Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Sinha, Kalyan Bidhan (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

Quantization of mathematical theories is now more than half a century old idea in mathe- matics. It goes back to Gelfand-Naimarks seminal paper [37] in 1943. As the name suggests noncommutative geometry is the quantization" of differential geometry. It is the study of noncommutative algebras as if they were algebras of functions on spaces like the commuta- tive algebras associated to affine algebraic varieties, smooth manifolds, topological spaces. One can trace its roots in the Gelfand-Naimark theorems (1943, 37]). In modern terminol- ogy their theorem says there is an antiequivalence between the category of (locally) compact Hausdorff spaces and (proper, vanishing at infinity) continuous maps and the category of (not necessarily) unital C-algebras and +-homomorphisms. In other words the entire topological information of a locally compact Hausdorffspace is encoded in the commutative C-algebra of continuous functions vanishing at infinity. This observation suggests an immediate extension of the notion of topological spaces by considering a not necessarily commutative C-algebra as the algebra of functions on some noncommutative space.This idea of extending classical notions to the domain of noncommutative algebras was exploited by Karoubi in the carly 70s. He showed that topological K-theory can be extended to Banach algebras. Next major breakthrough towards extending algebraic topological ideas in the noncommutative arena were the works of Brown, Douglas and Fillmore and Kasparov. Kasparov gave a unified approach towards extending the notion of analytical K-homology and topological K-theory.Strictly speaking all these developments were taking part in the realm of noncommutative topology. Noncommutative geometry took off in the hands of Connes with the introduction of cyclic (co)homology. It was introduced as an extension of the deRham cohomology of differentiable manifolds to the noncommutative setting, and serves as a natural target for the Chern character homomorphism from K-theory. At this point we should also mention that Boris Tsygan also independently arrived at the notion of cyclic homology. Novelty of the various constructions of Connes lies in the explicit nature of the pairing between cyclic cohomology and K-theory. He achieves this by lifting the notion of Dirac operators to the noncommutative arena. The essential features of geometry of spin manifolds are extended by the notion of spectral triples, which consists of a separable Hilbert space H, an involutive subalgebra A of the algebra of bounded operators, and D, a selfadjoint operator with compact resolvents deriving A in the sense that the commutator of D with every element of A is densely defined and admits a bounded extension. This operator D contains almost all the geometric information. With any closed Riemannian spin manifoid M there is associated a canonical spectral triple with A = C (M), the algebra of complex valued smooth functions on M, H = L?(M, S), the Hilbert space of square integrable sections of the irreducible spinor bundle over M and D, the Dirac operator associated with the Levi-Civita connection. For this spectral triple Connes has a recipe for getting back the usual differential calculus of forms on M. In fact the prescription given in the last chapter of his book [24] works for any spectral triple. In the general context we will call the calculus associated with a spectral triple the Connes-de Rham calculus. Connes extended metric notions like volume measure, connection, curvature etc.


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