Symplectic Geometry

We introduce the notions of flat translated chains of contactomorphisms and periodic flat translated chains of finite sequences of contactomorphisms, and extend to these notions the theorem of Viterbo (1992) on the multiplicity of periodic…

In this paper, we study obstructions to smoothing nodal orbicurves with orbighosts mapped into cyclic quotient singularities. As an application, we show, under some assumptions, that the orbifold Hecke algebra of a complex global quotient…

We prove that, given any contact $3$-manifold and any computable function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, there exists a defining contact form and a Poincar\'e section of its Reeb flow whose partially defined return map computes…

This paper is the first in a series following on our earlier work [arXiv:2205.14516, arXiv:2307.08180] studying the pair-of-pants product on fixed point Floer cohomology. In [arXiv:2205.14516, arXiv:2307.08180] we fully computed this…

Relative multisymplectic geometry studies smooth maps $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ in the mapping-cone complex of $F$, together with the associated Lie $n$-algebras of relative…

Associated to any smooth map $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ -- a \emph{relative $n$-plectic structure} -- is an $L_\infty$-algebra of relative observables $L_{\infty}(F,\varpi)$,…

We develop a theory of Hamiltonian group actions on cosymplectic manifolds. These odd-dimensional manifolds combine a codimension-one symplectic foliation with a distinguished Reeb direction, and arise naturally both in stable Hamiltonian…

Cosymplectic manifolds provide a natural geometric framework for codimension-one symplectic foliations and arise throughout geometry and mathematical physics. We develop an equivariant cohomological theory for cosymplectic manifolds,…

We study an extension of the algebra of the infrared to curve-valued potentials, focusing on the elliptic curve case. Given a finite configuration of points on an elliptic curve, we construct associated \(L_\infty\)- and…

We prove that any closed orientable hypersurface in a contact manifold of dimension five or greater is isotopic to a robustly non-convex hypersurface via an arbitrarily $C^0$-small isotopy. This strengthens a recent result of the first…

We give a proof of Conjecture $1.3$ of [SBEAS24].

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. In this article we provide a positive answer for the rotating Kepler problem by generalizing…

This survey article discusses the emerging interaction between conformally symplectic topology and dynamics, with a focus on recent developments in convex hypersurface theory.

Given an $L_\infty$-Kuranishi space introduced in \cite{Kim1}, we propose a notion called the Kuranishi chart category. Using the nerve of this category, together with a choice of atlas and a simplicial description of the covering of the…

We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of points in the zero locus of the Kuranishi section. We show that these objects collectively form a category…

We consider closed contact manifolds $(M,\xi)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb…

Tall complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,\omega,\Phi)$ such that $\frac{1}{2}\dim M -\dim T=1$ and the symplectic quotient at each moment value is a surface. The skeleton of a complexity one $T$-space is an important…

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma,\omega)$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov…

We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an…

In this paper, we establish three finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special actions, called GKM$_3$, which generalize smooth toric varieties. Specifically, we prove that, for…