2026-09-22T11:29:32Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/437852022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Tomasz Mrowka.
Lipyanskiy, Maksim
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2008-12-11T18:27:15Z
2008-12-11T18:27:15Z
2008
2008
http://hdl.handle.net/1721.1/43785
261137861
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (p. 81-82).
This dissertation is concerned with the foundations of a new approach to Floer theory. As opposed to the traditional approach, which can be viewed as a generalization of Morse theory to an infinite dimensional setting, our approach is a generalization of bordism to infinite dimensions. The key new insight, based on unpublished work of Tom Mrowka and Peter Ozsvath, is an understanding of how to axiomatize compactness in the infinite dimensional setting. We describe a general axiomatic framework for setting up a Floer theory of a polarized Hilbert space equipped with a functional. The resulting bordism theory can be seen as a refinement of the traditional Floer theory. By introducing cycles with corners, we demonstrate how the bordism theory leads to a geometric description of homology. We relate our geometric construction to the Morse-theoretic approach by indicating how one might compute the Floer homology of the space, if the associated functional is Morse. The general theory is illustrated in two examples: Seiberg-Witten-Floer homology and symplectic Floer theory for loops in Cn. We end by indicating various generalizations of the theory.
by Maksim Lipyanskiy.
Ph.D.
82 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.
A semi-infinite cycle construction of Floer homology
Thesis
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