Noncommutative geometry

In the early 1980s, Alain Connes showed how to extend both classical Hochschild homology and cyclic homology to the category of locally convex algebras, computing in particular the Hochschild homology of the Frechet algebra of smooth functions on a compact manifold X. This he showed was canonically isomorphic to the space of differential forms on X, thus generalizing a fundamental result of Hochschild-Kostant-Rosenberg. In the same paper, Connes also identified the periodic cyclic homology of the Frechet algebra of smooth functions on X with the de Rham cohomology of X,

HH(C(X)) = Ω(X)   and  HP(C(X)) = H(X).

Connes's theorem shows that cyclic homology is indeed a far-reaching generalisation of de Rham cohomology for noncommutative topological algebras, and serves as motivation for the joint work with D. Stevenson in [MS].

The central result in [MS] is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra C(X, EH) of stable continuous trace algebras having smooth manifolds X as their spectrum, and Dixmier-Douady invariant equal to HεH3(X,Z). More precisely, the Hochschild homology is identified with the space of differential forms on X, and a choice of bundle gerbe connection gives rise to an identification of the periodic cyclic homology with the twisted de Rham cohomology of X,

HH(C(X, EH)) = Ω(X)   and   HP(C(X, EH)) = H(X, H),

thereby generalizing the fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified in [MS] with the twisted Chern character.

Entire cyclic homology was introduced by Connes as a version of cyclic homology that is better suited to study the dual space of higher rank discrete groups and also certain infinite dimensional spaces occurring in the study of constructive field theory. In [MS2], the central result is the computation of the entire cyclic homology of the smooth subalgebras of stable continuous trace C*-algebras as above. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras,

HE(C(X, EH)) = HP(C(X, EH)).

By the main result in [MS], one concludes that upon choosing a bundle gerbe connection, the entire cyclic homology of the smooth subalgebra is isomorphic to the twisted de Rham cohomology of the spectrum.

I have also done research work on applications of noncommutative geometry to the fractional quantum Hall effect, and to string theory and duality.

February 2008 - January 2012, Editor in charge of "Global Analysis, Noncommutative Geometry, and the Mathematics of String Theory", Proceedings of the American Mathematical Society.

Recent invited talks in Noncommutative Geometry

Keynote speaker, Special session on Noncommutative Geometry and Operator Algebras, AustMS 2010 annual meeting, Brisbane, 27-30 September, 2010.

Invited speaker, Noncommutative Geometric Methods in Global Analysis, Hausdorff Research Institute for Mathematics, Bonn, Germany, June 29- July 4, 2009.

Invited speaker, Algebras, Operators and Noncommutative Geometry, ANU, Canberra, 1-5 December, 2008.

Invited speaker, Conference on Noncommutative Geometry, Hausdorff Research Institute for Mathematics, Bonn, Germany, July 27 - August 02, 2008.

Invited participant, Noncommutative Geometry workshop, MFO, Oberwolfach (Germany) 2-8 September 2007.

Invited speaker, Noncommutative Geometry conference at BIRS, Banff (Canada), April 8-13, 2006.

I co-organized a major international AMSI workshop entitled, Noncommutative Geometry and Index Theory, which was held at ANU, Canberra, 22 July - 1 August, 2005. I gave a talk on the paper [MS].


[HM11] K. Hannabuss, V. Mathai.
Nonassociative strict deformation quantization of C*-algebras and nonassociative torus bundles
Letters in Mathematical Physics,
102, No. 1 (2012), 107-123, [1012.2274]

[MM11] S. Mahanta, V. Mathai,
Operator algebra quantum homogeneous spaces of universal gauge groups,
Letters in Mathematical Physics,
97 (2011) 263-277. [1012.5893]

[HM10] K. Hannabuss and V. Mathai,
Parametrised strict deformation quantization of C*-bundles and Hilbert C*-modules,
Journal of the Australian Mathematical Society,
90 no. 01 (2011) 25-38, [1007.4696]

[MR] V. Mathai and J. Rosenberg
A noncommutative sigma-model,
Journal of Noncommutative Geometry,
5 no 2 (2011) 265-294, [0903.4241]

[HM09] K. Hannabuss and V. Mathai,
Noncommutative principal torus bundles via parametrised strict deformation quantization,
AMS Proceedings of Symposia of Pure Mathematics,
81 (2010) 133-148, [0911.1886]

[CM] P. Chakraborty and V. Mathai,
The geometry of determinant line bundles in noncommutative geometry,
Journal of Noncommutative Geometry,
3 no.4 (2009) 559-578. [0804.3232]

[MS2] V. Mathai and D. Stevenson,
Entire cyclic homology of stable continuous trace algebras,
Bulletin of the London Mathematical Society,
39 no.1 (2007) 71-75, [math.KT/0412485]

[MS] V. Mathai and D. Stevenson,
On a generalized Connes-Hochschild-Kostant-Rosenberg theorem,
Advances in Mathematics,
200 no. 2 (2006) 303-335. [math.KT/0404329]