Research:   I am interested in 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): I originally worked on this as a graduate student, discretizing exterior calculus of manifolds to create a combinatorial /geometric version on simplicial complexes. My currently active projects within DEC are: Discretization of nonlinear terms arising from wedge product of forms, discretization of tensors of various types, careful treatment of boundary conditions.
• 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 discrete exterior calculus and finite element exterior calculus. 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.

Industry: Before I became a Math faculty member I was a faculty member in CS and before that I worked as 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.

Publications     Citations     Software     Discrete Exterior Calculus (DEC)

Teaching (Spring 2023):

• MATH 490 Computational Mathematics
• MATH 481 Vector and Tensor Analysis

Graduate Students:

• Jung Soo (Ben) Park, Ph.D student in Mathematics, working with me since 2023.
• 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.
• Sean Shahkarami, 2013-2015.

Previous Recent Teaching (last few years):

• Fall 2022: MATH 423 Differential Geometry
• Spring 2022: MATH 490 Computational Mathematics
• Fall 2021: MATH 423 Differential Geometry
• Fall 2021: MATH 415 Linear Algebra
• Spring 2021: MATH 481 Vector and Tensor Analysis
• Spring 2021: MATH 490 Computational Mathematics
• Fall 2020: MATH 423 Differential Geometry
• Spring 2020: MATH 490 Computational Mathematics
• Spring 2020: MATH 461 (Two sections) Probability Theory
• Spring 2019: MATH 490 Computational Mathematics
• Spring 2019: MATH 519 Differentiable Manifolds II
• Fall 2018: MATH 518 Differentiable Manifolds I
• Spring 2018 (Second Half): MATH 595 Calculus on Meshes
• Spring 2018: MATH 461 Probability Theory

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)
• 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.