We continue the discussion of the basic objects in PCF theory like good (flat) points and exact upper bounds for sequences of functions in reduced products of regular cardinals. We will then give a proof of Shelah's 'Trichotomy' theorem.

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-trivial angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem, and we show that the two-dimensional water wave problem is solvable locally in time in this framework. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with $-\frac{\partial P}{\partial \bold{n}} \ge 0$, instead of the strong Taylor stability criterion $-\frac{\partial P}{\partial \bold{n}} \ge c > 0$. This work is partially joint with Rafe Kinsey.

The eigenvalues of the Laplacian encode fundamental
geometric information about a Riemannian metric. As an
example of their importance, I will discuss how they
arose in work of Cao, Hamilton and Illmanan, together
with joint work with Stuart Hall,  concerning stability
of Einstein manifolds and Ricci solitons. I will outline
progress on these problems for Einstein metrics with
large symmetry groups. We calculate bounds on the first
non-zero eigenvalue for certain Hermitian-Einstein four
manifolds. Similar ideas allow us estimate to the
spectral gap (the distance between the first and second
non-zero eigenvalues) for any toric Kaehler-Einstein manifold M in
terms of the polytope associated to M. I will finish by
discussing a numerical proof of the instability of the
Chen-LeBrun-Weber metric.

We show that for an immersed two-sided minimal surface in R^3,
there is a lower bound on the index depending on the genus and number of
ends. Using this, we show the nonexistence of an embedded minimal surface
in R^3 of index 2, as conjectured by Choe. Moreover, we show that the
index of an immersed two-sided minimal surface with embedded ends is
bounded from above and below by a linear function of the total curvature
of the surface. (This is joint work with Otis Chodosh)