Seminars Sorted by Series

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...

We will discuss the existence of rational (multi)sections and unirulings for projective families $f: X \to CP^1$ with at most two singular fibres. In particular, we will discuss two ingredients that are used to construct the above algebraic curves...

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two...

We construct a non-finite type four-dimensional Liouville manifold $M$ and describe a correspondence between certain birational transformations of the complex plane preserving a standard holomorphic volume form and symplectomorphisms of $M$. This...

I will explain a construction of a Legendrian version of embedded contact homology (ECH) for a sutured contact manifold Y along with a collection of Legendrians L contained in the boundary. The chain complex is generated by sets of Reeb orbits and...

I will talk about ongoing work with S. Venugopalan on computing disk potentials (which are counts of holomorphic disks with boundary on a Lagrangian) via multi-directional neck stretching. The focus will be on examples.

I will discuss some recent work showing that a generic area-preserving diffeomorphism of a closed surface has an equidistributed sequence of periodic orbits. The proof uses several properties of spectral invariants from periodic Floer homology...

If G is a Lie group whose adjoint representation preserves a nondegenerate symmetric bilinear form on its Lie algebra (e.g. a semisimple group) and F is the fundamental group of a closed oriented surface S, then the spaces of equivalence classes of...

I will discuss a recent work constructing quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere.  The existence of these quasimorphisms answers a question of Entov, Polterovich, and Py.  As an immediate...

Given a trivalent plane graph embedded in the Euclidean plane (up to isotopy), Treumann and Zaslow constructed and studied a certain associated Legendrian surface embedded in standard contact $R^{5}$, nowadays referred to as a Legendrian 2-weave...

We discuss the history of applying localization results in Lagrangian Floer homology to Heegaard Floer homology to obtain rank inequalities for various versions of the Heegaard Floer homology of branched double covers.  We then use recent...

Because of the presence of non-trivial automorphisms of stable maps, Gromov-Witten invariants of a general symplectic manifold are usually rational-valued. Realizing a proposal of Fukaya-Ono back in the 1990s, I will explain how to construct integer...

Most work on Lagrangian fillings of Legendrian knots to date has concentrated on orientable fillings, but instead I will present some first steps in constructions of and (especially) obstructions to the existence of (decomposable exact) non...

I shall present two billiard-like systems associated with a convex hypersurface in a symplectic space, the outer and an inner ones.  The talk will survey the known results and focus on open problems.

Arnold's famous conjecture on the numbers of fixed points of Hamiltonian diffeomorphisms on symplectic manifolds has motivated numerous important developments in geometry and topology, most notably the invention of Floer homology. In this talk I...

The Chekanov-Eliashberg dg-algebra gives an invariant of Legendrian submanifolds up to Legendrian isotopy and can be used to compute wrapped Fukaya categories of Weinstein manifolds. In this talk we introduce a type of surgery decomposition of...

An Anosov flow $\Phi$ on a closed $3$-manifold $M$ gives rise to a non-Weinstein Liouville structure on $V:=[-1,1] \times M$. Building upon the work of Hozoori, we establish a homotopy correspondence between Anosov flows and certain pairs of contact...

I will first review the recent construction of polyhedral Liouville domains, which are subdomains of a complex torus whose boundary dynamics encodes the singularities of a toric compactification. I will then report on work in progress aimed at...

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were...

A very affine hypersurface is the vanishing locus of a Laurent polynomial in a complex torus; its complement is also a very affine hypersurface, but in two subtly-different ways. The (partially) wrapped Fukaya categories of the hypersurface and its...

ECH capacities have found many applications to symplectic embedding problems, most of which in the toric setting. I will discuss a new application of ECH to studying optimal embeddings for non-toric rational surfaces. The key convex geometric...