辛几何
Co-oriented contact manifolds quite generally describe classical dynamical systems. Quantization is achieved by suitably associating a Schr\"odinger equation to every path in the contact manifold. We quantize the standard contact seven…
In this paper, we study the Floer theory of equivariant Lagrangian correspondences and apply it to derive precise relations between the disc potential of an invariant Lagrangian submanifold and that of its quotient, thereby addressing a…
For any Legendrian knot or link in $\mathbb{R}^3$, we construct an $L_\infty$ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The $L_\infty$ structure incorporates information from rational…
We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of…
The purpose of this paper is to present a fully algebraic formalism for the construction and reduction of $L_\infty$-algebras of observables inspired by multisymplectic geometry, using Gerstenhaber algebras, BV-modules, and the constraint…
In this note we analyze normalized symplectic capacities for two different notions of duality in Lagrangian products. Let $\Phi$ be a $n$-tuple of Young functions with Legendre transform $n$-tuple $\Phi^*$ and $K_{\Phi}$ the unit ball for…
We construct a uniformly bounded symplectic structure on $S^2 \times \mathbb{R}^4$ admitting embeddings by arbitrarily large balls. This provides a counterexample to a recent conjecture of Savelyev. We then prove the conjecture holds for a…
We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces (with their monotone symplectic forms) can be entirely explained in terms of rational sesquicuspidal symplectic curves.…
We define a new algebra associated to a Legendrian submanifold $\Lambda$ of a contact manifold of the form $\mathbb{R}_{t} \times W$, called the planar diagram algebra and denoted $PDA(\Lambda, \mathcal{P})$. It is a non-commutative,…
We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^{3}, \xi_{std})$. We allow our domains,…
The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for…
A brane in a symplectic manifold is a coisotropic submanifold $Y$ endowed with a compatible closed 2-form $F$, which together induce a transverse complex structure. For a specific class of branes we give an explicit description of branes…
In this paper we explain how to construct the EBK spectrum from the marked action spectrum and derive a minimax formula for concave toric domains. In the special case of the billiard on the disk we show that while the action spectrum is…
This paper introduces a new Floer homology for periodic Reeb orbits on the boundaries of Liouville domains. The construction of this Constrained Floer Homology (CFH) is based on the symplectic area functional, restricted to loops satisfying…
We study periodic orbits in a time-dependent two-center Stark-Zeeman system, which models the motion of a charged particle attracted by two fixed Coulomb centers and subject to external magnetic and time-dependent electric fields. A…
We prove that the homology class induced by the rational pseudocycle constructed via domain-dependent perturbations by Cieliebak and Mohnke coincides with the homology class induced by the virtual fundamental class defined through Kuranishi…
In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our…
We circumvent one of the roadblocks in associating Floer homotopy types to monotone Lagrangians, namely the curvature phenomena occurring in high dimensions. Given $N \ge 3$ and $R$ a connective $\mathbb E_1$-ring spectrum, there is a…
We introduce a new construction for tropical Lagrangian surfaces in $(\mathbb{C}^*)^2$. This construction makes the surfaces special Lagrangian, which gives a strong control over the asymptotic behavior of holomorphic disks near each…
We construct the Lagrangian Floer homotopy type, in the exact setting, as a spectrum parameterized over the moduli space of Maslov data. Our primary motivation for this construction is to provide stronger lower bounds for (possibly…