Related papers: Cyclotomic Structures in Symplectic Topology

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to…

In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…

Given a closed symplectic manifold $(M,\omega)$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ${\hbox{\it Ham}} (M,\omega)$ by means of the Hofer metric on ${\hbox{\it Ham}}…

We associate an invariant called the completed Tate cohomology to a filtered circle-equivariant spectrum and a complex oriented cohomology theory. We show that when the filtered spectrum is the spectral symplectic cohomology of a Liouville…

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the…

Given a Liouville manifold, we compute a Floer-homotopical invariant -- the complexification of the lift of symplectic cohomology to complex cobordism -- in terms of a classical Floer-theoretic invariant, namely, symplectic cohomology…

Given a closed connected symplectic manifold $(M,\omega)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{\mu_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism…

On a symplectic manifold $M$, the quantum product defines a complex, one parameter family of flat connections called the A-model or Dubrovin connections. Let $\hbar$ denote the parameter. Associated to them is the quantum $\mathcal{D}$ -…

Adapting the construction of global Kuranishi charts to the contact setting, we associate to any non-degenerate contact manifold a flow category based on Reeb orbits and moduli spaces of pseudo-holomorphic buildings. The construction lifts…

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for…

We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are…

We use quasimap Floer cohomology for varying symplectic quotients to resolve several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an {\em open} subset of…

We introduce here a natural functional associated to any $b \in QH_* (M, \omega)$: \emph{spectral length functional}, on the space of "generalized paths" in $ \text {Ham}(M, \omega)$, closely related to both the Hofer length functional and…

On any closed symplectic manifold we construct a path-connected neighborhood of the identity in the Hamiltonian diffeomorphism group with the property that each Hamiltonian diffeomorphism in this neighborhood admits a Hofer and spectral…

Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat…

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative…

We prove global equivariant refinements of Miller's stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode compatible…

Floer theory relates the dynamics of Hamiltonian isotopies and the homology of the ambient manifold. It was extended to similarly relate the dynamics of symplectic isotopies and the Novikov homology associated to their flux. We discuss this…

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an…

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on…