Georgia Institute of Technology College of Sciences

We review the basics of hyperbolic geometry in preparation for studying mapping class groups.

Consider a general linear Hamiltonian system u_t = JLu in a Hilbert space X, called the energy space. We assume that R(L) is closed, L induces a bounded and symmetric bi-linear form on X, and the energy functional has only finitely many negative dimensions n(L). There is no restriction on the anti-selfadjoint operator J except \ker L \subset D(J), which can be unbounded and with an infinite dimensional kernel space. Our first result is an index theorem on the linear instability of the evolution group e^{tJL}. More specifically, we obtain some relationship between n(L) and the dimensions of generalized eigenspaces of eigenvalues of JL, some of which may be embedded in the continuous spectrum. Our second result is the linear exponential trichotomy of the evolution group e^{tJL}. In particular, we prove the nonexistence of exponential growth in the finite co-dimensional center subspace and the optimal bounds on the algebraic growth rate there. This is applied to construct the local invariant manifolds for nonlinear Hamiltonian PDEs near the orbit of a coherent state (standing wave, steady state, traveling waves etc.). For some cases (particularly ground states), we can prove orbital stability and local uniqueness of center manifolds. We will discuss applications to examples including dispersive long wave models such as BBM and KDV equations, Gross-Pitaevskii equation for superfluids, 2D Euler equation for ideal fluids, and 3D Vlasov-Maxwell systems for collisionless plasmas. This work will be discussed in two talks. In the first talk, we will motivate the problem by several Hamiltonian PDEs, describe the main results, and demonstrate how they are applied. In the second talk, some ideas of the proof will be given.

The theory of étale cohomology provides a bridge between two seemingly unrelated subjects: the homology of braid groups (topology) and the number of points on algebraic varieties over finite fields (arithmetic). Using this bridge, we study two problems, one from topology and one from arithmetic. First, we compute the homology of the braid groups with coefficients in the Burau representation. Then, we apply the topological result to calculate the expected number of points on a random superelliptic curve over finite fields.