Public key cryptography is about 25 years old, and relies on number theory. We will discuss Diffie Hellman key exchange and ElGamal encryption, and some recent improvements on them. We show how number theory and algebraic geometry, and in particular the rationality of certain algebraic tori, can be used to give a deeper understanding of these improvements, and to give new cryptosystems.

The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a coupled PDE system between parabolic and hyperbolic phases, the latter inducing a loss of regularity. In this talk, I will outline the proof of the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. The functional framework employed is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.

I will talk on a generalization of classical Calabi's strong maximum (1957) in the framework of Dirichlet forms associated with strong Feller diffusion processes.
The proof is stochastic and the result can be applicable to a singular geometric space appeared in the measured Gromov-Hausdorff convergence (precisely in the convergence by spectral distance by Kasue Kumura) of compact Riemannian manifolds with uniform lower Ricci curvature and uniform upper diameter.

This is the second lecture of learning seminar
on the inverse scattering theory

In this talk, we demonstrate the existence of non-circular shape-invariant (self-similar) growing and melting two dimensional crystals. This work is motivated by the recent three dimensional studies of Cristini and Lowengrub in which the existence of self-similar shapes was suggested using linear analysis and dynamical numerical simulations. Here, we develop a nonlinear theory of self-similar crystal growth and melting. Because the analysis is qualitatively independent of the number of dimensions, we focus on a perturbed two-dimensional circular crystal growing or melting in a liquid ambient. Using a spectrally accurate quasi-Newton method, we demonstrate that there exist nonlinear self-similar shapes with k-fold dominated symmetries. A critical heat flux J_k is associated with each shape. In the isotropic case, k is arbitrary and only growing solutions exist. When the surface tension is anisotropic, k is determined by the form of the anisotropy and both growing and melting solutions exist. We discuss how these results can be used to control crystal morphologies during growth. This is joint work with Shuwang Li, Perry Leo and Vittorio Cristini.

This is joint work with Shuwang Li, Perry Leo and Vittorio Cristini.