Georgia Institute of Technology College of Sciences

I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

In this lecture I will outline an estimate on the indirect term of the Coulomb energy and finish the proof of Stability of Matter by showing that atoms in Thomas Fermi Theory do not bind.

It is an everyday observation that the internal energy of a piece of material is extensive, i.e., proportional to the number of atoms in this material. A celebrated result of Dyson and Lenard (1967) explains this fact on the basis of quantum mechanics, the fundamental theory that is the basis for the description of the material world. The proof of Dyson and Lenard was greatly simplified by Lieb and Thirring (1975) using Thomas Fermi theory and what is now called the Lieb-Thirring inequality. In these talks I explain the notion of Stability, give an outline of the Lieb-Thirring proof and explain a proof of the Lieb-Thirring inequality with good constants. If time permits I will talk about further developments, like systems interacting with magnetic fields.