Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dirac Eigenvalues of higher Multiplicity   Let M be a closed spin manifold of dimension at least three with a fixed
topological spin structure. For any Riemannian metric, we can construct the
associated Dirac operator. The spectrum of this Dirac operator depends on the
metric of course. In 2005, Dahl conjectured that M can be given a metric, for
which a finite part of the spectrum consists of arbitrarily prescribed
eigenvalues of arbitrary (finite) multiplicity. The only constraints one has to
respect are the exception of the zero eigenvalue (due to the Atiyah-Singer
index theorem) and in certain dimensions, the quaternionic structure of the
eigenspaces and also the symmetry of the spectrum. Dahl also proved his
conjecture in case all eigenvalues have simple multiplicities. The question, if
one can prescribe arbitrary multiplicities, or if the existence of eigenvalues
of higher multiplicity might somehow be topologically obstructed, has been open
ever since. In this thesis, we prove that on any closed spin manifold of
dimension m congruent 0, 6, 7 mod 8, there exists a metric for which at least
one eigenvalue is of higher multiplicity. For the proof, we introduce a
technique which "catches" the desired metric with a loop in the space of all
Riemannian metrics. We will construct such a loop on the sphere and transport
it to a general manifold by extending some classical surgery theory results by
B\"ar and Dahl. As a preparation, we will show that the Dirac spectrum can be
described globally by a continuous family of functions on the space of
Riemannian metrics and that the spinor field bundles with respect to the
various metrics assemble to a continuous bundle of Hilbert spaces. These
results might be useful in their own right.

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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: nlin