The Vanderbilt Initiative for Gravity, Waves, and Fluids is an interdisciplinary research venture providing mathematicians, physicists, and astrophysicists with the resources and space to connect and collaboratively work on problems of outstanding scientific merit, such as:

• General relativity: theoretical, mathematical, numerical, or experimental, including, but not restricted to, black holes, gravitational radiation, and multimessenger astrophysics.
• Fluid mechanics: theoretical, mathematical, numerical, or experimental, including, but not restricted to, relativistic fluids far from equilibrium.
• Evolution of partial differential equations related to fluids and gravity, including, but not restricted to, the geometric analysis of waves and fluids.
• The physics and mathematics of neutron star mergers and high-energy nuclear collisions.

VandyGRAF Seminar Series

All VandyGRAF talks will take place in the Chapel in the 17th&Horton building, unless indicated below.

---

Peter Olver, University of Minnesota

November 8, 12:00 pm - 1:30 pm

Two New Developments for Noether’s Two Theorems

Abstract: In the first part, I start by recalling the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg–de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler–Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler–Lagrange equations satisfy Noether dependencies; examples include general relativity, electromagnetism, and parameter-independent variational principles.

Noether’s First Theorem relates strictly invariant variational problems and conservation laws of their Euler–Lagrange equations.  The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. In the second part of this talk, I highlight the role of Lie algebra cohomology in the classification of the latter, and conclude with some provocative remarks on the role of invariant variational problems in fundamental physics.

Mainak Mukhopadhyay, Pennsylvania State University

November 15, 12:00 pm - 1:30 pm

James Dent, Sam Houston State University

November 22, 12:00 pm - 1:30 pm

Khwahish Kushwah, Universidade Federal Fluminense

December 6, 12:00 pm - 1:30 pm

---