Tuesday February 05, 2019 5:30 PM  6:30 PM Math Tower 5127
 Dennis Sullivan, Stony Brook University
Opportunities suggested by 3D geometrization1] 3D manifolds can be described via a reassembly of geometric pieces. These pieces arise from a century of results due to Poincare, Dehn, Alexandrov, Seifert, Milnor,.....to 1977 then Thurston, Hamilton, and Perelman,.... to 2007.
2] These results depend on topology plus a nonlinear PDE of the form time derivative term = a quadratic term plus a heat equation term with the evolving object being a metric or rather its Ricci curvature, but taken modulo the group of diffeomorphisms acting on metrics, thus the associated riemannian geometry or curvature. [Recall Ricci implies Riemann in 3D]
3] This progress by Perelman suggested the hamilton perelman PDE analysis might help in the theory incompressible fluid equations which also have the form time derivative term = a quadratic term plus a heat equation term with the evolving object being a 3D diffeomorphism and the nonlinear term being appropriately similar.
4] This description of 3D manifolds comes out in terms of compositions of a certain finite dimensional real algebraic set of diffeomorphisms of three space.
This set describing all closed three manifolds is derived from the limits of the evolving geometries of the hamilton perelman nonlinear PDE and seems to generate a group of more and more turbulent diffeomorphisms.
These seem general enough to describe the actual behaviour of fluids and gases in 3D.
Moreover,because they arise from an algebraically specific set of generators This group might be of help in the study of the specific solutions of the similar nonlinear PDEs for fluids.
Quaternions play a role , with a puzzling significance, in this description which is 7/8ths purely real algebraic and 1/8 an infinite cyclic covering space of such.
Warm Up Exercise:
To practice for the talk, try to compute pictorially the group of diffeopmorphisms of the upper half plane
{(x.y) ] y>0} generated by G= PSL{2,R}
acting by z = x+iy > az+b/cz+b
a,b,c,d real with positive determinant and Z/2Z generated by the flip (x,y)> (x,1/y).
It seems to be infinite dimensional and to give virtually ALL transformations approximately.
Hint: study the conformal distortion of the words in the free product of G and Z/2Z mapping into the set of bijections.
