Mathematics Colloquia and Seminars

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As  requested by the organizers, I'll tell stories and share memories of my career leading to four recent results with current collaborators Robin Young, Moritz Reintjes, Christopher Alexander, Heinrich Freistuehler  and Zeke Vogler.   (1) Discovery of a new set of equations for regularizing connections in Geometry (the RT-equations), which we recently used to establish a geometric level of regularity for metrics and connections in General Relativity and Geometry, (with Moritz Reinjes).  (2) The construction and proof of existence of the first time periodic solutions of compressible Euler equations, establishing a connection between the linear and nonlinear theories of sound.   The result establishes that the attenuation of oscillatory signals due to shock formation and subsequent shock wave  dissipation, can be overcome when one appropriately accounts for entropy and temperature variations, (with Robin Young).   (3)   A characterization of the radial instability of Friedmann spacetimes at the Big Bang, leading to a proof that under-dense solutions generically accelerate away from Friedmann spacetimes before they decay back to Friedmann spacetimes as time tends to infinity.   (Sound like Dark Energy?)  This is based on a self-similar formulation of the Einstein equations which represents the critical Friedmann spacetime as a rest point, allowing for eigenvalue analysis of the stability.  (With Christopher Alexander and Zeke Vogler.)  (4)   The derivation of a unique causal relativistic version of the classical Navier-Stokes equations which is "phenomenologically correct at shock waves" in the zero dissipation limit, like the classical version of the equations.   The result is obtained by formulating a precise set of first order equivalence transformations, appropriate for a first order theory of dissipation, and then establishing that there is a unique causal hyperbolic system of equations which is first order equivalent to both the Eckart and Landau equations, such that all weak and strong shocks admit shock profiles.  (With Heinrich Freistuehler, based on his definitive recent investigations.)  With special memories of my teacher, friend and long time collaborator Joel Smoller, and my friend and collaborator Eli Isaacson, who taught me so much. ``I get by with a little help from my friends."

There will be a Tea at 3:30pm and a Drinks Reception at 5-6pm