Seminars Sorted by Series

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...

In this talk, we first introduce the notion of a continuous cover of a manifold parametrised by any compact manifold endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the usual sum is replaced by...

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...

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.

I will explain the main ideas of a proof that for generic compatible almost complex structures in symplectic manifolds of dimension at least 6, closed embedded J-holomorphic curves of index 0 are always "super-rigid", implying that their multiple...

The group $\mathrm{SL}_n(\mathbb Z)$ (when $n > 2$) is very rigid, for example, Margulis proved all its linear representations come from representations of $\mathrm{SL}_n(\mathbb R)$ and are as simple as one can imagine. Zimmer's conjecture states...

In this talk I will introduce a way to associate a triangulated category of sheaves with a domain of $\mathbb R^{2n} \times S^1$. The cohomological information on the category side helps to detect the contact non-squeezing property of the domain on...

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...

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...

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...

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...

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...

Consider a Lagrangian torus fibration a la SYZ over a non compact base. Using techniques from arXiv:1510.04265, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity...

In this talk, I will describe an efficient way of compactifying moduli spaces of J-holomorphic maps relative to simple normal crossings (snc) symplectic divisors, including the holomorphic case. The primary goal of this construction is to define...

Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror...

We will discuss how to study the symplectic geometry of $2n$-dimensional Weinstein manifolds via the topology of a core $n$-dimensional complex called the skeleton. We show that the Weinstein structure can be homotoped to admit a skeleton with a...

Abstract: In this talk, I will give a new construction of the Morse-Bott cochain complex, where the underlying vector space is generated by the cohomology of the critical manifolds. This new construction has two nice features: (1) It requires the...

I will motivate the consideration of a special class of odd dimensional sphere bundles over symplectic manifolds. These bundles give a novel topological perspective for symplectic geometry. In particular, the symplectic A-infinity algebra recently...