Georgia Institute of Technology College of Sciences

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.

In two-dimensional critical percolation, the work of Aizenman-Burchard implies that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+\epsilon, for some positive \epsilon. No more precise lower bound has been given so far. Conditioned on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Morrow and Zhang to have volume n^4/3 on the triangular lattice. In 1992, Kesten and Zhang asked how, given the existence of an open crossing, the length of the shortest open crossing compares to that of the lowest; in particular, whether the ratio of these lengths tends to zero in probability. We answer this question positively.

We will discuss KAM results for symplectic and presymplectic maps. Firstly, we will study geometric properties of a symplectic dynamical system which will allow us to prove a KAM theorem in a-posteriori format. Then, a corresponding theorem for a parametric family of symplectic maps will be presented. Finally, using similar method, we will extend the theorems to presymplectic maps. These results appear in the work of Alishah, de la Llave, Gonzalez, Jorba and Villanueva.