MAE Seminar: Elasticity, Double forms, and Vector Bundles

Details

MAE Seminar: Elasticity, Double forms, and Vector Bundles
Date: April 9, 2026
Time: 12:50 PM Location: MAE-A 303

Dr. Anil Hirani
Professor
Department of Mathematics
University of Illinois at Urbana-Champaign

Abstract
Linearized elasticity and continuum models for crystal defects (dislocations, disclinations, point defects) fit neatly into some diagrams that first appeared in geometry and are now one of the hottest topics in some areas of numerical analysis. There are many other physical theories that appear in these diagrams, but I will focus on elasticity, especially the role of and geometric meaning of the incompatibility operator and the hessian. The diagrams not only encapsulate several physical theories, they are also leading to discoveries of new finite element spaces. The main actors in these diagrams are form valued differential forms, or double forms. Each row in the diagram is a sequence of vector bundles with connection. I will show a discretization of such rows that we have been developing and show some computations for biharmonic equation and linearized elasticity in the plane.

Biography
The speaker got his graduate degrees in Computer Science at Stanford and Caltech. He is currently Professor of Mathematics at University of Illinois at Urbana-Champaign before which he was in the CS department there. He has worked at Sun Microsystems as a computer graphics software engineer, at Sony Corporation as a researcher in image processing, and at NASA’s Jet Propulsion Laboratory in space mission design.

Faculty Host: Dr. James Fairbanks