I am a Professor in the Department of Mathematics at the University of Illinois at Urbana-Champaign. I did my Ph.D. in Computer Science with minors in Mathematics and Control and Dynamical Systems at Caltech and M.S. in Computer Science at Stanford. My undergraduate degree is from BITS Pilani in India. I am interested in the interplay between geometry, topology and computation. This relationship works in two ways. Geometric and topological invariants are critical to maintain in modern computational physics and engineering applications. And computation is increasingly helping to develop and disprove mathematical conjectures.

My current research is: (1) structure-preserving discretization of exterior calculus and differential geometry; and (2) and application of these in computational physics and engineering. I use and develop the two frameworks discrete exterior calculus (DEC) and finite element exterior calculus (FEEC), and the extensions of DEC to discrete differential geometry (DDG). These topics make contact with Hodge theory, algebraic topology, homological algebra, differential geometry, category theory, and numerical analysis, as well as topics in physics and engineering. To us, discretization means creating a combinatorial or finite-dimensional version of calculus and differential geometry on simplicial complexes that is suitable for computers. These discrete structures are mathematically fascinating in their own right, while being very useful in practical applications.

We have applied these structure-preserving ideas to the numerical solution of partial differential equations and the analysis of the resulting methods. Earlier in my career, I also brought this geometric perspective to computer graphics and astrodynamics. More recently, researchers in machine learning and AI have started to adopt the structure-preservation philosophy. Indeed, my current NSF grant (started June 2026) in collaboration with Nat Trask of UPenn explicitly combines machine learning with structure-preservation for PDEs. For more information on active and past projects, please see my research page.

RA Opportunity

I am seeking a graduate student (Mathematics, Physics, or Engineering) for a funded RA position starting Summer 2027. Current and prospective students are welcome. This position is supported by an NSF-funded grant: Geometric Scientific Machine Learning for PDEs with Tensorial Constraints. Ideal candidates will possess strong mathematical training and have an interest in geometry. Some familiarity with one or more of the following will be beneficial but is not required: numerical analysis, elementary algebraic topology, partial differential equations, differential geometry. The project will involve some scientific machine learning; prior experience is welcome but not strictly required. Depending on the candidate's background and interests, the work could lean towards more theoretical mathematical topics or more applied and computational topics. Interested students should email me their CV and a brief statement of interest (see my contact page).

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