### Harrison’s Geometric Calculus

#### Posted by Urs Schreiber

In

Jenny Harrison
*Lectures on Geometric Calculus*

arXiv:math-ph/0501001

Morris Hirsch # is quoted as having said the following:

A basic philosophical problem has been to make sense of “continuum”, as in the space of real numbers, without introducing numbers. Weyl [5] wrote, “The introduction of coordinates as numbers… is an act of violence”. Poincaré wrote about the “physical continuum” of our intuition, as opposed to the mathematical continuum. Whitehead (the philosopher) based our use of real numbers on our intuition of time intervals and spatial regions. The Greeks tried, but didn’t get very far in doing geometry without real numbers. But no one, least of all the Intuitionists, has come up with even a slightly satisfactory replacement for basing the continuum on the real number system, or basing the real numbers on Dedekind cuts, completion of the rationals, or some equivalent construction.

Harrison’s theory of chainlets can be viewed as a different way to build topology out of numbers. It is a much more sophisticated way, in that it is (being) designed with the knowledge of what we have found to be geometrically useful (Hodge star, Stokes’ theorem, all of algebraic topology,…), whereas the standard development is just ad hoc – starting from Greek geometry, through Newton’s philosophically incoherent calculus, Descarte’s identification of algebra with geometry, with additions of abstract set theory, Cauchy sequences, mathematical logic, categories, topoi, probability theory, and so forth, as needed. We could add quantum mechanics, Feynman diagrams and string theory! The point is this is a very roundabout way of starting from geometry, building all that algebraic machinery, and using it for geometry and physics. I don’t think chainlets, or any other purely mathematical theory, will resolve this mess, but it might lead to a huge simplification of important parts of it.

**The key idea of chainlets**

As Harrison discusses on p. 10, the distinguishing key ingredient and starting point of her approach is a certain new choice of *norm* with which to *complete* the vector space of polyhedral chains in $\mathbb{R}^n$ (definition of those on p. 15).

She calls this norm the *natural norm* – in muscial contrast to the *flat* and *sharp* norms on chains in $\mathbb{R}^n$ introduced by Whitney in the old days.

The elements in this vector space of chains completed with the *natural norm* are her **chainlets**. The precise definition of these is hidden on p. 32.

**Hint for readers of the lecture notes**. Maybe something went wrong in the text composition: when reading this start with chapter 1 on p. 15, which is called “Chainlets” but does not once mention the term. It ends with the definition of the *natural norm* on polyhedral chains at the bottom of p. 17 and with the observation that the obvious boundary operator is a bounded operator in the natural norm – in prop. 1.1 on p. 18.

Skip the standard discussion of standard differential forms in chapter 2 (unless you really don’t know yet what a differential form is) and skip for the moment also the beginning of chaper 3 and jump to corollary 3.6 on p. 32. The sentence below is the definition of chainlets. We are reminded that on them the boundary operator is continuous, which is the first goal to be achieved by the *natural norm*. The next paragraph states the second goal: Stokes’ theorem for the pairing of chainlets with suitable forms. Take that as the motivation and jump back to p. 30 now, where the “suitable forms” are defined as those having in turn bounded norm of certain kind (the notation for mass norm and derivative used there were defined on p. 16.)

All the norm gymnastics then ensures that the pairing appearing in Stokes theorem is well defined, with the structure of the proof being essentially trivial. Go back to the table on p. 10 to check how the *natural norm* is thus an improvement over the *sharp* and the *flat* norm.

A closer look at the space of chainlets is then taken from p. 33 on. A crucial statement is that on top of p. 36 (corollary 3.13), which gives an abstract characterization of the natural norm in terms of crucial properties.

**The point of the natural norm and of chainlets**

The point is, as far as I understand,
that the *natural norm* cures the deficiency of Whitney’s *sharp* and *flat* norms and admits the crucial ingeredients found in ordinary differential geometry: boundary, Hodge star, coboundary to exist on chains and their duals. The crucial theorems, such as Stokes’s and Green’s (p. 55) theorem, then hold and the crucial operators, such as curl, divergence, Laplace operators (both for cochains as well as on chains, see p. 48) exist essentially by construction.

But with that in hand, it turns out that the space of chainlets and their duals is much larger than that of the smooth chains and their smooth duals. For instance, the examples on p. 33 say that chainlets may be fractals or graphs or $L^1$-functions. This is then the topic of section 4: *Calculus on fractals*.

**Infinitesimals**

A crucial observation is, p. 43, that there are chainlets which are “infinitesimal”: the sequence of $k$-dimensional chains which are linear multiples of factor $(2^\ell)^k$ with a $k$-dimensional cube of side length $2^{-\ell}$ does converge to a chainlet as $\ell \to \infty$.

This chainlet behaves like a “Dirac delta $k$-form”, or rather the dual to that. And it is not just the limit of cubes becoming infinitesimal, but of any $k$-volume becoming infinitesimal (prop. 4.2, p. 44).

Using such Dirac delta-like chainlets, Harrison suggests a formalism of discrete calculus in the last section, starting on p. 75. I have not yet fully absorbed that, but I need to quit for now.

## Re: Harrison’s Geometric Calculus

Hello Urs,

Thanks for your interest. The paper you are reading was just a draft and is now obsolete. New vistas opened up as soon as it was written. I am working hard on a monograph to reflect the current state of the theory which has gone quite a bit further. If you can hold on a little while longer, I will be pleased to produce it, or at least the first few chapters.

Regards,

Jenny Harrison

Professor of Mathematics

University of California, Berkeley