Research:  
I am interested in the interplay between geometry/topology and algorithms. Current research is mainly in structure-preserving discretizations of exterior calculus and differential geometry. Discretization = combinatorial version of calculus and differential geometry on graphs and simplicial complexes. An example of structure preservation is the first theorem in paper on vector bundles. The structures are interesting in their own right and also useful in applications. Here is a list of current and recent research projects:
Discrete Exterior Calculus (DEC): I originally worked on this as a graduate student. Currently active projects: Vector bundle valued DEC (see next item), Discretization of nonlinear terms arising from wedge product of forms, discretization of tensors of various types, careful treatment of boundary conditions, discrete Poincare operators, A-infinty structures on simplicial complexes.
Discrete Differential Geometry: Combinatorial "differential" geometry using discrete vector bundles with connection on simplicial complexes.
Computational Topology: Computation of harmonic forms on hyperbolic 3-manifolds and hyperbolic surfaces using harmonic series expansion. Computation of harmonic forms in a given cohomology class on embedded surfaces using finite element exterior calculus.
Numerical Analysis: Numerical methods for PDEs using DEC, finite element exterior calculus, vector bundle valued DEC. Well-centered meshing. Conservative integrators for piecewise smooth dynamics.
Computational Physics: Navier-Stokes equations for fluid mechanics on simplicial surfaces. Two-phase flow on simplicial surfaces. Compressible flow. Elasticity. Flow in a porous medium.
Theoretical Computer Science: Structure of satisfiable and unsatisfiable propositional logic sentences using graph theory. Algorithms for finding optimal cycles in a homology class.
Industry: Before joing Math department I was a faculty member in CS. Before that I was a computer graphics software engineer (Sun Microsystems), image processing industrial researcher (SONY Corp.) and space mission designer (NASA Jet Propulsion Lab). I like to keep in touch with industry via collaboration and consulting.
Bingyan Liu, Ph.D student in Mathematics, working with me since 2022.
Chengbin Zhu, Ph.D student in Mathematics, working with me since 2022.
Siqi Jiao, Ph.D student in Mathematics, working with me since 2020.
Nikolas Wojtalewicz, Ph.D (Mathematics), 2022. "Discontinuous Differential Equations". (Co-advised with Andy Wan). Computer Vision Scientist at VSI, Inc.
Vaibhav Karve, Ph.D. (Mathematics), 2021. "Graphical Structure of Unsatisfiable Boolean Formulae". Senior Data Scientist at SimSpace.
Mark Schubel, Ph.D. (Physics), 2017. "Discretization of differential geometry for computational gauge theory". Senior Data Scientist at Apple.
Kaushik Kalyanaraman, Ph.D. (Computer Science), 2015. "Hodge Laplacians on simplicial meshes and graphs". Assistant Professor of Mathematics at Indraprastha Institute of Information Technology (IIIT) Delhi, India.
Evan VanderZee, Ph.D. (Mathematics), 2010. "Well-Centered Meshing". (Co-advised with Vadim Zharnitsky). Software Engineer at Argonne National Laboratory.
Andrew Colombi, Ph.D. (Computer Science), 2008. "Quick Evaluation of Small Body Gravitation". Co-founder and CTO of startup Tonic, San Fransisco.
Co-advised partially:
Seth Watts, Ph.D. (Mechanical Science and Engineering), 2013, advisor Daniel Tortorelli, co-advised 2010-2012, now at Lawrence Livermore National Laboratory.
Han Wang, Ph.D. (Mathematics), 2014, advisor Yuliy Baryshnikov, co-advised 2010-2013, now at Wells Fargo Bank.
Contact:   hirani at illinois dot edu, 375 Altgeld Hall,
(217) 333 2727
Mailing Address: Department of Mathematics, 1409 W. Green St., M/C 382, Urbana, IL 61801.
Page maintained by Anil Hirani.
Last modified: Tue Jan 14 16:44:12 CST 2025