Since the beginning of the subject, it has been speculated that Gromov-Witten invariants should admit refinements in complex cobordism. I will propose a resolution of this question based on joint work-in-progress with Abouzaid, building on recent advances in Symplectic Topology (FOP perturbations developed jointly with Xu) and functorial resolution of singularities from algebraic geometry.
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 is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids.
We discuss the shape invariant, a sort of set valued symplectic capacity defined by the Lagrangian tori inside a domain of ℝ4. Partial computations for convex toric domains are sometimes enough to give sharp obstructions to symplectic embeddings, but in general the shape is far from a complete invariant. We then consider continuous families of Lagrangian embeddings, and describe a seemingly close relation to stabilized symplectic embeddings.
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. Despite the algebraic nature of the talk, I promise pictures and examples.
Let H be any smooth function on ℝ4 and let Y be any compact and regular level set. I'll explain a proof that Y admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which moreover have dense union in Y. This improves on a recent result by Fish-Hofer.
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 such an embedding (regardless of the contact structure). I will review the ideas that lead to these results, and discuss recent extensions that can obstruct suitably convex embeddings in closed symplectic 4-manifolds, particularly rational complex surfaces.
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 addition that the flow is non-degenerate and the stable and unstable manifolds of all hyperbolic orbits intersect transversally, a condition which holds generically. I will explain recent joint work resolving this conjecture. Our results also apply to show that every Finsler metric on the two-sphere has either two or infinitely many prime closed geodesics, answering a question attributed to Alvarez Paiva, Bangert and Long.
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 (possibly) first generalization to non-compact fibrations, of Preissman's now classical theorem on non-existence of negative sectional curvature metrics on compact products. One additional goal of the talk is to use the above theory to motivate a very elementary but deep open problem in Riemannian geometry/dynamics concerning existence of Reebable and geodesible sky catastrophes. I will give a partial answer to this problem for surfaces.
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 negative contact structures. Additionally, when the foliation is taut, its contact approximations are (universally) tight. In this talk, I will present a converse result concerning the construction of taut foliations from suitable pairs of contact structures. I will also describe a comprehensive dictionary between the languages of foliations and of (pairs of) contact structures. Although taut foliations are usually considered rigid objects, this contact viewpoint reveals some degree of flexibility. As an application, I will show that taut foliations survive after performing large slope surgeries along transverse knots.
A magnetic system is the toy model for the motion of a charged particle moving on a Riemannian manifold endowed with a magnetic force. To a magnetic flow we associate an operator, called the magnetic curvature operator. Such an operator encodes together the geometrical properties of the Riemannanian structure together with terms of perturbation due to magnetic interaction, and it carries crucial informations of the magnetic dynamics. For instance, in this talk, we see how a level of the energy positively curved, in this new magnetic sense, carries a periodic orbit. We also generalize to the magnetic case the classical Hopf's rigity and we introduce the notion of magnetic flatness for closed surfaces.
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