Seminars Sorted by Series

For Legendrian and transverse links in the 3-sphere, Ozsvath, Szabo, and Thurston defined combinatorial invariants that reside in grid homology. Known as the GRID invariants, they are effective in distinguishing some transverse knots that have the...

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while...

We discuss recent developments in establishing uniform bounds on the spectral norm and related invariants in the absolute and relative settings. In particular, we describe new progress on a conjecture of Viterbo asserting such bounds for exact...

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two)...

In this survey talk, I will review the known constructions of mathematical theories of gauged sigma model and its relations with Gromov--Witten theory and FJRW theory. I will emphasize the analytic side, but will also mention related algebraic...

I'll outline the construction of an invariant for knots in homology 3-spheres which can be thought of as an SL(2,C) analog of Kronheimer and Mrowka's singular knot instanton homology. This invariant is similar to an invariant of 3-manifolds...

Analogous to Weinstein structures in symplectic geometry, there is a notion of convex structures in contact geometry. We discuss an explicit surgery theory for contact manifolds with convex structures, showing that they naturally decompose into...

I will describe a new family of symplectic capacities defined using rational symplectic field theory. These capacities are defined in every dimension and give state of the art obstructions for various "stabilized" symplectic embedding problems such...

It is conjectured that contact manifolds admitting flexible fillings have unique exact fillings. In this talk, I will show that exact fillings (with vanishing first Chern class) of a flexibly fillable contact (2n-1)-manifold share the same product...

I will discuss some current joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral J-curves) to study certain divergence free flows in dimension three. In particular, we show that...

I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information. I...

I will explain a polyfold proof, joint with Katrin Wehrheim, of the Arnold conjecture: the number of 1-periodic orbits of a nondegenerate 1-periodic Hamiltonian on a closed symplectic manifold is at least the sum of the Betti numbers. Our proof is a...

Certain `flexible' structures in symplectic and contact topology satisfy h-principles, meaning that their geometry reduces to underlying topological data. Although these flexible structures have no interesting geometry by themselves, I will show how...

Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results...

I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops. The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.I did not manage to solve this problem, but I...

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality...

We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case...

We will discuss work from https://arxiv.org/abs/1908.04227 on a homological mirror symmetry result for a complex genus 2 curve. We will first note how the result fits into the broader framework of HMS examples. Then we will describe the geometric...

I will explain how coisotropic submanifolds of symplectic manifolds can be distinguished among all submanifolds by a criterion ("local rigidity") related to the Hofer energy necessary to disjoin open sets from them. This criterion is invariant under...

I will report on a joint work with M. Abouzaid, S. Guillermou and T. Kragh. The nearby Lagrangian conjecture predicts that a closed exact Lagrangian submanifold in a cotangent bundle must be Hamiltonian isotopic to the zero-section. In particular...

We show evidence that homological mirror symmetry is a phenomenon that exists beyond the world of Kähler manifolds by exhibiting HMS-type results for a family of complex surfaces which includes the classical Hopf surface (S^1 x S^3). Each surface S...

An exact Calabi-Yau structure, originally introduced by Keller, is a special kind of smooth Calabi-Yau structures in the sense of Kontsevich-Vlassopoulos. For a Weinstein manifold, an exact Calabi-Yau structure on the wrapped Fukaya category induces...

Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After briefly reviewing convex surface theory in dimension three...

Symplectic embedding problems are at the core of symplectic topology. Many results have been found involving balls, ellipsoids and polydisks. More recently, there has been progress on problems involving lagrangian products and related domains. In...

The number of embedded pseudo-holomorphic curves in a symplectic manifold typically depends on the choice of an almost complex structure on the manifold and so does not lead to a symplectic invariant. However, I will discuss two instances in which...

I will explain a direct way for defining the Floer homotopy groups of a (framed) manifold flow category in the sense of Cohen Jones and Segal, which does not require any sophisticated tools from homotopy theory (in particular, the notion of a...

Suppose L is a link with n components and the rank of Kh(L;Z/2) is 2^n, we show that L can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and...

This talk will begin with a discussion of the string topology category of a manifold M; this was shown by Cohen and Ganatra to be equivalent as a Calabi-Yau category to the wrapped Fukaya category of T*M. In joint work with Ralph Cohen, we...

We will ask how many Lagrangian tori, say with an integral area class, can be `packed' into a given symplectic manifold. Similarly, given an arrangement of such tori, like the integral product tori in Euclidean space, one can ask about the...

The $C^0$ flux conjecture predicts that a symplectic diffeomorphism that can be $C^0$ approximated by a Hamiltonian diffeomorphism is itself Hamiltonian. We describe how the flux conjecture relates to new instances of the strong Arnol’d conjecture...

Chaidez and Edtmair have recently found the first examples of dynamically convex domains in $R^4$ that are not symplectomorphic to convex domains (called symplectically convex domains), answering a long-standing open question. In this talk we shall...

An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In the last years, interest in this problem has been revitalized since...

A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gurel, I will present a "forced existence" result...

An old open question in symplectic geometry asks whether all normalised symplectic capacities coincide for convex domains in the standard symplectic vector space. I will show that this question has a positive answer for smooth convex domains which...

Results concerning rigidity of Lagrangian submanifolds lie at the heart of symplectic topology, and have been intensively studied since the 1990s. An example for this phenomenon is the concept of Lagrangian Barriers, a form of symplectic rigidity...

In the study of Hamiltonian systems, integrable dynamics play a crucial role. Integrability, however, appears to be a delicate property that is not expected to persist under generic small perturbations. Understanding the essence of this fragility...

The dynamics associated with mechanical Hamiltonian flows with smooth potentials that include sharp fronts may be modeled, at the singular limit, by Hamiltonian impact systems: a class of generalized billiards by which the dynamics in the domain’s...

I'll explain joint work in progress with Abbondandolo and Kang concerning the Clarke dual action functional of convex domains and pseudoholomorphic planes. In dimension 4, I'll explain applications to the knot types of periodic Reeb orbits.

In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by positive and...

Dimitroglou-Rizell-Golovko constructs a family of Legendrians in prequantization bundles by taking lifts of monotone Lagrangians. These lifted Legendrians have a Morse-Bott family of Reeb chords. We construct a version of Legendrian Contact Homology...

I will give a construction of certain Q-valued deformation invariants of (in particular) complete non-positively curved Riemannian manifolds. These are obtained as certain elliptic Gromov-Witten curve counts. As one immediate application we give the...

In their 2001 paper, Hofer, Wysocki and Zehnder conjectured that every autonomous Hamiltonian flow has either two or infinitely many simple periodic orbits on any compact star-shaped energy level; in the same paper, the authors prove this assuming...

In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits...

Legendrian Contact Homology (LCH) was among the first, and is still among the most important, non-classical invariants of Legendrian knots. In this talk, I will tell a story that builds up ever more sophisticated analogues of Poincare Duality in LCH...

We discuss the shape invariant, a sort of set valued symplectic capacity defined by the Lagrangian tori inside a domain of $R^4$. Partial computations for convex toric domains are sometimes enough to give sharp obstructions to symplectic embeddings...

For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and...