Welcome!
Welcome to the Diffeomorphism and Geometry Group web
page. 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.