Let S be a compact, oriented surface with boundary. In work with Anton Alekseev, we found that the Teichmueller space of hyperbolic 0-metrics on S, up to isotopies fixing the boundary, is naturally a an infinite-dimensional symplectic manifold, and is a Hamiltonian Virasoro space. In recent work with Ahmadreza Khazaeipoul, we give a similar construction for the deformation space of convex projective structures with nondegenerate boundary, and  how that it is a Hamiltonian space for the groupoid integrating the Gelfand-Dikii Poisson structure.

In this talk I will present a formulation of the integration problem for ternary (and higher arity) Lie algebras, also known as Filippov algebras, based in examples from geometry and graph theory, and a direct parallel with ordinary (binary) Lie Theory. I will also discuss related open questions in the algebra of hypermatrices and generalized commutators. Lastly, I will introduce the computational techniques we are developing at the Wolfram Institute that may give us a chance to understand higher arity algebras in novel ways, and could lead to the solution of the higher arity integration problem.

The topological entropy of geodesic flows has been extensively studied since the foundational works of Dinaburg and Manning. It measures the exponential complexity of the geodesic flow of a Riemannian manifold, and there are several results connecting it to the geometry and topology of a Riemannian manifold. In this talk I will explain how recent advances in symplectic dynamics can be used to give a meaningful extension of the topological entropy to C^0-Riemannian metrics; i.e. Riemannian metrics which are continuous but not necessarily differentiable. Similarly, using contact geometry we will explain how we can talk in a meaningful way about the topological entropy of convex and starshaped polytopes in R^4, thinking of them as a C^0-contact form. This is joint work with Matthias Meiwes.

A tridiagonal isospectral manifold is a conjugacy class of symmetric matrices with simple spectrum. In 1984 Tomei showed how properties of the permutation group and the group of sign matrices (both Coxeter groups) entered in the study of the topology of the tridiagonal isospectral manifold. We shall recall these results and discuss the case of arbitrary spectrum.

This is joint work with C. Tomei