Seminars Sorted by Series

First, I will discuss a proof that a Lagrangian torus in \(\mathbb{CP}^2\) arising from a semitoric system described by Weiwei Wu coincides with the image in \(\mathbb{CP}^2\) of Chekanov's exotic Lagrangian torus in \(\mathbb R^4\). I will then...

The time evolution of an ideal incompressible fluid is described by the Euler equations. In this mostly speculative talk I will discuss a connection between stationary solutions of these equations and symplectic topology, as well as possible...

In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop...

Ideas of Kontsevich-Soibelman and Fukaya indicate that there is a natural rigid analytic space (the mirror) associated to a symplectic manifold equipped with a Lagrangian torus fibration. I will explain a construction which associates to a...

The Fukaya category of a fibration with singularities \(W: M \to C\), or Fukaya-Seidel category, enlarges the Fukaya category of \(M\) by including certain non-compact Lagrangians and asymmetric perturbations at infinity involving \(W\); objects...

An implicit atlas on a (moduli) space consists of certain auxiliary (moduli) spaces satisfying a precise set of axioms. We will summarize the construction of implicit atlases on moduli spaces of \(J\)-holomorphic curves, under the assumption of a...

Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. In this talk, I will construct an invariant of a contact structure on a 3-manifold with...

I will discuss some results, some older and some newer, on the general idea that in many birationally-Fano cases, generators of the Fukaya category seem to arise from transitions in the minimal model program. A specific result from a couple of years...

Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and...

We describe the recently observed relation between knot contact homology and open topological strings. This in particular gives two ways of looking at the augmentation variety from knot contact homology, which describe the relevant string theory at...

Witten conjectured a relationship between intersection theory on the moduli space of marked stable curves and the KdV integrable hierarchy. Alternatively, the KdV hierarchy can be rephrased in terms of a representation of half the Virasoro algebra...

More than twenty years ago, Witten proposed a remarkable conjecture connecting the geometry of moduli space of curve or Gromov-Witten theory of point to KdV integrable hierarchies (solved by Kontsevich). Since then, Witten's conjecture opened up a...

In this talk, I will explain the recent joint work with Bohui Chen and Anmin Li on virtual neighborhood techniques for a Fredholm system \((B, E, S)\), where \(E\) is a Banach vector bundle over a Banach manifold \(B\) with a Fredholm section. Using...

I will outline a way to extend the construction of Viterbo's transfer map in symplectic homology to nonexact embeddings of one Liouville domain into another. The approach is motivated by SFT, but at least parts of it can be implemented independently...

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the...

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...

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...

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...

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...

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....

I will report on recent joint work with L. Katzarkov and M. Kontsevich (arXiv:1409.8611) in which we construct Bridgeland stability conditions on partially wrapped Fukaya categories of surfaces. Stable objects in these stability conditions...

We show that for any symplectic manifold of dimension \(2n > 4\), there exists a symplectic hypersurface Poincare dual to some positive multiple of the symplectic form whose \((n - 2)\)th Betti number is as large as we like. The idea here is to find...

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 \(S^1\)...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...