Noncommutative Geometry and spectral invariants
Noncommutative geometry was developed by Alain Connes in the 1970s to generalize analytic tools to 'singular' spaces, e.g. the leaf space of foliations. It employs a large variety of techniques from operator algebras, K-theory, differential geometry, PDE theory, homological algebra, etc. to gain insight into problems where classical methods are not sufficient.
Noncommutative geometry techniques have been successfully applied in various areas of mathematics and mathematical physics, e.g. Number theory, Quantum physics, Representation theory and Harmonic analysis, Operator theory, etc.
Current Projects
These are the projects in Noncommutative "coarse" geometry I am working on right now.
1. Analytic Surgery sequence and relation to rho-invariants (joint work with M.-T. Benameur)
References:
(i) “The Higson-Roe exact sequence and $\ell^2$-eta invariants ”, Moulay-Tahar Benameur and Indrava Roy, Journal of Functional Analysis Vol. 268, Issue 4, 2015, pp. 974-1031 (58 pages). Journal Link ArXiv Link
(ii) “The Higson-Roe sequence for etale groupoids I. Dual algebras and compatibility with theBC map ”, Moulay-Tahar Benameur and Indrava Roy. ArXiV preprint, To appear in the Journal of Noncommutative Geometry, 2019
(iii) "The Higson-Roe sequence for etale groupoids II. Functoriality and Paschke duality for action groupoids", Moulay-Tahar Benameur and Indrava Roy, ArXiV preprint. To appear in the Journal of Noncommutative Geometry, 2019.
(iv) A Note on the Extended Pimsner-Popa-Voiculescu Theorem, Moulay-Tahar Benameur and Indrava Roy preprint 2019.
2. Secondary invariants for foliations via superconnections (joint work with S. Azzali and S. Goette), work in progress