Mathematics Colloquium, Spring 2004
Special Lecture presented by

On sets defined by polynomial inequalities

Let $P_1, \cdots, P_d$ be $d$ polynomials in $n$ variables, $d \geq n$. A "ball" (in $\mathbb R^n$ or $\mathbb C^n$) centered at $a$ with polyradius $\delta = (\delta_1, \cdots, \delta_d)$ is defined to be the set \[ B(a;\delta) = \left\{ x \, : \, \left| P_j(x) - P_j(a) \right| < \delta_j, \; 1 \leq j \leq d \right\}. \] Such balls occur naturally in many problems in harmonic and complex analysis, for instance in the context of certain strong maximal operators and in the study of the Bergman kernel on weakly pseudoconvex, non-decoupled domains. We obtain global estimates for the volumes of these balls, and use this to investigate some of these problems. This is joint work with Alexander Nagel of University of Wisconsin, Madison.

Wednesday, January 28, 2004, 245 Altgeld Hall, 4:00 p.m.