Related papers: A beginner's introduction to Fukaya categories

We investigate a possible theory of higher Fukaya categories associated to $n$-shifted symplectic stacks, where $n \geq 0$. We consider two paradigmatic cases, the shifted cotangent stack of a smooth manifold and the coadjoint stack of a…

There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship, with a view to applications in low-dimensional…

We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive…

This paper constructs and studies the Rabinowitz (wrapped) Fukaya category, a categorical invariant of exact cylindrical Lagrangians in a Liouville manifold whose cohomological morphisms, ``Rabinowitz wrapped Floer homology groups" measure…

These are the notes for a lecture series on Heegaard Floer homology, given by the first author at the R\'enyi Institute in January 2023, as part of a special semester titled ``Singularities and Low Dimensional Topology''. Familiarity with…

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $\mathcal{CO}_\lambda \colon \operatorname{QH}^*(X) \to \operatorname{HH}^*(\mathcal{F}…

In this article, the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory…

We define for an associative algebra an $A_{\infty}$ category whose objects are automorphisms of this algebra. This construction has some resemblance with Fukaya'a categories related to Floer cohomology.

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a…

Given an exact relatively Pin Lagrangian embedding Q in a symplectic manifold M, we construct an A-infinity restriction functor from the wrapped Fukaya category of M to the category of modules on the differential graded algebra of chains…

In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as…

Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the…

In this paper we use recollements to investigate partially wrapped Fukaya categories of surfaces with marked points. In particular, we show that cutting surfaces gives rise to recollements of the corresponding partially wrapped Fukaya…

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which…

We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. As applications, we demonstrate the existence of Hamiltonian…

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are…

In this paper, we construct a Fukaya category of any infinite type surface whose objects are gradient sectorial Lagrangians. This class of Lagrangian submanifolds is introduced by one of the authors in [Oh21b] which can serve as an object…

We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian…

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…

In part I, using the theory of $\infty$-categories, we constructed a natural ``continuous action'' of $\operatorname {Ham} (M, \omega) $ on the Fukaya category of a closed monotone symplectic manifold. Here we show that this action is…