Mathai's work is in the area of geometric analysis. His research interests are in 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.[11] 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.[12]
The Mathai–Quillen formalism appeared in Topology, shortly after Mathai completed his Ph.D. 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 shape 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.[citation needed]
