Emmy Noether Lectures

2016 Lecturer: Karen Smith

The Power of Noether’s Ring Theory in Understanding Singularities of Complex Algebraic Varieties

In one of the tremendous innovations of twentieth century mathematics, Emmy Noether introduced the rigorous definition of commutative rings and their homomorphisms. One of her main motivating examples was the ring of polynomial functions on a complex algebraic variety. The algebraic study of these rings can have deep geometric consequences for the corresponding variety. In this talk, I hope to explain one example of this phenomenon: namely, how reduction to prime characteristic can give us insight into the singularities of the corresponding algebraic variety. Of course, I will need to convince you that we gain something powerful in reducing modulo p, since we have given up all the tools of analysis in doing so. What we gain is the Frobenius operator on the ring, which raises elements to their p-th powers, and is a ring homomorphism in characteristic p. I hope to explain how the Frobenius operator is helpful in understanding the singularities. As an application, I will describe some work with Angelica Benito, Jenna Rajchgot and Greg Muller on the singularities of varieties that arise in the theory of cluster algebras in combinatorics.
Brief Biography:
Karen E. Smith is the Keeler Professor of Mathematics at the University of Michigan. She received a Bachelor’s degree in mathematics in 1987 from Princeton.   After a year of teaching high school, she went to the University of Michigan and received a PhD in 1993 with a thesis in commutative algebra.
In 1993, Smith went to Purdue for a year on a NSF postdoc and then became a Moore Instructor at MIT. After being promoted to Assistant Professor there, she moved to the University of Michigan in 1997, where she has been since. She received a Sloan Research Award in 1997, a Fulbright award in 2000, and was awarded the Ruth Lyttle Satter Prize in 2001. The prize citation cited her outstanding work in commutative algebra and in particular her work on “tight closure” and its applications to algebraic geometry. In 2014, she was an invited speaker at the International Congress held in Seoul, Korea.
Smith has served on the editorial boards of eight journals. She has graduated 16 PhD students, with 3 more currently, and is the Director of an NSF funded RTG program, which has supported 10 PhD students, 10 post-docs and 5 undergraduates each year since 2005.