Tag - Floer theory

Mohammed Abouzaid: Theory of bordisms

6th August, 2024

In this introductory lecture, which should be accessible to a general mathematical audience, I will review the classical bordism theory of manifolds, from its origin in Poincare's work, to the subsequent development by Pontryagin, Thom, Milnor, Wall, and Quillen among others.

Lecture 2: Bordism of orbifolds

An orbifold is a space with additional structure that describes it locally as the quotient of a manifold by a finite group. I will describe Pardon's recent result which reduces the study of orbifolds to the study of manifolds with Lie group actions. Then I will explain the relationship between equivariant and orbifold bordism, and formulation some structural properties of this theory.

Lecture 3: Bordism of derived orbifolds

The notion of a derived orbifold arises naturally in pseudo-holomorphic curve theory, and plays a central role in the emerging field of Floer homotopy. I will explain how it is related to the notion of "homotopical bordism" due to tom Dieck in the 1970s, and formulate some conjectures about its structure in the complex oriented case.

Egor Shelukhin: Symplectic Aspects of the Hilbert-Smith Conjecture and p-adic Actions

6th May, 2024

I will discuss a recent proof of new cases of the Hilbert-Smith conjecture for actions by homeomorphisms of symplectic nature. In particular, it rules out faithful actions of the additive p-adic group in this setting and provides further obstructions to group actions in symplectic topology. The proof relies on a new approach to this circle of questions combined with power operations in Floer cohomology and quantitative symplectic topology.

Yan-Lung Leon Li: Equivariant Lagrangian Correspondence and a Conjecture of Teleman

12th April, 2024

It has been a continuing interest, often with profound importance, in understanding the geometric and topological relationship between a Hamiltonian G-manifold Y and a symplectic quotient X. In this talk, we shall provide precise relations between their (equivariant) Lagrangian Floer theory. In particular, we will address a conjecture of Teleman, motivated by 3d mirror symmetry, on the 2d mirror construction of X from that of Y, which generalises Givental-Hori-Vafa mirror construction for toric varieties. The key technical ingredient is the Kim-Lau-Zheng’s equivariant extension of Fukaya’s Lagrangian correspondence tri-modules over equivariant Floer complexes.

Lenhard Ng: New Algebraic Invariants of Legendrian Links

25th March, 2024

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.

Thomas Mark: Constraints on Contact Type Hypersurfaces in Symplectic 4-Manifolds

4th March, 2024

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.

Francesco Morabito: Hofer Norms on Braid Groups and Quantitative Heegaard-Floer Homology

5th January, 2024

Given a Lagrangian link with k components it is possible to define an associated Hofer norm on the braid group with k strands. In this talk we are going to detail this definition, and explain how it is possible to prove non-degeneracy if k = 2 and certain area conditions on the Lagrangian link are met. The proof is based on the construction, using Quantitative Heegaard-Floer Homology, of a family of quasimorphisms which detect linking numbers of braids on the disc.

Han Lou: On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds

27th October, 2023

Arnold's conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this talk, I will show that the Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology.

Mohammed Abouzaid: Bordism and Floer theory

5th October, 2023

In the late 1980s Andreas Floer revolutionized low-dimensional and symplectic topology by discovering the existence an extension of Morse theory to an infinite-dimensional setting where the standard methods of variational calculus fail. While he foresaw that his theory should be able to encompass generalized homology theory (bordism, K-theory, ...), severe foundational difficulties prevented any significant progress on this question until two years ago. I will explain the advances that have been made on two fronts: (I) defining concrete models, in terms of equivariant vector bundles, for the moduli spaces that appear in Floer theory, and (II) understanding the geometric consequences of lifting Floer homology to generalized homology theories. I will end by formulating how the notion of derived orbifold bordism provides a universal receptacle for Floer's invariants, and its descendants.

Roman Krutowski: Maslov Index Formula in Heegaard Floer Homology

21st April, 2023

The formula introduced by Robert Lipshitz for Heegaard Floer homology is now one of the basic tools for those working with HF homology. The convenience of the formula is due to its combinatorial nature. In the talk, we will discuss the recent combinatorial proof of this formula.