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 viewpoints 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 makeup 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.