Anil N. Hirani
Research:  
I have always been attracted to the interplay between geometry/topology and algorithms. My main research area is structure-preserving discretizations of differential calculus on manifolds and of differential geometry. By discretization, I mean a combinatorial version of calculus and differential geometry. The structures preserved are some of the algebraic properties of spaces, operators and objects being discretized. (An example is the first theorem in the paper on vector bundles.) The structures in such discretizations are mathematically interesting in their own right and can be used for numerical solution of partial differential equations (PDEs), computation of geometric properties of spaces, and for many other applications. Current and recent research interests:
- Discrete Exterior Calculus (DEC): Discretization of nonlinear terms arising from wedge product of forms, discretization of tensors of various types.
- Discrete Differential Geometry: Combinatorial "differential" geometry using discrete vector bundles with connection on simplicial complexes.
- Numerical Analysis: Numerical methods for PDEs using discrete exterior calculus and finite element exterior calculus. Computation of harmonic forms on hyperbolic manifolds. Well-centered meshing. Conservative integrators for piecewise smooth dynamics.
- Computational Physics: Navier-Stokes equations for fluid mechanics on simplicial complexes. Two-phase flow. Elasticity. Flow in a porous medium. Einstein's equation for general relativity.
- Theoretical Computer Science: Structure of satisfiable and unsatisfiable propositional logic sentences using graph theory. Algorithms for finding optimal cycles in a homology class.
Publications    
Citations
   
Software    
Discrete Exterior Calculus (DEC)
Teaching (Spring 2023):
Graduate Students:
- 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 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.
- Sean Shahkarami, 2013-2015.
Previous Recent Teaching (last few years):
Short CV:
- 2022-, Professor, University of Illinois at Urbana-Champaign,
Department of Mathematics
- 2013-2022, Associate Professor, University of Illinois at Urbana-Champaign,
Department of Mathematics
- 2005-2013, Assistant Professor,
University of Illinois at Urbana-Champaign,
Department of Computer Science
- 2004-05, Senior Engineer,
Jet Propulsion Laboratory Guidance, Navigation and Control Section
- 2003-04, CIMMS
Postdoctoral Scholar,
California Institute of Technology
(Caltech),
Control and Dynamical Systems
- 2003, Ph.D,
California Institute of
Technology (Caltech),
(Advisor:
Jerrold E. Marsden),
Ph.D in Computer Science with minors
in Mathematics and
Control and Dynamical Systems
Ph.D Thesis, Discrete Exterior Calculus
( Abstract,
PDF,
BibTeX)
- Engineer, Sony Corporation
- Software Engineer, Sun Microsystems
- M.S, Computer Science (Theoretical track),
Stanford University
- Undergraduate degree, Computer Science
Birla Institute of Technology and Science (BITS), Pilani, India
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: Mon Jan 16 08:51:42 CST 2023