We discuss some ways to quantify how singular a given variety is. We introduce an analytic index of singularities, defined in terms of convergence of a certain integral. This provides a numerical measure of the singularity, sometimes called the log-canonical threshold. Remarkably, this numerical invariant can also be defined using reduction to characteristic p: for each p, we define a numerical measure of singularities called the F-pure threshold, which has striking fractal-like behavior. Amazing, taking the limit as p goes to infinity, we recover the analytically defined log canonical threshold. We will discuss these theorems and some deep open questions that remain at the frontier of this topic.
Karen Smith is the M. S. Keeler Professor of Mathematics at the University of Michigan. In 2001 she was awarded the Ruth Lyttle Satter Prize "for her outstanding work in commutative algebra, which has established her as a world leader in the study of tight closure, an important tool in the subject introduced by Hochster and Huneke. It is also awarded for her more recent work which builds new bridges between commutative algebra and algebraic geometry via the concept of tight closure.”
Smith is the recipient of a Sloan Research Award, a Fulbright award, and research grants from the National Science Foundation and the Clay Foundation. She has twice (in 2002–2003 and 2012–2013) helped organize a Special Year in Commutative Algebra at the Mathematical Sciences Research Institute (MSRI) in Berkeley CA, and in 2014, she was an invited speaker at the International Congress held in Seoul, South Korea. Smith has been on the editorial board of eight journals, is the director of an NSF funded Research Training Group (RTG) program, which has supported ten PhD students, ten post-docs and five undergraduates each year since 2005, and is co-author of An Invitation to Algebraic Geometry. She is featured in the book Complexities: Women in Mathematics edited by Bettye Anne Case and Anne M. Leggett.