This application proposes a period of mobility to place a researcher, currently making the transition to the research area of quantum topology, into a leading international research group in that field. The twin goals of the proposal are to provide the vehicle by which the applicant can complete his training in quantum topology, and to contribute new scientific results to the area. The scientific aim is to develop our understanding of quantum invariants of X-manifolds (that is, manifolds equipped with a map to a space X) and their role in quantum field theory. The importance of this topic stems from the fact that it is closely linked to Witten' s path integral formulation of topological quantum field theory and extends the framework and (rigorous) methods of Atria, Segal, Reshetikhin, Tureen and others to a more general setting. The focus will be on three aspects:
-To use shadow topology to define new invariants of X-manifolds
-To define and develop a theory of relative homogony QFT
-To study the geometry of HQFT and resulting applications The objectives will be met by combining the host' s expert knowledge of the methods and techniques of quantum groups, Lie bailers, humanization and topology with the researcher' s background knowledge in HQFT and grebes. Particular attention will be given throughout to examples based on quantum groups and to the case where the background space is the classifying space for G-bundles. Connections to mathematical physics will be pursued, especially to Churn-Simons gauge theory in the 2+1- dimensional case and to D-brine physics for the relative theory. By joining one of the top international research teams in the field, the applicant will: -Learn new methods and techniques in quantum groups and topology -Enhance links with the community -Diversify his expertise This will build on his existing knowledge and thus allow him to complete his training in quantum topology and establish himself in the area.
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