Welcome!
The Diffeomorphism and Geometry Group was formed as a joint
Physics and Mathematics effort between the late Prof. Thomas Branson
and Prof. Vincent Rodgers. The "Diagonal Sand" image
represents the two concepts of diffeomorphisms
and
geometry that are used in physics to mathematically
describe the forces in nature.
The rolling hills of sand on the left represent the concept of curvature which is known as geometry. Geometry is used in mathematical theories called gauge theories that are used to describe the strong nuclear force, the weak nuclear force, and electromagnetism. Geometry is also used in Einstein's theory of General Relativity which is the present day accepted mathematical theory of gravitation.
On the other hand, the smooth sand on the right corresponds to
the smooth coordinate transformations that are called diffeomorphisms
in
mathematics. Diffeomorphisms mathematically encode a
powerful physics principle that states that
physical events should not be tied to a particular point of
view. Diffeomorphisms provide the maps
from one point of view to another so these different view points can
agree on what has happened in a physical event.
Beneath it all are the tiny grains of sand. These tiny ingredients correspond to the quantum make up of the forces of Nature. A consistent mathematical description of these forces which includes their quantum nature is an active part of the research in theoretical physics.