Research Interest:  

• Numerical Analysis: Structure-preserving discretizations of partial and ordinary differential equations, e.g., Discrete Exterior Calculus (DEC) and finite element exterior calculus (FEEC).
• Discrete Differential Geometry: Combinatorial "differential" geometry using discrete vector bundles with connection on simplicial complexes.
• Computational Topology: Algorithms for optimal cycles in homology and cocycles in cohomology, computation of harmonic forms on hyperbolic manifolds.
• Computational Physics: Navier-Stokes equations for fluid mechanics on simplicial complexes, two-phase flow, elasticity, porous media flow, Einstein equation of general relativity.

Publications     Citations     Software     Discrete Exterior Calculus (DEC)

Current Teaching (Spring 2021):

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

Current Ph.D students:

• Siqi Jiao
• Nikolas Wojtalewicz
• Vaibhav Karve

Former Ph.D students and postdoc:

• Mark Schubel, Ph.D (Physics) 2017. Now at Apple.
• Kaushik Kalyanaraman, Ph.D (CS) 2015. Now Assistant Professor of Mathematics at IIIT-Delhi, India.
• Han Wang, Ph.D (Mathematics) 2014. Now postdoc at University of Texas, San Antonio.
  Advisor: Yuliy Baryshnikov. Coadvised 2010-2013.
• Seth Watts, Ph.D (Mechanical Engineering) 2013. Now at Lawrence Livermore Lab.
  Advisor: Dan Tortorelli. Coadvised during 2010-2012.
• Evan VanderZee, Ph.D (Mathematics), 2009. Now at Argonne National Lab.
  Coadvised with Vadim Zharnitsky.
• Andrew Colombi, Ph.D (CS) 2008. First job after Ph.D: Palantir, now a founder of a startup in Silicon Valley.
• John Jossey (Postdoc), September 2006 - August 2007. Now Senior Manager at RBC Capital Markets in Canada.

Computational Bootcamp: Learn Python programming for scientific computing in 10 days.

Previous Recent Teaching (last 3 years):

• 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

Awards:

• 2012 : Best Paper Award (Second Place) in Solid and Physical Modeling 2012, for paper Delaunay Hodge Star.
• 2007 : NSF CAREER Award, Division of Mathematical Sciences (Algebraic Topology and Exterior Calculus in Numerical Analysis)

Short CV:

• 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 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.