Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodifferential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under sufficiently small perturbations. By developing the machinery of distributions and in particular Sobolev spaces, this paper addresses this more specific part of the famous Theorem from a completely analytic approach. We first prove the regularity of elliptic operators, then the. | The Invariance of the Index of Elliptic Operators Constantine Caramanis Harvard University April 5 1999 Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem which states among other things that the space of elliptic pseudodifferential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under sufficiently small perturbations. By developing the machinery of distributions and in particular Sobolev spaces this paper addresses this more specific part of the famous Theorem from a completely analytic approach. We first prove the regularity of elliptic operators then the finite dimensionality of the kernel and cokernel and finally the invariance of the index under small perturbations. cmcaram@ 1 Acknowledgements I would like to express my thanks to a number of individuals for their contributions to this thesis and to my development as a student of mathematics. First I would like to thank Professor Clifford Taubes for advising my thesis and for the many hours he spent providing both guidance and encouragement. I am also indebted to him for helping me realize that there is no analysis without geometry. I would also like to thank Spiro Karigiannis for his very helpful critical reading of the manuscript and Samuel Grushevsky and Greg Landweber for insightful guidance along the way. I would also like to thank Professor Kamal Khuri-Makdisi who instilled in me a love for mathematics. Studying with him has had a lasting influence on my thinking. If not for his guidance I can hardly guess where in the Harvard world I would be today. Along those lines I owe both Professor Dimitri Bertsekas and Professor Roger Brockett thanks for all their advice over the past 4 years. Finally but certainly not least of all I would like to thank Nikhil Wagle Allison Rumsey Sanjay Menon .
