Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vu Ngoc's Conjecture on focus-focus singular fibers with multiple
  pinched points   We classify, up to symplectomorphisms, a neighborhood of a singular fiber of
an integrable system (which is proper and has connected fibers) containing $k >
1$ focus-focus critical points. Our result shows that there is a one-to-one
correspondence between such neighborhoods and $k$ formal power series, up to a
$(\mathbb{Z}_2 \times D_k)$-action, where $D_k$ is the $k$-th dihedral group.
The $k$ formal power series determine the dynamical behavior of the Hamiltonian
vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 \colon
(M, \omega) \to \mathbb{R}$ of the integrable system on the symplectic manifold
$(M,\omega)$ via the differential equation $\omega(X_{f_i}, \cdot) =
\mathop{}\!\mathrm{d} f_i$, near the singular fiber containing the $k$
focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.

22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys