**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.

Page maintained by Anil Hirani. Last modified: Thu Jan 21 10:53:10 CST 2021