We develop basic properties of generators of J_{\lambda^+}(a). Then we present the theorem of Shelah that under certain circumstances, max(pcf(a)) has the largest possible cardinality.

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Abstract:

(Joint work with Injo Hur)

 The KdV equation has many connections in many different parts of mathematics and physics. For example, it is of critical importance in the inverse spectral theory of the Schrödinger operator, since it describes a way to evolve a Schrödinger operator that keeps its spectrum invariant. This evolution is known as the KdV hierarchy.

Another useful perspective on the inverse spectral theory of the Schrödinger operator is that of the Herglotz m-function. To each Schrödinger operator we associate a holomorphic function from the upper half-plane to itself, such that the limiting behavior of this function on the real line determines the spectrum of the Schrödinger operator. 

We combine these two perspectives on inverse spectral theory, and introduce a version of the KdV hierarchy that applies to all holomorphic functions from the upper half-plane to itself, not just the ones that are associated to a Schrödinger operator. This approach suggests a way to unify a large class of isospectral evolutions for many different operators.

Mutual stationarity is a notion of infinite products of stationary sets introduced by Foreman and Magidor. The assertion that an infinite sequence of stationary sets is mutually stationary has a natural model theoretic interpretation and can be viewed as a strengthening of the Loewenheim-Skolem property.