Georgia Institute of Technology College of Sciences

We will discuss a proof that finite energy solutions to the defocusing cubicKlein Gordon equation scatter, and will discuss a related result in thefocusing case. (Don't worry, we will also explain what it means for asolution to a PDE to scatter.) This is joint work with Rowan Killip andMonica Visan.

This talk is about a random Schroedinger operator describing the dynamics of an electron in a randomly deformed lattice. The periodic displacement configurations which minimize the bottom of the spectrum are characterized. This leads to an amusing problem about minimizing eigenvalues of a Neumann Schroedinger operator with respect to the position of the potential. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. This is joint work with Jeff Baker, Frederic Klopp, Shu Nakamura and Guenter Stolz.

I will discuss linear extremal problems in the Bergman spaces $A^p$ ofthe unit disc and a theorem of Ryabykh about regularity of thesolutions to these problems. I will also discuss extensions I havefound of Ryabykh's theorem in the case where $p$ is an even integer.The proofs of these extensions involve Littlewood-Paley theory and abasic characterization of extremal functions.

Let C[-1, 1] be the space of continuous functions on [-1, 1], and denote by \Delta^2 the set of convex functions f \in C[-1, 1]. Also, let E_n(f) and En^{(2)}_n(f) denote the degrees of best unconstrained and convex approximation of f \in \Delta^2 by algebraic polynomials of degree < n, respectively. Clearly, E_n(f) \le E^{(2)}_n (f), and Lorentz and Zeller proved that the opposite inequality E^{(2)}_n(f) \le CE_n(f) is invalid even with the constant C = C(f) which depends on the function f \in \Delta^2. We prove, for every \alpha > 0 and function f \in \Delta^2, that sup{n^\alpha E^{(2)}_n(f) : n \ge 1} \le c(\alpha)sup{n^\alpha E_n(f): n \ge 1}, where c(\alpha) is a constant depending only on \alpha. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases s \le 1 and s \ge 2.