Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Dirichlet to Neumann operator on invariant differential
  forms and equivariant cohomology   In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian
manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to
Neumann (DN) operator $\Lambda$ on all forms on the boundary and they prove
that the real additive de Rham cohomology structure of the manifold in question
is completely determined by $\Lambda$. This shows that the DN map $\Lambda$
inscribes into the list of objects of algebraic topology. In this paper, we
suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie
algebra of $G$ and the corresponding vector field $X_M$ on $M$, one defines
Witten's inhomogeneous coboundary operator $d_{X_M} = d+\iota_{X_M}$ on
invariant forms on $M$. The main purpose is to adapt Belishev and
Sharafutdinov's boundary data to invariant forms in terms of the operator
$d_{X_M}$ and its adjoint $\delta_{X_M}$. In other words, we define an operator
$\Lambda_{X_M}$ on invariant forms on the boundary which we call the $X_M$-DN
map and using this we recover the long exact $X_M$-cohomology sequence of the
topological pair $(M,\partial M)$ from an isomorphism with the long exact
sequence formed from our boundary data. We then show that $\Lambda_{X_M}$
completely determines the free part of the relative and absolute equivariant
cohomology groups of $M$ when the set of zeros of the corresponding vector
field $X_M$ is equal to the fixed point set $F$ for the $G$-action. In
addition, we partially determine the mixed cup product (the ring structure) of
$X_M$-cohomology groups from $\Lambda_{X_M}$. These results explain to what
extent the equivariant topology of the manifold in question is determined by
the $X_M$-DN map $\Lambda_{X_M}$. Finally, we illustrate the connection between
Belishev and Sharafutdinov's boundary data on $\partial F$ and ours on
$\partial M$.

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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: comp-gas