# Varghese Mathai

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Mathai Varghese is a mathematician and an Australian Research Council (ARC) Discovery Outstanding Research Fellow at the University of Adelaide. His most influential contribution to date is the Mathai-Quillen formalism, [1] which he formulated together with Daniel Quillen, and which has since found applications in index theory and topological quantum field theory. He became a full professor in 2006. He became Director of the Institute for Geometry and its Applications in 2009. In 2011, he was elected Fellow of the Australian Academy of Science.[2] In 2013, he was appointed the (Sir Thomas) Elder Chair in Mathematics [3] and was elected Fellow of the Royal Society of South Australia.

Mathai received his BA at the Illinois Institute of Technology. He then proceeded to the Massachusetts Institute of Technology, where he was awarded a doctorate under the supervision of Daniel Quillen, a Fields medallist.

Mathai's work may be considered to fall under the ambit of geometric analysis. His research interests are in $L^2$ analysis, index theory and noncommutative geometry. He currently works on mathematical problems that have their roots in physics, for example, topological field theories, fractional quantum Hall effect, and D-branes in the presence of B-fields. The main focus of his research is on the application of noncommutative geometry and index theory to mathematical physics, with particular emphasis on string theory. His current work on index theory is ongoing joint work with Richard Melrose and Isadore Singer, on the fractional analytic index and on the index theorem for projective families of elliptic operators. [4] His current work on string theory is ongoing joint work with Peter Bouwknegt, Jarah Evslin, Keith Hannabuss and Jonathan Rosenberg, on T-duality in the presence of background flux.[5]

The Mathai–Quillen formalism appeared in Topology shortly after Mathai completed his PhD. Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions for the fermionic analogue of a harmonic oscillator with source term. In particular, they obtained a nice Gaussian shaped representative of the Thom class in cohomology, which has a peak along the zero section. Its universal representative is obtained using the machinery of equivariant differential forms.

Mathai was awarded the Australian Mathematical Society Medal in 2000.[6] From August 2000 to August 2001, he was also a Clay Mathematics Institute Research Fellow and Visiting Scientist at the Massachusetts Institute of Technology. From March to June 2006, he was a Senior Research Fellow at the Erwin Schrödinger Institute in Vienna.

## Selected publications

• Mathai, Varghese and Quillen, Daniel. (1986) "Superconnections, Thom classes and equivariant differential forms". Topology 25 (1), 85–110.
• Bouwknegt, Peter, Evslin, Jarah and Mathai, Varghese. (2004) "T-duality: Topology Change from H-flux". Communications in Mathematical Physics 249 (2), 383–415.
• Mathai, Varghese, Melrose, Richard B. and Singer, Isadore M.. (2006) "Fractional Analytic Index". Journal of Differential Geometry 74 (2), 265–292.

## Notes

1. ^ Mathai, Varghese and Quillen, Daniel. (1986) "Superconnections, Thom classes and equivariant differential forms". Topology 25 (1), 85–110.
2. ^ "Fellows elected in 2011". Australian Academy of Sciences.
3. ^
4. ^ Mathai, Varghese, Melrose, Richard B. and Singer, Isadore M.. (2006) "Fractional Analytic Index". Journal of Differential Geometry 74 (2), 265–292.
5. ^ Bouwknegt, Peter, Evslin, Jarah and Mathai, Varghese. (2004) "T-duality: Topology Change from H-flux". Communications in Mathematical Physics 249 (2), 383–415.
6. ^ "Christine O'Keefe and Mathai Varghese share the Australian Mathematical Society 2000 Medal". Australian Mathematical Society.