Talks - Anil N. Hirani

Lectures on Discrete Exterior Calculus (DEC)

In addition to conference and seminar talks, I have given multi-lecture introductions to discrete exterior calculus. A three-lecture series was delivered in November 2025 at the International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research, in Bengaluru, India, as part of the program Geometry, Mechanics and the Physics of Growth. Video recordings of those lectures are available. A considerably expanded five-lecture series was delivered in August–September 2026 at the Mathematical Institute, Oxford University, hosted by GeoFEM and Kaibo Hu. Videos and written notes for the Oxford lectures will be posted when available.

2026: Oxford Lectures on Discrete Exterior Calculus; Details
2025: ICTS-TIFR Lectures on Discrete Exterior Calculus; Details

Selection of recent talks

Many of my talks are chalkboard talks, but here are some recent talks with slides or video:

Discrete Vector Bundles with Connection (viewed as a combinatorial theory allowing extension of sheaf neural networks to accommodate nonzero curvature) Poster
Discrete Vector Bundles with Connection and Čech Cohomology Slides
Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions Slides
Contraction Wedge Duality and Double Forms Slides
Boundary Dual and Codifferential in DEC Slides

Selected Invited & Contributed Talks

A fuller record, without links. Slides and videos, where they exist, are linked in the lecture series and recent talks sections above.

2026: Poster: Discrete Vector Bundles with Connection (viewed as a combinatorial theory allowing extension of sheaf neural networks to accommodate nonzero curvature). Mathematical Foundations of AI: The Erlangen Hub Conference 2026, Oxford, UK.
: Discrete Vector Bundles with Connection and Čech Cohomology. Workshop on Discretization of Differential Complexes, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Vienna, Austria.
: Elasticity, Double forms, and Vector Bundles. Department of Mechanical and Aerospace Engineering Seminar, University of Florida, Gainesville.
: BGG for DEC in two dimensions with applications to biharmonic equation and planar elasticity. Workshop on Finite Element Tensor Calculus, Tsinghua Sanya International Mathematics Forum (TSIMF), Sanya, China.
: Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions. International Workshop on Structure-Preserving Finite Element Methods, The Institute of Computational Mathematics and Scientific/Engineering Computing, Beijing, China.
2025: Lecture series on Discrete Exterior Calculus (3 lectures). Geometry, Mechanics and the Physics of Growth, International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research, Bengaluru, India.
: Contraction Wedge Duality and Double Forms. Special Session on Numerical PDEs and Geometry, AMS Fall Central Section Meeting, St. Louis, Missouri.
: Discrete Exterior Calculus: Overview and Perspective. Discrete Exterior Calculus: Differential Geometry and Applications. NSF Institute for Mathematical and Statistical Innovation (IMSI). Chicago, Illinois.
: Tutorial Introduction to Exterior Covariant Derivatives of Double Forms. Geometric Mechanics, Structure Preserving Discretization, and Discrete Differential Geometry on General Fibered Manifolds, SUNY Polytechnic Institute, Utica, New York.
: Subdivision functor and vector bundle valued DEC. BIRS Workshop on Geometric Mechanics Formulations for Continuum Mechanics, Banff, Canada.
: Boundary dual and codifferential in DEC. SIAM CSE, Fort Worth, Texas.
2024: Marsden Memorial Lecture: Calculus and geometry on graphs and meshes. Distinguished Lecture Series of Pacific Institute for the Mathematical Sciences, UC Merced, California.
: Some Properties of Wedge Product in Discrete Exterior Calculus. NSF CompMath PI meeting, University of Washington at Seattle.
: Bianchi Identity for Discrete Vector Bundles. SIAM Annual Meeting, Spokane, Washington.
: Discrete Exterior Calculus: Introduction to calculus on graphs and meshes via examples. Discovery Partners Institute, Chicago, Illinois.
2023: Wedge product and naturality in discrete exterior calculus. Applied Mathematics Seminar, Department of Mathematics, University of Hawaii at Mānoa, Hawaii.
: Wedge Product and Naturality in Discrete Exterior Calculus. Sandia National Laboratory, Albuquerque, New Mexico.
: Naturality of Discrete Exterior Derivative and Discrete Wedge Product in DEC. University of Jyväskylä, Finland.
2022: Discrete vector bundles with connection and curvature. Hilbert Complexes: Analysis, Applications, and Discretizations, MFO Oberwolfach, Germany.
: Averaging property of a discrete wedge product. Minisymposium on Conservative and Geometric Discretizations, CAIMS Annual Meeting, UBC Okanagan, Canada.
: Discrete vector bundles with connection. Workshop on Discrete Shapes, Center of Mathematical Sciences and Applications, Harvard University, Massachusetts.
2021: Structure-preserving discretization of differential calculus and geometry (online). Kyushu-Illinois Strategic Partnership Colloquium, UIUC and Kyushu University, Japan.
2020: Discretizations of exterior calculus and differential geometry. Mathematics Seminar Series, Indraprastha Institute of Information Technology Delhi, New Delhi, India.
: Discretizations of exterior calculus and differential geometry. Department of Mathematics, Indian Institute of Science Education and Research, Pune, India.
2019: Discrete vector bundles with connection and the first Chern class. Geometry, Compatibility and Structure Preservation in Computational Differential Equations, Newton Institute for Mathematical Sciences, Cambridge University.
: Conservative methods for piecewise smooth dynamics. Annual Meeting of the Canadian Society of Applied and Industrial Mathematics.
2016: Discrete exterior calculus discretization of Navier-Stokes equations on surface meshes. Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Japan.
: Discretizations of exterior calculus. Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, Japan.
2015: Introduction to PyDEC. Workshop on Geometry and Symmetry based Mathematical and Computational Methods, SUNY Polytechnic Institute, Utica, New York.
2014: New spaces for DEC. Geometric Discretization Methods and Adaptivity, Canadian Mathematical Society (CMS) Winter Meeting.
: Lectures on discretizations of exterior calculus. King Abdullah University of Science and Technology (KAUST), Saudi Arabia.
: First steps toward DEC Poisson convergence proofs. Structure-Preserving Discretizations of Partial Differential Equations, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota.
2012: Applied topology and matrix methods. AMS Special Session on Applied Topology, AMS Fall Central Sectional Meeting, Columbus, Ohio.
: Numerical PDEs and applied topology. Center for Computational Mathematics Seminar, Department of Mathematics, University of California at San Diego.
: Applied topology and numerical PDEs. Applied Mathematics and Analysis Seminar, Department of Mathematics, Duke University, Durham, North Carolina.
: Applied topology and matrix methods. ECE Distinguished Seminar, Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina.
: Discrete exterior calculus and its connection to finite element exterior calculus. ICERM, Brown University, Providence, Rhode Island.
: Optimization problems in computational topology. Applied Mathematics and PDE Seminar, Department of Mathematics, University of California at Davis, California.
: Relative torsion in homology computations. Oberwolfach Workshop on Triangulations, Mathematisches Forschungsinstitut Oberwolfach, Germany.
: Algebraic topology for computations. Topology Seminar, Department of Mathematics, Stony Brook University.
: Algebraic topology for computations. Topology Seminar, Department of Mathematics, City University of New York.
: Torsion in computations. Dagstuhl Seminar on Applications of Combinatorial Topology to Computer Science, Schloss Dagstuhl, Germany.
: Holes, handles, and tunnels. Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland.
: Hodge decomposition: applications and numerics. Scientific and Statistical Computing Seminar, Departments of Computer Science, Mathematics, and Statistics, University of Chicago, Illinois.
2011: Optimization, knots, and differential equations. Workshop on Computational Topology, Fields Institute, Toronto, Canada.
: Optimal homology cycles and knots. Certified Geometry Processing Minisymposium, SIAM Conference on Geometric & Physical Modeling.
: Darcy flow using discrete exterior calculus. High Order Mimetic Discretizations Minisymposium, 7th International Congress on Industrial and Applied Mathematics.
: Optimal (co)homologous (co)chains. Department of Computer Science, University of Texas at Austin.
: Latest developments in discretization of exterior calculus. X-Computational Physics Division, Los Alamos National Laboratory.
2010: Optimal homologous cycles, total unimodularity, and linear programming. 42nd ACM Symposium on Theory of Computing (STOC'10), Cambridge, MA, June 6–8.
: Computational topology and linear programming. Departments of Computer Science and Mathematics, Stanford University.
: Computational topology and linear programming. Graphics, Vision and Interaction Seminar, Harvard University.
: Some PyDEC applications. Workshop on Discrete Differential Geometry for Multiphase Flow Problems, Indiana University Purdue University Indianapolis.
: Homologous chains, total unimodularity, and linear programming. Department of Computer Science, Caltech.
2009: Implementation of discrete exterior calculus. Guest Lecture in Topics in Scientific Computing, University of Chicago.
: Implementation of Hodge theory for simplicial complexes. Machine Learning Summer School, University of Chicago.
: Stiffness and Mass Matrices for Whitney Forms. Finite Element Circus, Newark, DE, April 24–25.
: Algorithms for Exterior Calculus. Finite Element Methods in Engineering and Science (FEMTEC 2009), Lake Tahoe, CA, January 5–9.
2008: Structure Preserving Approximations of Conservative Forces for Application to Small Body Dynamics. AAS/AIAA Astrodynamics Specialist Conference, Honolulu, HI, August 18–21.
: Software and algorithms for exterior calculus. Department of Mathematics Colloquium, Northern Illinois University.
: Calculus on meshes. Department of Computer Science and Engineering Seminar, Ohio State University.
: Calculus on meshes. Department of Mathematics Seminar, Mumbai, Indian Institute of Technology Bombay, India.
2006: Approximations of Distant Retrograde Orbits for Mission Design. AAS/AIAA Spaceflight Mechanics Meeting, Tampa, FL, January 22–26.
: Dynamical Systems and Space Mission Design. Department of Aerospace Engineering AE 590 Seminar, University of Illinois at Urbana-Champaign.
2005: Surface Structure of an Invariant Manifold of a Halo Orbit. AAS/AIAA Astrodynamics Specialists Conference, Lake Tahoe, CA, August 7–11.
2004: Discrete exterior calculus and the averaged template matching equations. Mathematics in Brain Imaging, Institute for Pure and Applied Mathematics, UCLA.
: Discrete exterior calculus. Oberwolfach Seminar on Discrete Differential Geometry, Mathematisches Forschungsinstitut Oberwolfach, Germany.
: Discrete exterior calculus: applications in mechanics and computer science. Los Alamos National Laboratory.
: Discrete exterior calculus: applications in computer science. Applied Mathematics and Mechanics, Sandia National Laboratory.
2003: Discrete Exterior Calculus. Mimetic Discretizations of Continuum Mechanics, San Diego, CA, July 9–11.
: Discrete Exterior Calculus. SIAM Conference on Applications of Dynamical Systems (DS03), Snowbird, UT, May 27–31.
: Discrete Exterior Calculus and its Applications. SIAM Conference on Computational Science and Engineering (CSE03), San Diego, CA, February 13.
: Template matching, EPDIFF and discrete exterior calculus. Center for Nonlinear Studies, Los Alamos National Laboratory.
2002: Analysis of 1D Template Matching Equations. 9th International Conference on Hyperbolic Problems (HYP2002), Pasadena, CA, March 25–29.
2001: Averaged Template Matching Equations. Energy Minimization Methods in Computer Vision and Pattern Recognition, Sophia-Antipolis, France, September 3–5.
1996: Combining Frequency and Spatial Domain Information for Fast Interactive Image Noise Removal. ACM SIGGRAPH, New Orleans, LA, August 4–9.