The time t maps of Hamiltonian flows are symplectic. The order n Taylor series approximation with respect to initial conditions of such a map is symplectic through terms of order n. Given an order n Hamiltonian symplectic map, there are a variety of procedures, called symplectification methods, which produce exactly symplectic maps with Taylor series that agree with the initial Taylor map throu...

The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(...

Let J be the Julia set of a conformal dynamics f . Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out...

Given an abstract triangulation of a torus, there is a unique point in moduli space which supports a circle packing for this triangulation. We will describe combinatorial deformations analogous to the process of conformal welding. These combinatorial deformations allow us to travel in moduli space from any packable torus to a point arbitrarily close to any other torus we choose. We also provide...