For every prime power q and positive integer g, we let N_q(g) denote the maximum value of #C(F_q), where C ranges over all genus-g curves over F_q. Several years ago Kristin Lauter and I used a number of techniques to improve the known upper bounds on N_q(g) for specific values of q and g. The key to many of our improvements was a numerical invariant attached to non-simple isogeny classes of abelian varieties over finite fields; when this invariant is small, any Jacobian in the given isogeny class must satisfy restrictive conditions. Now Lauter and I have come up with a better invariant, which allows us to make even stronger deductions about Jacobians in isogeny classes. In this talk, I will explain how we have been able to use this new invariant, together with arguments about short vectors in Hermitian lattices over imaginary quadratic fields, to pin down several more values of N_q(g) and to improve the best known upper bounds in a number of other cases.

We discuss modeling and numerical issues for geometric evolution laws, such as mean curvature flow and surface diffusion and relate the approach to the treatment of PDEs on surface. Results for the evolution of curves on surfaces, so called geodesic evolution laws will also be shown. The numerical treatments include parametric finite elements, level set methods as well as phase field approximations.

We prove that stochastic layer of the standard map has full
Hausdorff dimension for sufficiently large topologically generic parameters.