I will present an application of Borel Uniformization theorem for "large sections", and if time permits give a proof of the theorem.

In 2003, the National Security Agency (NSA) payed Certicom, a small Canadian security company, 25 million dollars for the right to use Certicom's elliptic curve cryptography technology. We will introduce elliptic curves and their applications to cryptography and computer security, and suggest why the NSA paid so much. Then we will describe the computationally important "point counting problem", which is necessary for efficient elliptic curve cryptography. We will survey some recent research that "counts points" on certain elliptic curves.

I will discuss a representation for the magnetization field of the critical two-dimensional Ising model in the scaling limit as a random filed using an ensemble of measures on the plane associated with renormalized cluster areas.

The renormalized areas come from the scaling limit of critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. The representation is based on the interpretation of the lattice magnetization as the sum of the signed areas of clusters. If time permits, potential extensions, including to three dimensions, will also be discussed. The talk will be based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.