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(12 Mar 2019)


ill titleWelcome!

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
ill_1in 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.