2026-09-18T19:23:40Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/601932022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Richard B. Melrose.
Kottke, Christopher N. (Christopher Nicholas)
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2010-12-06T17:36:14Z
2010-12-06T17:36:14Z
2010
2010
http://hdl.handle.net/1721.1/60193
681923895
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 101-102).
In this thesis, I investigate the index of Callias type operators on asymptotically conic manifolds (also known as asymptotically locally Euclidean manifolds or scattering manifolds) and give an application to the moduli space of magnetic monopoles on these spaces. The index theorem originally due to C. Callias and later generalized by N. Anghel and others concerns operators of the form ... is a family of Hermitian invertible matrices. The first result is a pseudodifferential version of this index theorem, in the spirit of of the K-theoretic proof of the Atiyah-Singer index theorem, using the theory of scattering pseudodifferential operators. The second result is an extension to the case where [Iota] has constant rank nullspace bundle at infinity, using a b-to-scattering transition calculus of pseudodifferential operators. Finally I discuss magnetic monopoles, which are solutions to the Bogomolny equation ... principal bundle over a complete 3-manifold, and I show how the previous results can be applied to compute the dimension of the moduli space of monopoles over asymptotically conic manifolds whose boundary is homeomorphic to a disjoint union of spheres.
by Christopher N. Kottke.
Ph.D.
102 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Index theorems and magnetic monopoles on asymptotically conic manifolds
Thesis
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<Abstract>In this thesis, I investigate the index of Callias type operators on asymptotically conic manifolds (also known as asymptotically locally Euclidean manifolds or scattering manifolds) and give an application to the moduli space of magnetic monopoles on these spaces. The index theorem originally due to C. Callias and later generalized by N. Anghel and others concerns operators of the form ... is a family of Hermitian invertible matrices. The first result is a pseudodifferential version of this index theorem, in the spirit of of the K-theoretic proof of the Atiyah-Singer index theorem, using the theory of scattering pseudodifferential operators. The second result is an extension to the case where [Iota] has constant rank nullspace bundle at infinity, using a b-to-scattering transition calculus of pseudodifferential operators. Finally I discuss magnetic monopoles, which are solutions to the Bogomolny equation ... principal bundle over a complete 3-manifold, and I show how the previous results can be applied to compute the dimension of the moduli space of monopoles over asymptotically conic manifolds whose boundary is homeomorphic to a disjoint union of spheres.</Abstract>
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