Karen Uhlenbeck | The Noether Theorems in Geometry: Then and Now

Math Science Literature Lecture, 12/9/21

Speaker: Karen Uhlenbeck (Institute for Advanced Study)

Title: The Noether Theorems in Geometry: Then and Now

Abstract: The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will mention only briefly their importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant in geometric analysis: how they underlie standard techniques and why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of Bill Thurston. Some applications to integrals on a domain a hyperbolic surface leave open possibilities for applications to integrals on domains which are locally symmetric spaces of higher dimension. The talk finishes with an example or two from the literature.

[hyperduality]

Symmetry is dual to conservation -- the duality of Noether's theorem.
Divergence (entropic) is dual to convergence (syntropic).
Energy is duality, duality is energy -- physics.
The conservation of duality (energy) will be known as the 5th law of thermodynamics! -- Generalized duality.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Forces are dual:-
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- tension, stresses are dual.
Cause is dual to effect -- forces correspond to the correlation or conserving the duality of cause & effect.
If effects are correlated with causes then this implies or synthesizes the concept of forces -- duality and correlation.
Correlation always requires two dual perspectives.
"Always two there are" -- Yoda.
Perpendicularity or orthogonality = duality!
The inner or dot product is dual to the wedge or cross product (forms) -- Maxwell's equations.
The Christian cross is composed of two perpendicular lines -- duality.
Christians have been worshipping duality for thousands of years.
Duality (energy) creates reality.