Seminars Sorted by Series

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...

The contact connected sum is a well-understood operation for contact manifolds. I will focus on the 3-dimensional case and the Weinstein 1-handle model for the contact connected sum. I will discuss how pseudo-holomorphic curves in the...

I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to...

Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in...

I will show that in symplectic space smooth strongly convex bodies with all characteristics planar or all outer billiard trajectories planar are affine symplectic images of a ball.

The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. This function can have a property called an infinite...

I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of S^3 along T(2,q) torus knots to deduce information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages...

In this talk I will discuss a Bennequin type inequality for symplectic caps of $S^3$ with standard contact structure. This has interesting applications which can help us to understand the smooth topology of symplectic caps and smoothly embedded...

Hutchings used Embedded Contact Homology to show the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: if the asymptotic mean action on the boundary is greater than the Calabi invariant, then the...

Embedded contact homology (ECH) is a diffeomorphism invariant of three-manifolds due to Hutchings, defined using a contact form. This very diffeomorphism invariance makes it quite useful when studying contact dynamics, because it is possible to...

I will discuss coproduct structures and why we care about them. Then I will define a new coproduct structure on symplectic cohomology and indicate how to compute it in an example.

Given an area-preserving surface diffeomorphism, what can one say about the topological properties of its periodic orbits? In particular, a finite set of periodic orbits gives rise to a braid in the mapping torus, and one can ask which isotopy...

While much is known about the algebraic structure of the group of volume preserving homeomorphisms of a manifold when the dimension of the manifold is at least three, the structure of these groups in the two-dimensional case has long been mysterious...

Contact homology is a Floer-type invariant for contact manifolds, and is a part of Symplectic Field Theory. One of its first applications was the existence of exotic contact structures on spheres. Originally, contact homology was defined only for...

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a...

In the 1960's, Burgess proved a subconvexity bound for Dirichlet L-functions. However, the quality of this bound was not as strong, in terms of the conductor, as the classical Weyl bound for the Riemann zeta function. In a major breakthrough, Conrey...

Given a Shimura variety, we can construct two kinds of automorphic local systems, i.e., local systems attached to algebraic representations of certain associated algebraic group. The first is based on the classical complex analytic construction...

Machine learning algorithms are ubiquitous in most scientific, industrial and personal domains, with many successful applications. As a scientific field, machine learning has always been characterized by the constant exchanges between theory and...

Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one...

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and...

We will discuss some recent constructions of "exotic" Lagrangian tori in simple symplectic manifolds such as $\mathbb{CP}^2$ (work of Renato Vianna) and $\mathbb R^6$ that are not Hamiltonian isotopic to previously known examples, inspired by wall...

We study symplectic invariants of the open symplectic manifolds $X$ obtained by plumbing cotangent bundles of spheres according to a plumbing tree. We prove that certain models for the Fukaya category $\mathcal F(X)$ of closed exact Lagrangians in...

I will describe how the notion of Lagrangian cobordism, introduced by Arnold in 1980, offers a systematic perspective on the study of Lagrangian topology. There are three aspects that will be emphasized: the relations with the triangulated structure...

We produce examples of non-trivial Hamiltonian fibrations that are not detected by previous methods (the characteristic classes of Reznikov for example), and improve theorems of Reznikov and Spacil on cohomology-surjectivity to the level of...

A point $q$ in a contact manifold $(M,\xi)$ is said to be a translated point of a contactomorphism $\phi$, with respect to a contact form $\alpha$ for $\xi$, if it is a "fixed point modulo the Reeb flow", i.e. if $q$ and $\phi(q)$ are in the same...