The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2009 to the Russian-French mathematician Mikhail Leonidovich Gromov (65) for “his revolutionary contributions to geometry”. The Abel Prize recognizes contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of NOK 6,000,000 (close to € 700,000, US$ 950,000). Mikhail L. Gromov will receive the Abel Prize from His Majesty King Harald at an award ceremony in Oslo, Norway, May 19.

*Here, courtesy my colleague Kapil Paranjape, is a short layman description of the work that got Gromov the prize, followed by a link to a longer exposition.*

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 showed how we can study geometric properties without retaining homogeneity or continuity.

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.

For a longer (layman) exposition on the subject see here.

April 17, 2009 at 12:03 am |

Since the article started with classical gauge theories, it could be of interest to state whether Gromov’s coarse structures have any relevance to quantum field theories (realized as lattice field theories, for example). There are not quite resolved questions about analogues of classical structures such as instantons in such theories.

April 17, 2009 at 12:11 pm |

As one would have said after a talk — “Good question!”

One could wonder whether the appropriate “background” for quantum

field theories should be a “non-Euclidean lattice” in order to more

closely mimic curved space.

One problem is that in a purely hyperbolic lattice correlations weaken

much faster than in usual lattices and so it might make the physics

less interesting. So an appropriate lattice may be something like

SL(n,Z) for n>2 which is not hyperbolic but semi-hyperbolic (called

CAT(0) among the cognoscenti).

Also, the people who are looking for gauge theory in curved space,

want gravity and so (I think!) they are usually looking for positive

curvature — which leads to compact-ness. Compact simply-connected

objects are not interesting from a coarse geometry point of view

since the loose constants of quasi-isometry can “swallow” all compact

behaviour.

There may be work on this in the literature (some of it by Gromov!)

but I am not really “in” the area, so I don’t know.