Advanced search
Start date

Monoidal geometries


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)

Articles published in Agência FAPESP Newsletter about the research grant:
Articles published in other media outlets (0 total):
More itemsLess items

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)
CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. SYMPLECTIC MICROGEOMETRY III: MONOIDS. Journal of Symplectic Geometry, v. 11, n. 3, p. 319-341, . (10/15069-8)
CABRERA, ALEJANDRO; DHERIN, BENOIT. Formal Symplectic Realizations. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, n. 7, p. 1925-1950, . (10/15069-8, 10/19365-0)
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, . (10/15069-8)
DHERIN, BENOIT; MENCATTINI, IGOR. Deformations of momentum maps and G-systems. Journal of Mathematical Physics, v. 55, n. 11, . (10/15069-8, 10/19365-0)
CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. Integration of Lie algebroid comorphisms. PORTUGALIAE MATHEMATICA, v. 70, n. 2, p. 113-144, . (10/15069-8, 10/19365-0)
DHERIN, BENOIT; MENCATTINI, IGOR. G-systems and deformation of G-actions on R-d. Journal of Mathematical Physics, v. 55, n. 1, . (10/15069-8, 10/19365-0)

Please report errors in scientific publications list by writing to: