Anil N. Hirani
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. Recently, I made an excursion outside my main area into the structure of satisfiable and unsatisfiable propositional logic sentences using the tools of graph theory. Current and recent research interests:
- Numerical Analysis: Numerical methods for PDEs using discrete
exterior calculus (DEC) and finite element exterior calculus (FEEC). Well-centered meshing. Computation of harmonic forms on hyperbolic manifolds. Conservative integrators for piecewise smooth dynamics.
- Discrete Differential Geometry: Combinatorial "differential" geometry using discrete vector bundles with connection on simplicial complexes.
- 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
- Theoretical Computer Science: Structure of satisfiable and unsatisfiable propositional logic sentences using graph theory. Algorithms for finding optimal cycles in a homology class.
Teaching (Spring 2022):
(Tentative) Teaching next semester (Fall 2022):
- MATH 423: Differential Geometry
- Chengbin Zhu, Ph.D student in Mathematics, since 2022.
- Siqi Jiao, Ph.D student in Mathematics, since 2020.
- Nikolas Wojtalewicz, Ph.D student in Mathematics, Defended April 2022.
- Vaibhav Karve, Ph.D. (Mathematics), 2021, "Graphical Structure of Unsatisfiable Boolean Formulae," now Senior Data Scientist at SimSpace.
- Mark Schubel, Ph.D. (Physics), 2017, "Discretization of differential geometry for computational gauge theory," now Senior Data Scientist at Apple.
- Kaushik Kalyanaraman, Ph.D. (Computer Science), 2015, "Hodge Laplacians on simplicial meshes and graphs," now 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, now Software Engineer at Argonne National Laboratory.
- Andrew Colombi, Ph.D. (Computer Science), 2008, "Quick Evaluation of Small Body Gravitation," now Co-founder and CTO of startup company Tonic, San Fransisco.
- 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.
Bootcamp: Learn Python
programming for scientific computing in 10 days.
Previous Recent Teaching (last few years):
- 2012 : Best Paper Award (Second Place) in Solid and Physical Modeling
2012, for paper Delaunay
- 2007 : NSF CAREER Award, Division of Mathematical Sciences
(Algebraic Topology and Exterior Calculus in Numerical Analysis)
- 2013-, 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
California Institute of Technology
Control and Dynamical Systems
- 2003, Ph.D,
California Institute of
Jerrold E. Marsden),
Ph.D in Computer Science with minors
in Mathematics and
Control and Dynamical Systems
Ph.D Thesis, Discrete Exterior Calculus
- Engineer, Sony Corporation
- Software Engineer, Sun Microsystems
- M.S, Computer Science (Theoretical track),
- 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: Tue May 10 11:08:23 CDT 2022