Geometry working seminar

Talk 1. Jon Herman, Pure Mathematics Department, University of Waterloo

“Noether’s Theorem under the Legendre transform”

First I will give a proof of Noether’s theorem as stated in Hamiltonian mechanics. I will then show how the Legendre transform, Φ, gives a one-to-one correspondence between the Hamiltonian and Lagrangian statements of Noether’s theorem. That is, continuous symme- tries and their resulting conserved quantities as so defined in a Lagrangian system (M,L) give the corresponding continuous symmetries and conserved quantities in the Hamiltonian system (T∗M,ω,L ◦ Φ). Conversely, in some given cotangent bundle, T∗M, equipped with a Hamiltonian H and a canonical symplectic 2-form, the continuous symmetries and con- served quantities induce corresponding continuous symmetries and conserved quantities in the underlying Lagrangian system (M, H ◦ Φ−1).

Talk 2. Matthew Beckett, Pure Mathematics Department, University of Waterloo

“Monopoles and S1-invariant instantons” Abstract

We continue where we left off last time, describing a correspondence between monopoles on R3 \ {0} and S1-invariant instantons on R4 \ {0}. I will go into more detail about what S1-invariance means and then describe the correspondence and work through an example.