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Monoidal geometries

Abstract

The main idea behind this research proposal is the replacement of the vector space category by another symmetric monoidal category in order to do geometry (i.e. to understand and classify additional geometric structures on smooth manifolds) in the spirit of Alain Connes's noncommutative geometry. The commutative monoids of this category replacing the vector spaces should play the role that the commutative algebras of functions on the manifold play in noncommutative geometry. The first step consists of establishing a Gelfand-Naimark type duality between smooth manifolds and the commutative monoids. The second step is to understand noncommutative monoids in terms of some additional geometric structures present on smooth manifolds. We call geometric structures that arise this way "monoidal geometries". The last step is to classify the geometric structures by classifying their corresponding monoids. An immediate first goal is to show that Poisson geometry is a monoidal geometry using, as a symmetric monoidal category, the microsymplectic category recently constructed by A. Cattaneo, A. Weinstein and the applicant in the recent article "Symplectic microgeometry I: micromorphisms", J. Sympl. Geom. Volume 8, Number 2, 1--19, 2010. (AU)

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Scientific publications (6)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
CABRERA, ALEJANDRO; DHERIN, BENOIT. Formal Symplectic Realizations. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, n. 7, p. 1925-1950, 2016. Web of Science Citations: 0.
DHERIN, BENOIT; MENCATTINI, IGOR. Deformations of momentum maps and G-systems. Journal of Mathematical Physics, v. 55, n. 11 NOV 2014. Web of Science Citations: 0.
DHERIN, BENOIT; MENCATTINI, IGOR. G-systems and deformation of G-actions on R-d. Journal of Mathematical Physics, v. 55, n. 1 JAN 2014. Web of Science Citations: 1.
CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. SYMPLECTIC MICROGEOMETRY III: MONOIDS. Journal of Symplectic Geometry, v. 11, n. 3, p. 319-341, SEP 2013. Web of Science Citations: 5.
CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. Integration of Lie algebroid comorphisms. PORTUGALIAE MATHEMATICA, v. 70, n. 2, p. 113-144, 2013. Web of Science Citations: 5.
CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. Symplectic microgeometry II: generating functions. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 42, n. 4, p. 507-536, DEC 2011. Web of Science Citations: 8.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.