## Discrete differential geometry

Various communities of researchers have thought about this class of problems; there are results scattered through journals on image processing, geometric mechanics, numerical analysis, graph theory, machine learning, general relativity (and more?). Besides its use in various domains of application, discrete geometry also provides fresh insights into classical, continuous geometry. (See, e.g., Regge’s discussion of Bianchi identities in his original paper on Regge calculus.)

Discrete exterior calculus, cohomology, etc.

- Discrete exterior calculus by Desbrun, Hirani, Leok, Marsden
- Discrete Poincare lemma by Desbrun, Leok, Marsden

Discrete Riemannian geometry, connections, etc.

- A Discrete Theory of Connections on Principal Bundles by Leok, Marsden, Weinstein.
- General Relativity Without Coordinates by Regge (1961) (Introduces Regge calculus.)
- The geometry of classical Regge calculus by Barrett (1987)
- Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice by Schmidt, Kohler (2001)

Discrete spectral geometry

- References on graph laplacians, etc.

Convergence of discrete geometric quantities to their continuous counterparts.

- Add references.

Discrete geometric mechanics, discrete field theory

- Discrete Euler-Poincaré and Lie-Poisson Equations by Marsden, Pekarsky, Shkoller (1999)
- Multisymplectic geometry, variational integrators, and nonlinear PDEs by Marsden, Patrick, Shkoller.
- Discrete Geometric Mechanics for Variational Time Integrators by Stern, Desburn
- Variational Integrators, a post by Rod Carvalho with relevant references
- Jerrold Marsden on Discrete Mechanics and Optimal Control
- Differential Geometry in Computational Electromagnetics, PhD thesis by Eric Alan Forgy

General links

- The Discrete Differential Geometry Forum at Columbia University
- Applied Geometry Lab at Caltech

People

I removed the post dedicated to Marsden’s talk a few days ago. You might want to update your post and replace this (broken) link with the following:

Jerrold Marsden on Discrete Mechanics and Optimal Control

Since you seem to be a physicist, you may find it interesting to take a look at Eric Forgy’s PhD thesis:

Differential Geometry in Computational Electromagnetics [pdf]

Rod CarvalhoMarch 30, 2010 at 12:31 am

The 2nd URL is obviously wrong. The right one is:

http://phorgyphynance.files.wordpress.com/2008/06/final.pdf

Rod CarvalhoMarch 30, 2010 at 12:33 am

Thanks Rod, I’ve added the references.

mkzMarch 30, 2010 at 3:13 pm

The link to Eric’s thesis is incorrect. The correct one is the one on my 2nd comment.

Rod CarvalhoMarch 30, 2010 at 4:18 pm

I actually opened his thesis and then added the link, but still managed to add the wrong link, apparently. Thanks.

mkzMarch 30, 2010 at 4:59 pm

This is a great post. I’m sure I’ll find the links very useful in my own research. Thanks!

David HorganAugust 12, 2013 at 7:16 pm