorem from my thesis: Every loose Legendrian is in a positive loop amongst Legendrian embeddings. The idea is that we add wrinkles to a loose Legendrian and rotate the wrinkles positively then resolve the wrinkles without changing the isotopy class of the initial Legendrian. Finally, I will give some application, especially we can define a strong partial oder on the universal cover of groups of contactomorphisms.

##### Log geometric techniques for open invariants in mirror symmetry

**This is a joint Algebraic Geometry and Symplectic Geometry seminar. Please note different room (322) and start time (4:30). **We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry. The talk will start with a smooth overview of the use of logarithmic geometry in the Gross-Siebert program. We then will discuss various illustrations of the use in open invariants, including a description of the symplectic Fukaya category via certain stable logarithmic curves. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. However, the results are expected to generalise to higher dimensional Calabi-Yau manifolds.

##### Relative quantum product and open WDVV equations

The standard WDVV equations are PDEs in the potential function that generates Gromov-Witten invariants. These equations imply relations on the invariants, and sometimes allow computations thereof, as demonstrated by Kontsevich-Manin (1994). We prove analogous equations for open Gromov-Witten invariants that we defined in a previous work. For (CP^n, RP^n), the resulting relations allow the computation of all invariants. The formulation of the open WDVV requires a lift of the big quantum product to relative cohomology. Surprisingly, this brings us to use moduli spaces of disks with geodesic conditions. This is joint work with Jake Solomon. No prior knowledge of the subjects mentioned above will be assumed.

##### Gromov-Witten theory of locally conformally symplectic manifolds and the Fuller index

We review the classical Fuller index which is a certain rational invariant count of closed orbits of a smooth vector field, and then explain how in the case of a Reeb vector field on a contact manifold $C$, this index can be equated to a Gromov-Witten invariant counting holomorphic tori in the locally conformally symplectic manifold $C \times S^1$. This leads us to prove a certain variant of the classical Seifert conjecture for the odd dimensional spheres.

##### C^\infty closing lemma for three-dimensional Reeb flows via embedded contact homology

I will explain an application of embedded contact homology (ECH) to Reeb dynamics. Specifically, I prove so-called $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds. The key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M.Asaoka) will be also presented.

##### Symplectic homology for cobordisms

Symplectic homology for a Liouville cobordism (possibly filled at the negative end) generalizes simultaneously the symplectic homology of Liouville domains and the Rabinowitz-Floer homology of their boundaries. I intend to explain a conceptual framework within which one can understand it, and give a sample application which shows how it can be used in order to obstruct cobordisms between contact manifolds. Based on joint work with Kai Cieliebak and Peter Albers.

##### Liouville sectors and local open-closed map

I will describe joint work-in-progress with Sheel Ganatra and Vivek Shende. We introduce a class of Liouville manifolds with boundary, called Liouville sectors, in which Floer theory is well behaved (a condition on the characteristic foliation of the boundary controls holomorphic curves from escaping). A corollary of this setup is a local-to-global argument for verifying Abouzaid's generation criterion for the wrapped Fukaya category. We verify this criterion for Liouville sectors associated to Nadler's arboreal singularities, and so any Liouville manifold covered by such sectors has a finite generating collection of Lagrangians (conjecturally, all Weinstein manifolds admit such a cover).