Seminars Sorted by Series

In this survey talk I will describe developments in the study of the planar circular restricted 3-body problem that were made possible through the use of pseudo holomorphic curves, following the theory developed by Hofer, Wysocki and Zehnder.

We will discuss several examples where ideas from persistent homology have applications to the study of the coarse geometry of groups of Hamiltonian diffeomorphisms, equipped with the Hofer norm..

The key idea is to consider Hamiltonian Floer...

I will talk about Ginzburg and Gurel's work on Hamiltonian pseudo-rotations of projective spaces, in particular, their proof of no fixed point of a pseudo-rotation of projective space is isolated as an invariant set.

I would like to discuss three topics concerning applications of holomorphic curve methods to the planar circular restricted three-body problem. The first is the existence of direct orbits and a conjecture due to Birkhoff. The second is the use of...

I will show that a closed, negatively curved Riemannian manifold of 1/4 pinched negative curvature has constant negative curvature if and only if the Lyapunov spectrum of its geodesic flow is the same as that of a hyperbolic manifold. The Lyapunov...

One of the most useful tools in quantifying the complexityof a dynamical system are the concepts of topological and metricentropy. While relevant in several contexts, entropy plays aparticularly important role in describing the behaviour of...

We consider the elliptic restricted three-body problem as a perturbation of the circular problem, with the perturbation parameter being the eccentricity of the orbits of the primaries. We show that for every suitably small, non-zero perturbation...

I will discuss some technical details regarding properties of feral pseuodoholomorphic curves, specifically those properties arising from stretching constructions and as components of limit buildings arising from attempts to compactify certain...

Semitoric systems are a type of 4-dimensional integrable system which has been classified by Pelayo-Vu Ngoc in terms of five invariants, one of which is a family of polygons generalizing the Delzant polygons which classify 4-dimensional toric...

A symplectic Lie groupoid is a Lie groupoid with a multiplicative symplectic form. We take the perspective that such an object is symplectic manifold with an extra categorical structure. Applying the machinery of Floer theory, the extra structure is...

In this talk, which reports on joint work with Jungsoo Kang, we formulate a local systolic inequality for Reeb flows and Hamiltonian diffeomorphisms, which we establish in low dimension. As a consequence, we derive bounds for the minimal magnetic...

For G a Lie group acting on a symplectic manifold and preserving a pair of Lagrangians, we use techniques from infinity category theory to construct a G-equivariant Floer homology of L0 and L1 without equivariant transversality. We give a sample...

Given a Weinstein domain $M$ and a compactly supported, exact symplectomorphism $\phi$, one can construct the open symplectic mapping torus $T_\phi$. Its contact boundary is independent of $\phi$ and thus $T_\phi$ gives a Weinstein filling of $T_0...

An outstanding problem in smooth ergodic theory is the estimation from below of Lyapunov exponents for maps which exhibit hyperbolicity on a large but non- invariant subset of phase space. It is notoriously difficult to show that Lypaunov exponents...

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...