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# Classifying spaces and tensor products for manifolds with $\R$-actions

 Grant number: 19/08834-4 Support Opportunities: Regular Research Grants Duration: February 01, 2022 - January 31, 2024 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade Principal Investigator: Luiz Roberto Hartmann Junior Grantee: Luiz Roberto Hartmann Junior Principal researcher abroad: Gerardo A Mendoza Institution abroad: Temple University, United States Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil Associated research grant: 18/23202-1 - Spectral invariants on pseudomanifolds, AP.R

Abstract

We propose to do research in two questions of a topological nature that arose from the study of complex structures on manifolds with conical singularities. Both questions (pertaining to classifying spaces and tensor products) are motivated by the same principle that underlies non-commutative geometry: knowing a lot about an object (for instance topological), and having another of a different nature (for instance algebraic) but with some similarities, to what extent can one prove more properties of the later object because the former object has them. The prototype of this is the Gelfand-Naimark theorem identifying commutative $C^*$ algebra with the space of continuous functions on some (certain) compact space, which can be translated into the statement that knowing the space of continuous functions as an algebraic object means one knows the underlying topological space (hence everything else: homology and cohomology, possibly $C^\infty$ structures). (AU)