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

##### Symplectic fillings and star surgery

**This is a joint Symplectic Geometry-Topology seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations called star surgery to construct examples of exotic 4-manifolds.

##### Superconformal simple type and Witten's conjecture on the relation between Donaldson and Seiberg-Witten invariants

We shall discuss two new results concerning gauge-theoretic invariants of "standard" four-manifolds, namely closed, connected, four-dimensional, orientable, smooth manifolds with $b^1=0$ and $b+\geq 3$ and odd. We first describe how the SO(3) monopole link-pairing formula from Feehan and Leness (2002) implies that all standard four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data.

##### Equivariant structures in mirror symmetry

When a variety X is equipped with the action of an algebraic group G, it is natural to study the G-equivariant vector bundles or coherent sheaves on X. When X furthermore has a mirror partner Y, one can ask for the corresponding notion of equivariance in the symplectic geometry of Y. The infinitesimal notion (equivariance for a single vector field) was introduced by Seidel and Solomon (GAFA 22 no. 2), and it involves identifying a vector field with a particular element in symplectic cohomology. I will describe the analogous situation for a Lie algebra of vector fields, and discuss the application of this theory to mirror symmetry of flag varieties. In this situation, we expect to find a close connection to the canonical bases of Gross-Hacking-Keel. This talk is based on joint work with Yanki Lekili and Nick Sheridan.

##### Joint Columbia-IAS-Princeton Symplectic Seminar: Symplectic embeddings from concave toric domains into convex ones

**This is a Joint Columbia-IAS-Princeton Symplectic Seminar. **Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one ellipsoid into another. We explain why ECH capacities give a sharp obstruction to embedding any "concave toric domain" into a "convex" one. We also explain why the ECH capacities of any concave or convex toric domain are determined by the ECH capacities of a corresponding collection of balls. Some of this is joint work with Keon Choi, David Frenkel, Michael Hutchings, and Vinicius Ramos.

##### Joint Columbia-IAS-Princeton Symplectic Seminar: Beyond ECH capacities

**This is a Joint Columbia-IAS-Princeton Symplectic Seminar. ** ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain” (see previous talk). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. In this talk we explain how more refined information from ECH gives stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain.

##### On the Gromov width of polygon spaces

After Gromov’s foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold \((M, \omega)\) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in \((M, \omega)\). I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in \(\mathbb{R}^3\) with edges of lengths \((r_1,\ldots, r_n)\). Under some genericity assumptions on lengths \(r_i\), the polygon space is a symplectic manifold. After introducing this family of manifolds, I will concentrate on the spaces of 5-gons and calculate their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.

##### C^0-characterization of symplectic and contact embeddings

Symplectic and anti-symplectic embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C^0-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C. Sikorav and Y. Eliashberg). The aforementioned rigidity is again an easy consequence.

##### Cyclic homology and S^1-equivariant symplectic cohomology

In this talk, we study two natural circle actions in Floer theory, one on symplectic cohomology and one on the Hochschild homology of the Fukaya category. We show that the geometric open-closed string map between these two complexes is S^1-equivariant, at a suitable chain level. In particular, there are induced maps between equivariant homology theories, natural with respect to Gysin sequences, which are isomorphisms whenever the non-equivariant map is.

##### Gauged linear $\sigma$-model and gauged Witten equation

This is a joint work with Gang Tian. I will talk about the analytical properties of the classical equation of motion in gauged linear $\sigma$-model, which we call the gauged Witten equation. This is a generalization of the Witten equation in Landau-Ginzburg A-model (Fan-Jarvis-Ruan, Witten) and the symplectic vortex equation (Mundet, Cieliebak-Gaio-Salamon). We will also discuss a mathematical definition of the correlation function, using the moduli space of gauged Witten equation, when the curve is fixed.

##### On normal crossings symplectic divisors

I will describe purely symplectic notions of normal crossings divisor and configuration. They are compatible with the existence of the desired auxiliary almost Kahler structures, provided ``existence" is suitably interpreted. These notions lead to a multifold version of Gompf's symplectic sum construction. They also imply that Brett Parker's work on exploded manifolds concerns a multifold version of the usual symplectic sum (or degeneration) formula for Gromov-Witten invariants. This talk will be about general symplectic topology, not GW-invariants. This is joint work with Mark McLean and Mohammad Tehrani.

##### TBA - McGibbon

##### Symplectic forms in algebraic geometry

Imposing the existence of a holomorphic symplectic form on a projective algebraic variety is a very strong condition. After describing various instances of this phenomenon (among which is the fact that so few examples are known!), I will focus on the specific topic of maps from projective varieties admitting a holomorphic symplectic form. Essentially, the only such maps are Lagrangian fibrations and birational contractions. I will motivate why one should care about these types of maps, and give many examples illustrating their rich geometry. I will not assume background material, beyond a basic course in algebraic geometry.

##### Path Products in Projective Space

We compute the mod 2 homology of the space of paths in CP^n with endpoints in RP^n, and its algebra structure with respect to the Pontryagin-Chas-Sullivan product . Our method combines Morse theory with geometry. Joint work with Alexandru Oancea.

##### Symplectic homology via Gromov-Witten theory

Symplectic homology is a very useful tool in symplectic topology, but it can be hard to compute explicitly. We will describe a procedure for computing symplectic homology using counts of pseudo-holomorphic spheres. These counts can sometimes be performed using Gromov-Witten theory. This method is applicable to a class of manifolds that are obtained by removing, from a closed symplectic manifold, a symplectic hypersurface of codimension 2. This is joint work with Samuel Lisi.

##### Existence of Lefschetz fibrations on Stein/Weinstein domains

**Please note special day, time and location. **I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

##### The symplectic displacement energy

To begin we will recall Banyaga's Hofer-like metric on the group of symplectic diffeomorphisms, and explain its conjugation invariance up to a factor. From there we will prove the positivity of the symplectic displacement energy of open subsets in compact symplectic manifolds, and then present examples of subsets with finite symplectic displacement energy but infinite Hofer displacement energy. The talk is based on a joint project with Augustin Banyaga and David Hurtubise.

##### Disc filling and connected sum

In my talk I will report on recent work with Hansjörg Geiges about a strong connection between the topology of a contact manifold and the existence of contractible periodic Reeb orbits. Namely, if the contact manifold appears as non-trivial contact connected sum and has non-trivial fundamental group or torsion-free homology, then the existence is ensured. This generalizes a result of Helmut Hofer in dimension three.

##### From Knots to Clusters: the Path via Sheaves

**Please note special day, time and location.** Given a Legendrian knot, I will construct a category of constructible sheaves, invariant under Legendrian isotopy up to equivalence. The constructble sheaves live on the front plane and are stratified by the front diagram of the Legendrian knot. I will then prove that the "rank-one" subcategory is equivalent to a category of augmentations of the Chekanov-Eliashberg differential graded algebra. The proof boils down to local calculations, which are easy to describe. Next I will apply a similar construction to alternating strand diagrams (such as those generated by bipartite graphs) on surfaces and show how the mooduli spaces of rank-one objects are cluster varieties.

##### IAS-CU - Bendersky-Gitler conference: Nearby Lagrangians are simply homotopic

**Please note special location. **This is a report on joint work in progress with T. Kragh, wherein we prove that a closed exact Lagrangian in a cotangent bundle is simply homotopy equivalent to the base. I will explain the two main ingredients of the proof: (i) realising the Whitehead torsion of the projection to the base as the torsion of a Floer theoretic map and (ii) using a large Hamiltonian deformation to deform the Floer complexes in such a way that this torsion can be shown to be trivial by an action filtration argument.

##### IAS-CU - Bendersky-Gitler conference: Non-Hamiltonian actions with isolated fixed points

**Please note special time and location. **Let a circle act symplectically on a closed symplectic manifold $M$. If the action is Hamiltonian, we can pass to the reduced space; moreover, the fixed set largely determines the cohomology and Chern classes of $M$. In particular, symplectic circle actions with no fixed points are never Hamiltonian. This leads to the following important question: What conditions force a symplectic action with fixed points to be Hamiltonian? Frankel proved that Kahler circle actions with fixed points on Kahler manifolds are always Hamiltonian. In contrast, McDuff constructed a non-Hamiltonian symplectic circle action with fixed tori.