PICS Colloquium with Anil Hirani and Christopher Eldred

This will be a joint colloquium, with two speakers, each with a 30 minute time slot. 

Anil Hirani Abstract: TBA

Anil Hirani Bio:

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 have also worked outside academia, at Sun Microsystems, SONY Corporation, and NASA’s Jet Propulsion Laboratory.

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.

Chris Eldred Abstract: TBA

Chris Eldred Bio:

Chris Eldred got his PhD in Atmospheric Science from Colorado State University in Fort Collins, CO. After this, he did two postdocs in France, studying applications of geometric mechanics formulations (GM) and structure-preserving discretizations (SP) to atmospheric modeling. He is currently a staff scientist at Sandia National Laboratories in Albuquerque, NM, where he works on GM/SP methods for a variety of fluid dynamical systems including geophysical fluid dynamics, compressible flow and plasma/fusion. His other research interests include uncertainty quantification, scientific machine learning and optimal control/designs, especially the application of ideas from GM/SP to these areas.