Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Stratified Spaces of Real Polynomials & Trajectory Spaces of
  Traversing Flows   This paper is the third in a series that researches the Morse Theory,
gradient flows, concavity and complexity on smooth compact manifolds with
boundary. Employing the local analytic models from \cite{K2}, for
\emph{traversally generic flows} on $(n+1)$-manifolds $X$, we embark on a
detailed and somewhat tedious study of universal combinatorics of their
tangency patterns with respect to the boundary $\d X$. This combinatorics is
captured by a universal poset $\Omega^\bullet_{'\langle n]}$ which depends only
on the dimension of $X$. It is intimately linked with the combinatorial
patterns of real divisors of real polynomials in one variable of degrees which
do not exceed $2(n+1)$. Such patterns are elements of another natural poset
$\Omega_{\langle 2n+2]}$ that describes the ways in which the real roots merge,
divide, appear, and disappear under deformations of real polynomials. The space
of real degree $d$ polynomials $\mathcal P^d$ is stratified so that its pure
strata are cells, labelled by the elements of the poset $\Omega_{\langle d]}$.
This cellular structure in $\mathcal P^d$ is interesting on its own right (see
Theorem \ref {th4.1} and Theorem \ref {th4.2}). Moreover, it helps to
understand the \emph{localized} structure of the trajectory spaces $\mathcal
T(v)$ for traversally generic fields $v$, the main subject of Theorem \ref
{th5.2} and Theorem \ref {th5.3}.

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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg