The relationship of knot theory and physical theory is intimately tied with a mystery discovered by Herman Weyl in the early part of the 20th century. Weyl was so impressed with his observation that he suggested building a Geometry that would unify his idea with General Relativity to make a unified field theory.  The Weyl theory did not quite succeed. 

Yet Weyl's did succeed by the quantum reformulation of Fritz London who placed it in a quantum context. Experimental verification came much later with the Aharonov Bohm effect (circa 1950) mixing quantum theory and electromagnetism.  Theoretical influence of this idea came with the generalization to Yang-Mills theory and the Standard Model of Particle Physics. In the 1980's Ed Witten suggested the use of the Weyl idea of measuring physics along space curves to  produce invariants such as the Jones polynomial in knot theory. Topological Quantum Field Theory was born. This is the story of a revolution in knot theory that can give us a perspective on scientific revolutions in general and on the sustainability of both scientific and social structures. 

In order to tell this story we will begin with the fundamental ideas of knot theory. These ideas speak to the essence of sustainability of structure. A knot is a kind of topological pattern in a rope. We are all familiar with it. But consider that if you knot your arms and hold an unknotted rope between your hands, then you can unfold the arms and find that the knot has been transferred to a knot in the rope! The knot is an example of Pattern Integrity, as Buckminster Fuller liked to say. The knot is not a thing, but it needs material substrate to be sustained. Just so, the key theme in the discussion of sustainability in our world is how we choose the patterns to sustain, how they are related to fundamental science and mathematics, and how we are to find the material forms that will sustain them. We will discuss science, mathematics and sustainability in this talk.