
At the end of the 1950's, it was felt that there was a good
mathematical theory of the geometry of (classical/non-quantum)
physical systems. Broadly, this could be called the study of
connections on principal bundles on manifolds or, to use physics
terminology, the study of gauge fields; we will refer to these as
Cartan geometries below. The _qualitative_ properties of such
geometries can be obtained by studying their topological invariants;
which can be thought of as properties that do not change under
continuous operations like stretching. (Topology is sometimes called
"rubber-sheet" geometry).

This became the background in which an enormous number of beautiful
theories (like cobordism, Index theory and so on) were studied 
through the 1960's, 70's and 80's.

This formulation of geometry required the physical system to have an
infinitesimal homogeneity (in other words, the laws of physics are to
be given by differential equations involving tensors and spinors).
From a mathematical perspective strong notions of continuity, such as
differentiability, were essential to this approach to geometry.

> ![Mikhail Gromov](gromov_small.jpg)

Mikhail Gromov[^1] showed how we can study geometric properties without
retaining homogeneity or continuity.

[^1]: Image taken from [the web site](http://www.abelprisen.no/en/)
for the Abel Prize.

The key mathematical definition is that of _quasi-isometry_. Gromov's
definition allows us to "tear" space and "re-stitch" it differently
provided that these operations are small in comparison to the scales
at which we want to examine the space; the resulting geometry
still shares some "coarse" geometric properties with the older
one. In particular, it is possible to detect whether the geometry is
negatively curved (i.e. like the non-Euclidean geometry of Bolyai
and Lobachevsky). In addition, one can study the quasi-symmetries of
the geometry (which are quasi-isometric transformations of space to
itself). This leads to rigidity results that bind groups of symmetries
more tightly with the kinds of spaces that they can act on.

There are a number of physical systems (ensemble systems like
sand-piles and glass or biological systems) that do not exhibit the
kind of homogeneity that Cartan geometries have. It certainly
_seems_ as if Gromov's coarse structures are more applicable in such
cases. Further refinements are required before one can design and
carry out experiments that will confirm these expectations.

For those interested in the interface between geometry and physical
systems, the 3G technology of Geometry, Groups and Gromov is worth
exploring![^2]

[^2]: I have written a [longer article](http://www.imsc.res.in/~kapil/geometry/gromov) that gives a
layman's introduction to Geometry before and after Gromov.

