Monoidal geometries

• Grant number: 10/15069-8
• Support Opportunities: Research Grants - Young Investigators Grants
• Start date: February 01, 2011
• End date: July 31, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Benoit Richard Umbert Dherin
• Grantee: Benoit Richard Umbert Dherin
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• City of the host institution: São Carlos
• Associated researchers: Daniel Levcovitz ; Igor Mencattini ; Mauro Spreafico ; Pedro Paulo de Magalhaes Rios
• Associated scholarship(s): 10/19365-0 - Monoidal geometries, BP.JP

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)

Scientific publications

(7)

(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. (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, 2013. (10/15069-8, 10/19365-0)

DHERIN, BENOIT; MENCATTINI, IGOR. Deformations of momentum maps and G-systems. Journal of Mathematical Physics, v. 55, n. 11, NOV 2014. (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, DEC 2011. (10/15069-8)

CATTANEO, ALBERTO S.; DHERIN, BENOIT; WEINSTEIN, ALAN. SYMPLECTIC MICROGEOMETRY III: MONOIDS. Journal of Symplectic Geometry, v. 11, n. 3, p. 319-341, SEP 2013. (10/15069-8)

CABRERA, ALEJANDRO; DHERIN, BENOIT. Formal Symplectic Realizations. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, v. 2016, n. 7, p. 26-pg., 2016-01-01. (10/19365-0, 10/15069-8)

DHERIN, BENOIT; MENCATTINI, IGOR. G-systems and deformation of G-actions on R-d. Journal of Mathematical Physics, v. 55, n. 1, JAN 2014. (10/15069-8, 10/19365-0)