Southern California Number Theory Day
See http://math.uci.edu/~krubin/scntd for more details

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model.

By reducing the problem to a discontinuous infinite dynamical system, we construct all periodic solutions with exactly two switchings on the period and study their stability. In the problem under consideration, the hysteresis gap (the difference between the switching temperatures) is of especial importance.

If the hysteresis gap is large enough, then the constructed periodic solution is in fact unique and globally stable. For small values of hysteresis gap coexistence of several periodic solutions with different stability properties is proved to be possible.

This is a joint work with Pavel Gurevich.

I shall present known results on the following topics in the contexts of associative, Lie, and Jordan algebras:

1. Derivations and cohomology of finite dimensional algebras

2. Structure and continuity of derivations on operator and Banach algebras

3. Continuous cohomology of operator algebras, including perturbation theory and the role of complete boundedness

My purpose is to provide the background for a study of cohomology of Banach triple systems (associative, Lie, and Jordan), which currently exists only in finite dimensions, and minimally at that.

The classical Cauchy integral is a fundamental object of complex analysis whose analytic properties are intimately related to the geometric properties of its supporting curve.

In this talk I will begin by reviewing the most relevant features of the classical Cauchy integral. I will then move on to the (surprisingly more involved) construction of the Cauchy integral for a hypersurface in
$\mathbb C^n$.

I will conclude by presenting new results joint with E. M. Stein concerning the regularity properties of this integral and their relations with the geometry of the hypersurface.

(Time permitting) I will discuss applications of these results to the Szeg\H o and Bergman projections (that is, the orthogonal projections of the Lebesgue space $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).