### The search for discrete differential geometry

#### Posted by Urs Schreiber

At first it may look like a trivial problem whose solution should have been known for ages, being used all over the place for applications in mathematics, theoretical physics and engineering. But surprisingly, it has apparently not fully been understood yet. I am talking about the adaption of the full machinery of differential geometry to discrete spaces. In particular, I am being told that the metric aspects of such a theory still puzzle many researchers, the Hodge star operator, for instance, being notoriously hard to come by on general discrete spaces.

A while ago Eric Forgy has convinced me that it may be worthwhile to think about these issues, and after a very intesive collaboration we came up with what looks like an interesting approach to discrete metric differential geometry to us. Now we are trying to communicate our results with other researchers in the field.

Since most of this exchange is currently going on by e-mail and since this puts severe restrictions on the amount of true interaction and maybe cross-fertilization as soon as more than two people are involved, I was wondering if maybe we’d need some sort of discussion forum. This blog entry is supposed to be the entry point to such a discussion. To get started, I list some relevant liks to the current literature below. The list won’t be comprehensive at all at the moment, but I am planning to update it as we go along.

Ok, so here are some links to people and texts currently dealing with discrete differential geometry. This is just a very rough first approximation to a comprehensive link list. I am going to flesh this out as soon as I find the time.

First of all, Dimakis and Mueller-Hoissen have written many papers on dicrete differential geometry. For the moment I’ll just cite this list.

Jenny Harrison says she has been developing a new discrete theory of exterior calculus using *chainlets*. I don’t know if the new results are already published.

There is an interesting talk by Sullivan about his ideas on discrete geometry.

Alain Bossavit is an expert on discrete electromagnetism using finite elements and Whitney forms.

A. Sitarz has papers on Noncommutative Geometry apporaches to discrete spaces, for instance Noncommutative Geometry of Finite Groups.

In Berlin there is a ‘Sonderforschungsbereich’ concerned with Discrete Differential Geometry, Quantum Field Theory, and Statistical Mechanics. See their list of publications.

Our own notes (which are still in a draft stage) are

E. Forgy, U. Schreiber Discrete Differential Geometry on $n$-Diamonds.

More links, some of which need to be incorporated into this list, can be found here of course.

## Re: The search for discrete differential geometry

Hi Urs! :)

The relationship between string theory and discrete differential geometry is not obvious to me, but a lot of the papers by Dimakis and Mueller-Hoissen mention the word now and then :)

I hope that the other hosts do not mind if we use this powerful forum to discuss our work out in the open. If anyone besides us ends up reading this, let me just quickly say that Urs and I typically end up writing several emails to each other each day. It is usually me coming up with some whacky idea and then he’ll turn it into something fairly intelligible. It’s great fun! :)

Urs is a graduate student studying string theory. I somehow managed to get him hooked on discrete differential geometry, and he is now set on applying it toward string theory. So this is not really as off topic as it might seem.

Until any of the other gracious hosts object, we’ll probably try this forum out as a way to communicate our ideas to one another and to anyone else who might find what we have to say interesting.

It should go without saying that we welcome any and all comments along the way. There is a great potential to have some really nice discussion here.

Best wishes,

Eric