Commutativity of Lie groups

By Omar Antolín, Universidad Nacional Autónoma de México.
Language: English.

Given a Lie group G, we can think of the space of homomorphisms Hom(Z^n, G) as the space of n-tuples of elements of G that commute pairwise. These spaces are more subtle than one might think, and even basic invariants such as the number of connected components can lead to surprising results. Fixing G and varying n we can construct what is known as the classifying space for commutativity in G. I will describe what is known about these classifying spaces, whose study has just begun.

Complete Intersection Injective Dimensions

By Sean Sather-Wagstaff, Clemson University.
Language: English.

The complete intersection dimension of a module M over a commutative local Noetherian ring R is a homological invariant that detects many properties like projective resolution properties of modules over formal complete intersection rings. It was introduced by Avramov, Gasharov, and Peeva in 1997. In this talk, we will introduce a new variant of this, the Hom complete intersection dimension (Hom-CI-id for short), that detects many properties like injective resolution properties of modules over formal complete intersection rings. For instance, every module over a formal complete intersection ring has finite Hom-CI-id; conversely, if the residue field has finite Hom-CI-id, then the ring is a formal complete intersection. Also, the Hom-CI-id satisfies versions of the Bass formula, the Chouinard formula, and Bass’ conjecture. This is joint work with Jon Totushek.

Some non-associative algebraic structures in Low dimensional topology

By Mohamed Elhamdadi, University of South Florida.
Language: English.

We will introduce some non-associative algebraic structures called racks and motivated by the study of knots in the 3-space and knotted surfaces in 4-space. We will define a cohomology theory for these structures and will explain its use in constructing invariants of knots. We will explain how these structures give an enhancement when considered in the category of topological spaces or Lie groups.