arXiv research

We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…

We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality f…

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

The paper finds non-isotopic exact Lagrangians in symplectic manifolds with $\mathbb{C}^*$-actions.

Constructs a cyclic, filtered, strictly unital curved $A_{\infty}$ category for Lagrangian submanifolds and develops Floer theory.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.

Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.

Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence …

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

Floer homotopy theory applies to Lagrangians, overcoming curvature issues.

We study holomorphic discs with boundary on a Lagrangian submanifold $L$ in a Kaehler manifold admitting a Hamiltonian action of a group $K$ which has $L$ as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in $\mathbf{CP}^3$ …

Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted version of Floer cohomology in the invariant set. We then present applications to S…

Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and o…

New construction of Fukaya-Seidel categories using complex gradient flow equation.

We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor s…

This paper computes fixed point Floer cohomology for Dehn twists on surfaces.

Generalizes Floer homotopy via Morse-Bott theory.

This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove th…

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…

This is a research monograph on symplectic cohomology (disguised as an advanced graduate textbook), which provides a construction of this version of Hamiltonian Floer cohomology for cotangent bundles of closed manifolds. The focus is on the aspects of the theory that have been neglected in the literature: (1) the base …

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $φ$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hatφ$ such that neither $\hatφ$ nor $\hatφ^2$…

Study Lagrangian Floer theory in smooth divisor complements.

New method for Lagrangian Floer homology groups using flow trees.

Study on singularities of Lagrangian immersions with applications in Floer theory.

In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…

Modified proof constructs holomorphic quilts on closed surfaces.

Quantifies closeness of special Lagrangians under Floer conditions.

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conject…

Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.

Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Sei…

Defines s-manifolds and s-manifolds with corners for symplectic applications.

In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the S…

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and t…

We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already ava…

Develops Floer cohomology for 4-manifolds with involutions and links.

The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.

The paper explores the Thomas-Yau conjecture using holomorphic curves and Floer theory.

The study connects knot Floer homology and Lagrangian cobordisms.

We formulate certain sufficient conditions for the symplectic monodromy of an isolated quasihomogeneous singularity to be of infinite order in the relative symplectic mapping class group of the Milnor fibre and give a proof using Maslov classes, stability theory for Lagrangian folds resp. stable Morse theory for genera…

Proves spectral sequence for real Heegaard Floer homology.

Lecture notes on Heegaard Floer homology for beginners.