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Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence of objects in the Fukaya category. However, there are currently no known non-trivial examples of monotone Lagrangian cobordisms with two ends. We look at an…
We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used…
Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic…
We develop Lagrangian Floer Theory for exact, graded, immersed Lagrangians with clean self-intersection using Seidel's setup. A positivity assumption on the index of the self intersection points is imposed to rule out certain (but not all)…
We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth orientable manifold. The main new idea is the proof that a cotangent fibre generates the…
We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module…
This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the…
In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\Sigma$ via the topological Fukaya category. We prove that the…
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}…
We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland…
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding…
A classical way to construct a Lagrangian in a symplectic manifold $\Sigma$ is to let $\Sigma$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $\Sigma$…
In this article, we study the singularities of Lagrangian immersions into Cartesian product of surfaces. After applying a Hamiltonian isotopy in the Weinstein tubular neighbourhood of the Lagrangian immersion, the singular points of the…
Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_\infty$ endomorphism algebra of such a Lagrangian is…
We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms,…
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line…
We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable…
Given, in the Lagrangian torus fibration $R^4\to R^2$, a Lagrangian submanifold $L$, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies…
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…
We study Lagrangian cobordism groups of oriented surfaces of genus greater than two. We compute the immersed oriented Lagrangian cobordism group of these surfaces. We show that a variant of this group, with relations given by unobstructed…