Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projected and near-projected embeddings   A stable smooth map $f:N\to M$ is called "$k$-realizable" if its composition
with the inclusion $M\subset M\times\Bbb R^k$ is $C^0$-approximable by smooth
embeddings; and a "$k$-prem" if the same composition is $C^\infty$-approximable
by smooth embeddings, or equivalently if $f$ lifts vertically to a smooth
embedding $N\to M\times\Bbb R^k$.
  It is obvious that if $f$ is a $k$-prem, then it is $k$-realizable. We refute
the long-standing conjecture that the converse is always true. Namely, for each
$n=4k+3\ge 15$ there exists a stable smooth immersion $S^n\to\Bbb R^{2n-7}$
that is $3$-realizable but is not a $3$-prem.
  We also prove the converse in a wide range of cases. A $k$-realizable stable
smooth fold map $N^n\to\Bbb R^{2n-q}$ is a $k$-prem if $q\le n$ and $q\le
2k-3$; or if $q<n/2$ and $k=1$; or if $q\in\{2k-1,\,2k-2\}$ and $k\in\{2,4,8\}$
and $n$ is sufficiently large.

31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con