My research transcends traditional subject boundaries and involves intersections between Theoretical Physics, Mathematics and Machine Learning. I enjoy developing new connections between these subjects and my work has resulted in new physical and mathematical results.

A large part of my research has been at the intersection of Supersymmetric Quantum Field Theories (idealized versions of the Quantum Field Theory describing our real world) and the mathematics of Complex Integrable Systems like the Hitchin Integrable System. In joint work with J. Distler and R. Donagi, we initiated the study of the Hitchin Integrable System as a family over the Deligne-Mumford moduli space of curves and proved some new mathematical results about the Hitchin Base (as a global family) and its degeneration in the nodal limit.  In our work, we also proposed a conjectural solution to the Deligne-Simpson problem that involves the “OK condition” – a condition whose origins are in the study of certain monopole operators in the associated Supersymmetric Field Theories. This suggests a new approach to solving a fundamental mathematical problem.

In an earlier work with J. Distler, we had outlined how the theory of sheets (a beautiful part of the theory of complex semi-simple Lie algebras) offers the natural language to describe the mass deformed versions of a large class of Superconformal Field Theories. In the Wilsonian sense, studying the conformal theories together with their mass deformations is an important part of understanding the larger world of N=2 theories. For SU(2) N=2 theories, this program was carried out in detail by Seiberg-Witten in the 1990s and our work can be taken as a starting point for a similar study for this larger class of theories.

In work with J. Teschner, I worked on developing an alternate approach to connect the Geometric Langlands Program, Supersymmetric Field Theories and 2d Conformal Field Theories using the Alday-Gaiotto-Tachikawa correspondence.

More recently, I have developed an interest in the theory of machine learning and have ongoing projects devoted to understanding machine learning using various tools from mathematics and physics.