Symplectic Geometry
We show that for a closed Legendrian submanifold in a 1-jet bundle, if there is a sheaf with compact support, perfect stalk and singular support on that Legendrian, then (1) the number of Reeb chords has a lower bound by half of the sum of…
The Rotating Kepler Problem (RKP) arises as a fundamental model in celestial mechanics, appearing as a limiting case of the circular restricted three-body problem. It offers a tractable yet rich framework for studying periodic orbits,…
Let G be a complex reductive group and D a finite subset of a compact Riemann surface X. It was shown in [BJ] that the moduli space of G-characters of the complement of D in X has a natural Poisson structure. We show that the moduli space…
It is known that the union of fibers over elliptic singularities of an almost toric fibered (ATF) closed symplectic four-manifold forms a symplectic log Calabi-Yau (LCY) divisor. In this paper, we show the converse: any symplectic LCY…
Under Floer theoretic conditions, we obtain quantitative estimates on the closeness (Hausdorff distance, flat norm, F-metric) between two Lagrangians, depending on the smallness of Lagrangian angles. Some applications include a strong-weak…
We extend the family of capacities given by McDuff and Siegel by including a constraint $\ell$ on the number of positive asymptotically cylindrical ends of curves showing up in the definition. We prove a generalized computation formula for…
The Mikami-Weinstein theorem is a generalization of the classical Marsden-Weinstein-Meyer symplectic reduction theorem to the case of symplectic groupoid actions. In this paper, we introduce the notion of a cosymplectic groupoid action on a…
In this article, we define an index of Maslov type for general symplectic paths which have two arbitrary end points. This Maslov-type index is a partial generalization of the Conley-Zehnder-Long index in the sense that the degenerate set of…
We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over…
We show that the action of spacetime vector fields on the variational bicomplex of general relativity has a homotopy momentum map that extends the map from vector fields to conserved currents given by Noether's first theorem to a morphism…
Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows,…
In earlier work, we constructed counterexamples to Lagrangian Poincar\'e recurrence for many toric symplectic four manifolds. Here we provide a few more examples extending the family of counterexamples to include all non-monotone toric…
A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be…
We adapt and generalize McDuff's method of probes from toric system to so called semitoric integrable systems and apply it to study the (non)displaceability properties of the fibers of $3$ examples of semitoric integrable systems.
We study the closure of a complex subtorus in a toric manifold. If the closure of the complex subtorus is a smooth complex submanifold in the toric manifold, then the subtorus action on such submanifold is Hamiltonian. In this case, we may…
In this article, circular arcs are considered both individually and as elements of a piecewise circular curve. The endpoint parameterization proves to be quite advantageous here. The perspective of symplectic geometry provides new vectorial…
Assume $M$ to be $\mathbb R^2$ or a closed surface of genus $g \geq 1$ and $\omega$ a symplectic form on $M$. Let $\varphi: M \to M$ be a symplectomorphism with hyperbolic fixed point $x$ and transversely intersecting stable and unstable…
Given a Liouville manifold $M$, we introduce an invariant of $M$ that we call the Heegaard Floer symplectic cohomology $SH^*_\kappa(M)$ for any $\kappa \ge 1$ that coincides with the symplectic cohomology for $\kappa=1$. Writing $\hat{M}$…
For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This…
We consider the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$. Previous work suggests that determining the entirety of $c_b(a)$ for…