# Seminars & Events for Symplectic Geometry Seminar

##### Fukaya categories and variation of symplectic form

I hope to talk more about how to find generators for Fukaya categories using symplectic version of the minimal model program in examples such as symplectic quotients of products of spheres and moduli spaces of parabolic bundles.

##### The simplification of caustics

We present a full hh-principle (relative, parametric, C0C0-close) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are a refinement of the holonomic approximation lemma and the construction of a local wrinkling model for Lagrangian and Legendrian submanifolds.

##### The many forms of rigidity for symplectic embeddings

We look at the following chain of symplectic embedding problems in dimension four.

**$E(1,a) \to Z^4(A) E(1,a) \to C^4(A) E(1,a) \to P(A,bA) (b \in \mathbb{N}_{\geq 2}) E(1,a) \to T^4(A)$**

Here $E(1,a)$ is a symplectic ellipsoid, $Z^4(A)$ is the symplectic cylinder $D^2(A) \times \mathbb{R}^2, C^4(A) = D^2(A) \times D^2(A)$ is the cube and $P(A,bA) = D^2(A) \times D^2(bA)$ the polydisc, and $T^4(A) = T^2(A) \times T^2(A)$, where $T^2(A)$ is the 2-torus of area $A$. In each problem we ask for the smallest $A$ for which $E(1,a)$ symplectically embeds. The answer is very different in each case: total rigidity, total flexibility with a hidden rigidity, and a two-fold subtle transition between them. The talk is based on works by Cristofaro-Gardiner, Frenkel, Latschev, McDuff, Muller, and myself.

##### The stabilized symplectic embedding problem

I will discuss some recent work (mostly joint with Dan Cristofaro-Gardiner and Richard Hind) on the stabilized symplectic embedding problem for ellipsoids into balls. The main tools come from embedded contact homology.

##### On Zimmer's conjecture

The group $Sl_n({\mathbb Z})$ (when $n > 2$) is very rigid. For example, Margulis proved all its linear representations come from representations of $Sl_n({\mathbb R})$ and are as simple as one can imagine. Zimmer's conjecture states that certain "non-linear" representations (group actions by diffeomorphisms on a closed manifold) come also from simple algebraic constructions. For example, conjecturally the only action on $Sl_n({\mathbb Z})$ on an $(n-1)$ dimensional manifold (up to some trivialities) is the one on the $(n-1)$ sphere coming projectivizing natural action of $Sl_n({\mathbb R})$ on ${\mathbb R}^n$. I'll describe some recent progress on these questions due to A. Brown, D. Fisher and myself.

##### Sheaves and contact non-squeezing in $R^{2n} \times S^1$

**Please note special time (9:30 a.m.) **In this talk I will introduce a way to associate a triangulated category of sheaves with a domain of $R^{2n} \times S^1$. The cohomological information on the category side helps to detect the contact non-squeezing property of the domain on the topology side.

##### Contact non-squeezing in ${\mathbb R}^{2n} \times S^1$ by other means

In this talk, following right after Chiu's, I will summarize two other tools capable of detecting the non-squeezing property of pre-quantized balls in ${\mathbb R}^{2n} \times S^1$. One of these is a ${\mathbb Z}_k$-equivariant version of contact homology, the other is in terms of generating functions.

##### Symplectic field theory and codimension-2 stable Hamiltonian submanifolds

Motivated by the goal of establishing a "symplectic sum formula" in symplectic field theory, we will discuss the intersection behavior between punctured pseudoholomorphic curves and symplectic hypersurfaces in a symplectization. In particular we will show that the count of such intersections is always bounded from above by a finite, topologically-determined quantity even though the curve, the target manifold, and the symplectic hypersurface in question are all noncompact.

##### Lagrangian Floer theory in symplectic fibrations

Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fiber Lagrangians. The primary example that we have in mind is the manifold of full flags in ${\mathbb C}^3$ which fibers as ${\mathbb P}^1 \to {\rm Flag}({\mathbb C}^3) \to {\mathbb P}^2$, and a Lagrangian $T^3$ that fibers over the Clifford torus in ${\mathbb P}^2$. It turns out that one can prove the usual transversality and compactness results when the base is a rational symplectic manifold and the fibers are monotone. Assuming that we have a solution to the Maurer--Cartan equation, we then write down a Leray--Serre type spectral sequence which computes the Floer cohomology of the fibered lagrangian.

##### Floer theory in spaces of stable pairs over Riemann surfaces

I will report on joint work with Andrew Lee, which explores the notion that spaces of stable pairs over Riemann surfaces (in the sense of Bradlow and Thaddeus) could form a natural home for a "non-abelian" analog of Heegaard Floer homology for 3-manifolds - just as the g-fold symmetric product is the home of Heegaard Floer homology - thereby circumventing the problems with singularities that beset instanton-type theories. In an initial foray into this area, we set up a theory not for Heegaard splittings but for fibered 3-manifolds, based on fixed-point Floer homology. We show that, when the fiber has genus 1, it contains the expected information from the Seiberg-Witten Floer theory of the fibered 3-manifold.

##### String topology coproduct: geometric and algebraic aspects

The string topology coproduct is an intersection type operation, originally described by Goresky-Hingston and Sullivan, which considers transverse self-intersections on chains of loops in a smooth manifold and splits loops at these intersection points. The geometric chain level construction of string topology operations involves deforming chains to achieve certain transversality conditions and these deformations introduce higher homotopy terms for algebraic compatibilities and properties. The chain level theory for the coproduct is much more subtle than other operations and thus expected to depend on finer information of the underlying manifold.