Seminars Sorted by Series

In this talk I will present an approach to constructing finite energy foliations by pseudo-holomorphic curves with prescribed asymptotic orbits in the symplectization of a mapping torus. The idea is that so called maximally unlinked braids of...

In this talk, we discuss our work in progress about how degeneration of the divisor at infinity into a normal crossing divisor affects the symplectic homology of an affine variety. From an anti-surgery picture, by developing an anti-surgery formula...

In recent joint work with Borman and Eliashberg, a new definition of overtwisted contact structures was given for high-dimensional contact manifolds, which were then classified up to isotopy. However, the definition is fairly cumbersome, so much so...

Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez's theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted...

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplectic cochain complex of a Liouville domain and obtain two periodic symplectic...

The Atiyah-Bott localization formula has become a valuable tool for computation of symplectic invariants given in terms of integrals on the moduli spaces of holomorphic stable maps. In contrast, the ``open'' moduli spaces, of stable maps of marked...

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the...

In recent work of Braden, Licata, Proudfoot, and Webster, a "symplectic duality" was described between pairs of module categories $O(M)$, $O(M')$ associated to certain pairs of complex symplectic manifolds $(M, M')$. The duality generalizes the...

In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one...

How small is the smallest period of a closed trajectory of a Reeb flow? In this talk I will present recent answers to instances of this question in three-dimensions which reveal connections between systolic and symplectic geometry. I will present...

In this talk we focus on Legendrian presentations of Weinstein manifolds; in particular, we discuss loose Legendrian embeddings and their absolute analogues, flexible Weinstein manifolds and overtwisted contact structures. The required definitions...

I will introduce a "low-area" version of Floer cohomology of a non-monotone Lagrangian submanifold and prove that a continuous family of Lagrangian tori in $\mathbb CP^2$, whose Floer cohomology in the usual sense vanishes, is Hamiltonian non...

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to...

We compare absolute and relative Gromov-Witten invariants with the basic contact vector for very positive divisors. For such divisors, one might expect that these invariants are the same up to a natural multiple. We show that this is indeed the case...

A much studied combinatorial object associated to a polytope is its "Ehrhart function". This is typically studied for integral or rational polytopes. The first part of the talk will be about joint work with Li and Stanley extending this theory to...

I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I...

I'll outline the construction and computation of a Floer homology theory for contact manifolds which are prequantization spaces over monotone symplectic manifolds, and of the spectral invariants resulting therefrom, and present some applications...

In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether $C^0$-small positive loops exist. We give a...

After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called "subflexible", which are not flexible but become so after attaching some Weinstein handles. For example, the standard...

We present some homological mirror symmetry statements for the singularities of type $T_{p,q,r}$. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types $A$, $D$ and $E$. We will consider some symplectic...

In this talk I will define and explain some basic notions from stable homotopy theory, and illustrate how it relates to (and refines) the notion of Floer homology in some simple cases. I will also discuss what extra kind of information this...

Given a Legendrian surface in a contact one jet space, there is a local combinatorial DGA associated to the cell decomposition of the base projection of the Legendrian. The DGA is quasi-isomorphic to the Legendrian contact DGA. Using this...

I'll outline recent results with Steven Sivek classifying the Stein fillings, up to topological homotopy equivalence, of the canonical contact structure on the unit cotangent bundle of a surface. The proof begins with Li, Mak and Yasui's technology...

We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either...

We define an invariant of contact structures in dimension three based on the contact invariant of Ozsvath and Szabo from Heegaard Floer homology. This invariant takes values in $\\mathbb Z_{\\geq0}\\cup\\{\\infty\\}$, is zero for overtwisted contact...

McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers...

The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish...

We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes (joint work with Mustafa Korkmaz) which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small...

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...

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...

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...

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

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...

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...

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...

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...

Let $n > 1$. Given two maps of an $n$-dimensional sphere into Euclidean $2n$-space with disjoint images, there is a $\mathbb Z/2$ valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on...

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...

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

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...

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...

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...

We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry.

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...

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...

$C^r$ closing lemma is an important statement in the theory of dynamical systems, which implies that for a $C^r$ generic system the union of periodic orbits is dense in the nonwondering domain. $C^1$ closing lemma is proved in many classes of...

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...

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...

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.

We present a full $h$-principle (relative, parametric, $C^0$-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...