Contact Information
314C Harker Hall, MC-382
1405 W. Green Street
Urbana, IL 61801
Research Description
My current research is: (1) Structure-preserving discretization of exterior calculus and differential geometry; and (2) Applications 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.
Education
PhD, Caltech, 2003
Awards and Honors
NSF CAREER Award, "Algebraic Topology and Exterior Calculus in Numerical Analysis," 2007–2012
Additional Campus Affiliations
Professor, Mathematics
External Links
Recent Publications
Hirani, A. N., Wan, A. T. S., & Wojtalewicz, N. (2024). CONSERVATIVE INTEGRATORS FOR PIECEWISE SMOOTH SYSTEMS WITH TRANSVERSAL DYNAMICS. Journal of Computational Dynamics, 11(2), 135-152. https://doi.org/10.3934/jcd.2023009
Schubel, M. D., Berwick-Evans, D., & Hirani, A. N. (2024). Averaging property of wedge product and naturality in discrete exterior calculus. Advances in Computational Mathematics, 50(4), Article 84. https://doi.org/10.1007/s10444-024-10179-8
Hirani, A. N., Kalyanaraman, K., Wang, H., & Watts, S. (2023). Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus. Computational Geometry: Theory and Applications, 109, Article 101937. https://doi.org/10.1016/j.comgeo.2022.101937
Wang, M., Jagad, P., Hirani, A. N., & Samtaney, R. (2023). Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method. Journal of Computational Physics, 488, Article 112245. https://doi.org/10.1016/j.jcp.2023.112245
Karve, V., & Hirani, A. N. (2020). The complete set of minimal simple graphs that support unsatisfiable 2-CNFs. Discrete Applied Mathematics, 283, 123-132. https://doi.org/10.1016/j.dam.2019.12.017