Seminars Sorted by Series

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...

We will explain a definition of open Gromov-Witten invariants on the rational elliptic surfaces and explain the connection of the invariants with tropical geometry. For certain rational elliptic surfaces coming from meromorphic Hitchin system, we...

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to $(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk...

Knot contact homology is a knot invariant derived from counting holomorphic curves with boundary on the Legendrian conormal to a knot. I will discuss some new developments around the subject, including an enhancement that completely determines the...

One of the key problem in mirror symmetry is that singular fibres in higher dimensional SYZ fibrations of sufficiently interesting spaces (e.g. the quintic 3-fold) have bad singularities. This makes it unreasonable to directly define their...

We introduce the notion of critical toric surfaces, with a focus on the so-call "critical toric blow-ups". These are symplectic manifolds with a toric action on the complement of an isotropic Lagrangian skeleton. We explain some applications of this...

I will report on recently discovered relations between closed Gromov-Witten theory of a Fano variety and open Gromov-Witten theory of Lagrangian submanifolds contained in it. The focus will be on the result saying that the quantum period of a Fano...

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan...

(Joint work with J. Palmer) Suppose that $\phi_t: L \to X, t \in [0,T]$ is a family of Lagrangian immersions flowing under a Maslov flow such as a reverse mean curvature flow, or reverse mean curvature flow coupled to a K\"ahler-Ricci flow as...

Ribbon graphs capture the topology of open Riemann surfaces in an elementary combinatorial form. One can hope this is the first step toward a general theory for open symplectic manifolds such as Stein manifolds. We will discuss progress toward such...

(Joint work with Chris Woodward) Consider a Lagrangian submanifold $\bar L$ in a GIT quotient $\bar X = X//G$. Besides the usual Fukaya $A_\infty$ algebra $Fuk(\bar L)$ defined by counting holomorphic disks, another version, called the quasimap...

I will present the proof of the split-generation of the compact Fukaya categories of certain Milnor fibers by vanishing cycles. The proof is based on the verification of the Koszul duality between compact and wrapped Fukaya $A_\infty$ algebras of...

Consider a Calabi-Yau manifold which arises as a member of a Lefschetz pencil of anticanonical hypersurfaces in a Fano variety. The Fukaya categories of such manifolds have particularly nice properties. I will review this (partly still conjectural)...

In a joint work with B. Chantraine, P. Ghiggini, and R. Golovko we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. half-dimensional) handles in a Weinstein handle decomposition...

For given a Lagrangian in a symplectic manifold, one can consider deformation of A-infinity algebra structures on its Floer complex by degree 1 elements satisfying the Maurer-Cartan equation. The space of such degree 1 elements can be thought of as...

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is...

We discuss interactions between quantum mechanics and symplectic topology including a link between symplectic displacement energy, a fundamental notion of symplectic dynamics, and the quantum speed limit, a universal constraint on the speed of...

We will discuss the first steps in an approach to proving homological mirror symmetry for Looijenga pairs through tropical Lagrangian sections. Namely, we will see how to construct these Lagrangian sections from tropical data corresponding to line...

30 years ago I proved that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. I also optimistically claimed at the same paper that similar methods could be used to prove a multi-parametric version: the space of tight...

The Gross-Siebert program suggests that mirror symmetry is mediated by the combinatorial data of a dual pair of integral affine manifolds with singularities and polyhedral decomposition. Much is now understood about the passage from the...

In this talk, as a continuation of my talk in the Members’ Colloquium but with a specialized audience in mind, I will discuss in more detail some of the general geometric and dynamical structures underlying the theoretical aspects of the restricted...

The 2011 PhD thesis of Farris outlined a scheme to show that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a $Z/2Z$-graded group to the exterior algebra of the homology of its base. In addition to providing an overview...

Legendrian torus knots were classified by Etnyre and Honda. I will explain the classification of Legendrian torus links. In particular, I will describe restrictions on the Legendrian torus knots that can be realized as the components of a Legendrian...

We describe two extensions, called the virtual Morse-Bott index and circle-equivariant virtual Morse-Bott index, of the classical Morse-Bott index of a Morse-Bott function on a smooth manifold to the setting of (a) suitably defined analytic...