While conic sections and spheres are smooth varieties, in general, a variety can have singular points—places where it is pinched or intersects itself. In this first talk, we will discuss Hironaka's famous theorem on resolution of singularities—a technique to "get rid" of the singular points. We introduce a class of singular varieties called rational singularities that are important because they are well-approximated by their resolutions, and explain how one can use "reduction modulo p" to characterize them.
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.