A glimpse of Hamiltonian mechanics, integrable systems, and Newton-Okounkov bodies
This will be an expository talk on both the history of, and one recent development in, the theory of integrable systems. We will start the story with the mathematical (a.k.a. symplectic- or Poisson-geometric) formulation of classical Hamiltonian mechanics. From there, armed with the notion of "conserved quantities", we can define the notion of integrable systems. To illustrate how this can work in practice, we'll describe two of the most famous examples, (1) the Gel'fand-Zeitlin system on spaces of isospectral Hermitian matrices, and (2) the symplectic toric manifolds. From there, time permitting, we'll take a detour into some complex algebraic geometry, and briefly describe a recent, much more abstract, method for proving the existence of integrable systems using the theory of valuations and Newton-Okounkov bodies.