Algebra Seminar

The Algebra Seminar provides graduate students, advanced undergraduate students, and faculty a forum to discuss and/or present Research, Problems, Applications, or Exposition in Algebra. For more information contact Justin Lynd.

Spring 2026

For the Spring 2026 semester we will meet on Wednesdays from 12 noon - 1:00 mostly on zoom. We may need to change the time and date on occasion to accommodate visitors.
For more information or connection details contact contact Justin Lynd.

1. Wednesday 28 January 2026 (12:00-1:00 on Zoom) contact Justin Lynd for the zoom link
   Models for rational (∞, 1)-categories
   Eleftherios Chatzitherodoridis
   University of Virginia
   Abstract: An (∞, 1)-category is a category enriched in spaces, possibly weakly. Our understanding of (∞, 1)-categories has been advanced thanks to the development of various models for (∞, 1)-categories, that is, mathematical objects that exhibit the structure of an (∞, 1)-category. Two such models are complete Segal spaces, as introduced by Rezk, and Segal categories, as developed from the homotopical perspective by Bergner.
   We introduce rational (∞, 1)-categories, which are (∞, 1)-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We produce two models for rational (∞, 1)-categories, rational complete Segal spaces and rational Segal categories. Our argument works for enrichment in general localizations of spaces, such as the v_n-periodic spaces of Heuts in unstable chromatic homotopy theory.
2. Wednesday 25 February 2026 (12:00-1:00 on Zoom) contact Justin Lynd for the zoom link
   Extended weak order on a Coxeter group
   Grant Barkley
   University of Michigan
   Abstract: The weak order is a partial ordering of the elements of a Coxeter group. For finite Coxeter groups, such as the symmetric group, the weak order is a lattice, meaning that any two elements have a least upper bound and a greatest lower bound. This property fails for infinite Coxeter groups, which include the affine symmetric group or a free product of Z/2s. Matthew Dyer introduced a natural completion of the weak order, which adds more objects to form a poset called the extended weak order. The new objects are "biclosed sets" of roots in a root system. Conjecturally, the extended weak order is always a lattice. We will introduce these objects and discuss progress on this and other conjectures.
3. Wednesday 25 March 2026 (12:00-1:00 in-person) Maxim Doucet Hall room 211
   Isotypic blocks that are not $p$-permutation equivalent
   John McHugh
   University of Denver
   Abstract: Two important types of "equivalences" that can exist between blocks of finite group algebras are isotypies (defined by Broue) and $p$-permutation equivalences (defined by Boltje and Perepelitsky). If a $p$-permutation equivalence exists between two blocks, then one can construct an isotypy between them. I will give examples which show that the reverse construction is not always possible. Along the way I will review some of the fundamental invariants one can attach to a block, such as fusion systems and Kulshammer-Puig classes.
4. Wednesday 1 April 2026 (12:00-1:00 on Zoom) contact Justin Lynd for the zoom link
   Root Systems of Brink-Howlett Groupoids
   Harrison Gimenez
   Notre Dame
   Abstract: In one of their papers, Brink and Howlett introduced a class of groupoids that are built from the data of a Coxeter system. These groupoids, called Brink-Howlett groupoids, are a horizontal categorification of the notion of a Coxeter system. In Brink and Howlett's original paper, they attached a root system to each object of the groupoid, but each such root system was only acted on by a specific subgroup of the associated vertex group. There was no transfer of roots between root systems at distinct objects. In this talk, I will present the first half of an upcoming joint paper with Matthew Dyer on how one can attach structures that behave like root systems to each object of a Brink-Howlett groupoid. These root system-like structures, whose elements we call "roots", allow for morphisms of the groupoid to transfer "roots" to other "roots" at different objects. The collection of these root system-like structures attached to each object of a groupoid is called a signed groupoid-set. I will analyze a particular signed groupoid-set that allows for a generalization of the Tits cone to a specific class of Brink-Howlett groupoids. This generalization of the Tits cone will satisfy a theorem that is analogous to a theorem associated to the Tits cone of a Coxeter system.
5. Wednesday 29 April 2026 (12:00-1:00 on Zoom) contact Justin Lynd for the zoom link
   Carlos Tapp Monfort
   Rutgers