Devin Murray

Title: Introduction to right-angled Artin groups

Speaker: John Calabrese (Rice)

Title:

From Hilbert’s Nullstellensatz to quotient categories

Abstract:

A common theme in algebraic geometry is the interplay between algebra and geometry. In this talk I will discuss a few “reconstruction theorems”, in which the algebra determines the geometry.

Speaker: Dusko Pavlovic (UMH ICS)

Title: From Data Analysis to Dedekind-MacNeille Completions of Categories

Abstract: Solutions of mathematical problems are well-known to help with practical applications (to the point of being “unreasonably effective”, as Wigner put it). It is less well-known that practical applications sometimes help solving long standing mathematical problems. I will tell a story of this second kind.

The Dedekind-MacNeille completion of a poset is the smallest complete lattice that contains it, or equivalently the largest complete lattice where each element is both a meet and a join of the elements of the poset. Dedekind devised it to reconstruct the reals as a completion the rationals, and MacNeille generalized it to arbitrary posets. When posets are generalized to categories (so that the partial ordering a<b is expanded into the morphisms a->b), then meets and joins become limits and colimits, and the obvious task arises: generalize the Dedekind-MacNeille completion to categories. The task is thus to embed any given category into a category with all small limits and colimits, in such a way that any limits and colimits that already existed are preserved, and that any new objects that are added are both limits and colimits from the original category. This task was formulated already in the 50s, and it was listed as the most important open problem in Lambek’s 1966 “Completions of Categories” (volume #24 of Springer LNM). Stunningly, in 1972, Isbell proved that already the group Z_4, viewed as a one-object category, cannot be embedded into a bicomplete category where each object is both a limit and a colimit of diagrams built from copies of Z_4. But since the inductive process of adjoining limits to a category obviously settles at various bicompletions, and since it is easy to see that some of these bicompletions must be minimal, Isbell’s negative result just expanded the question: What are minimal bicompletions of categories, and which properties make them minimal? The question remained open for more than 40 years, or almost 60, depending how you count.

In this talk I will sketch the answer at which we arrived in 2015. It emerged as a special case of a matrix bicompletion construction, developed in a data analysis project. In the meantime, the practical applications of the result have expanded, but some of the mathematical repercussions, and most of the algorithmic issues, have not been settled.

Speaker: Michael Yampolsky (U. Toronto)

Title: Renormalization of rotation domains

Abstract: Renormalization has emerged as a key theme in low-dimensional dynamics. I will start with a review of renormalization of maps of the circle, and will then discuss applications of renormalization theory to the study of Siegel disks in one and two complex variables.

Speaker: Robin Deeley (U Colorado)

Title: Minimal dynamical systems

Abstract: A self-homeomorphism of a compact Hausdorff space is called minimal if each of its orbits is dense. I will discuss the following question: given a compact Hausdorff space does there exist a minimal homeomorphism on it? Although the answer is no, a similar question has a positive answer for any finite CW-complex. I will also discuss a number of explicit examples of minimal dynamical systems. All of our constructions are motivated by questions in C*-algebra theory. Nevertheless no knowledge of C*-algebras is required for the talk. This is joint work with Ian Putnam and Karen Strung.

Speaker: Rohit Nagpal (U. Wisconsin)

Title: S_{infty}-equivariant modules over polynomial rings in infinitely many variables

Abstract: Let R be the polynomial ring k[x_1, x_2, ldots] in countably many variables. Cohen proved that S_{infty}-stable ideals in R satisfy the ascending chain condition. This makes the category of smooth equivariant R-modules a noetherian category. Smooth modules over R appear naturally in algebraic statistics and chemistry, and so it makes sense to study this category in detail. The first step in this direction is to understand its spectrum. In this talk, we describe this spectrum. In particular, we show that the S_{infty}-stable ideal I_n generated by n-variable discriminants is in this spectrum. We show that every nonzero S_{infty}-stable ideal must contain I_n for some large n, and so these ideals are of primary interest. We also mention some new algebraic properties of discriminants. This is a part of an ongoing project with Andrew Snowden.