arXiv research

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

Proves Arnol'd's chord conjecture for conormal bundles.

Paper establishes an isomorphism between Fukaya category and bordered knot Floer homology.

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…

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…

This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip $M \setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M \setminus K)$ with the induced kinetic energy H…

We define a torus algebra for Heegaard Floer homology.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

Study volume growth in Milnor fibers using real Lagrangians.

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…

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshit…

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

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…

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…

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 …

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

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…

A mathematical isomorphism connects Floer homology to DAHA representations.

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…

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

Study computes SL(2,C) Floer cohomology for surgeries on knots.

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 theory constructs filtrations on quantum cohomology for symplectic manifolds.

Develops Floer theory for 3-manifold covers using equivariant structures.

Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.

We give a presentation for the Floer cohomology ring $HF^*(Σ\times S^1)$, where $Σ$ is a Riemann surface of genus bigger than one, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of flat SO(3)-connections of odd degree over $Σ$. We study the spectrum of the action o…

Study Morse theory on loop spaces and Hecke algebras.

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

This is the first of a series of two articles where we construct a version of wrapped Fukaya category $\mathcal W\mathcal F(M\setminus K;H_{g_0})$ of the cotangent bundle $T^*(M \setminus K)$ of the knot complement $M \setminus K$ of a compact 3-manifold $M$, and do some calculation for the case of hyperbolic knots $K …

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

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

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-d…

Introduces linear K-systems for Hamiltonian Floer theory.

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…

Extends equivariant contact structure results to mod p L-spaces.

This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. This paper describes what is involved in the construction.

Floer cohomology is computed for certain elements of the mapping class group of a surface $Σ$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $Σ$. The computations cover a certain class of pseudo-Anasov maps.

New invariant recovers known contact element and considers finite coverings.

Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.

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

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane---that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory---is a cohomological field theory which describes the topology of the moduli space o…

New invariant connects symplectic fillings and contact structures.

We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance.

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

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 …

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ a…

This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as mo…

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