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2018-19 Milliman Lectures

Karen E. Smith
Karen E. Smith, University of Michigan

Karen E. Smith, University of Michigan

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.


Singularities in Algebraic Geometry: Resolving and Measuring using reduction modulo p

In these three talks, I hope to share some of the beauty of an ancient field of mathematics called Algebraic Geometry, and some of the excitement of modern techniques used to investigate it.

Algebraic geometry is the study of algebraic varieties, or geometric shapes described by polynomial equations. You already know many examples, such as the circle, whose polynomial equation is x2 + y2 = 1, or a sphere. Algebraic varieties are ubiquitous throughout mathematics and its applications to science and engineering. Not only do they naturally arise in important contexts—the set of all rigid transformations of space, for example, can be given the structure of an algebraic variety—but also all kinds of complicated behavior can be described (or approximated) by polynomials. At the same time, polynomials are relatively easy to manipulate by hand or machine, so algebraic geometry is a tool for scientists, engineers and even artists, as well as a rich source of examples for mathematicians in almost any field.

Lecture I: Resolutions of Singularities

March 5, 2019 | 4-5pm | Physics-Astronomy Auditorium A102

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.

Lecture 1

 

Lecture II: Measuring Singularities

March 6, 2019 | 4-5pm | Communications Building 120

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.

Lecture 2

 

Lecture III: Non-Commutative Resolutions of Singularities

March 7, 2019 | 4-5pm | Communications Building 226

A foundational result of Hilbert underlies all of algebraic geometry: an algebraic variety can be understood more or less completely by fully understanding its ring of regular (polynomial) functions. As we saw in the first two lectures, questions about varieties can be translated into corresponding questions about their coordinate rings—which are always commutative Noetherian rings—and then attacked with tools of commutative algebra. On the other hand, many of these algebraic tools make sense even for non-commutative rings. In the final talk, we consider what it might mean to have a "non-commutative" variety, and what it might mean that it is "smooth." We will introduce Van den Bergh's idea of a non-commutative resolution of singularities, and again, show how working over a field of prime characteristic can provide insight into this problem.

   

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