Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symplectic structure perturbations and continuity of symplectic
  invariants   This paper studies how symplectic invariants created from Hamiltonian Floer
theory change under the perturbations of symplectic structures, not necessarily
in the same cohomology class. These symplectic invariants include spectral
invariants, boundary depth, and (partial) symplectic quasi-states. This paper
can split into two parts. In the first part, we prove some energy estimations
that control the shifts of symplectic action functionals. These directly imply
positive conclusions on the continuity of spectral invariants and boundary
depth, in some important cases including any symplectic surface $\Sigma_{g
\geq1}$ and any closed symplectic manifold $M$ with $\dim_{\mathcal K} H^2(M;
\mathcal K) = 1$. This follows by applications on some rigidity of the subsets
of a symplectic manifold in terms of heaviness and superheaviness, as well as
on the continuity property of some symplectic capacities. In the second part,
we generalize the construction in the first part to any closed symplectic
manifold. In particular, to deal with the change of Novikov rings from
symplectic structure perturbations, we construct a family of variant Floer
chain complexes over a common Novikov-type ring. In this set-up, we define a
new family of spectral invariants called $t$-spectral invariants, and prove
that they are upper semicontinuous under the symplectic structure
perturbations. This implies a quasi-isometric embedding from $(\mathbb
R^{\infty}, |-|_{\infty})$ to $({\widetilde {\rm Ham}}(M, \omega), |-|_{\rm
Hofer})$ under some dynamical assumption, imitating a result from Usher.

13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th