AWM-MAA Etta Z. Falconer Lecture 2014

Lecturer: Marie A. Vitulli, University of Oregon

From Algebraic to Weak Subintegral Closure in Algebra and Geometry

Abstract: As students of algebra we quickly learn that for the purpose of solving polynomial equations the field of rational numbers is inadequate. We soon become acquainted with algebraic extensions of the rationals and later in our studies meet the fields of algebraic numbers, real numbers, and complex numbers, the latter as the algebraic closure of the real field. As students of commutative algebra we learn about integral extensions of rings and their properties and consequences in the study of algebraic varieties and schemes. Again, for some purposes, integral extensions do not accomplish all that we had hoped for. Much more recently geometers and algebraists introduced the twin theories of weak normality and seminormality for commutative rings and algebraic varieties to address some of these deficiencies. In this talk we outline the history of the twin theories with an emphasis on the recent developments in the area over the past fifteen years. For clarity of exposition we will focus our attention on the characteristic 0 case where the theories merge into one.