Algebra and Number Theory Seminars

$Tiling$

Group theory is the formal mathematical study of symmetry. Groups are among the foundational objects composing abstract algebra, yet they also pervade nearly every discipline in pure mathematics as well as many areas of science and engineering. One striking result of group theory shows that there are exactly 17 different types of planar symmetry. This image illustrates one of these types of symmetry in a section of tilework at the Alhambra Palace in Granada, Spain. This particular symmetry is characterized by 3-fold rotational symmetry with no reflections (Photo credit The_Alhambra_and_The_Alcazar).

The Algebra and Number Theory Seminar is structured to include talks on a broad range of mathematical areas that are of interest to algebraists and number theorists, including analytic and algebraic number theory, algebra, combinatorics, algebraic and arithmetic geometry, cryptography, representation theory, and more. Talks are given by a variety of local, national, and international speakers in number theory and related areas.

See below for upcoming seminars or access the seminar archive.

Organizers

Mary Flahive, Clayton Petsche, Thomas Schmidt and Holly Swisher

Timing

We traditionally meet every Tuesday at 11:00 a.m.

Shuffle-compatibility for statistics on words, parking functions, and set partitions

Tuesday, April 21, 2026 – 11:00 am

Zoom

Speaker: Spencer Daugherty

Abstract: Gessel and Zhuang introduced the concept of shuffle-compatibility of statistics on permutations to describe statistics whose multiset of values on the shuffles of two disjoint permutations is determined exactly by the size and statistic values of the two permutations being shuffled. Shuffle-compatibility implies the existence of an algebraic structure on the equivalence classes induced by the statistic. For example, the descent set statistic is shuffle-compatible, and the algebra on the equivalence classes it induces is isomorphic to the Hopf algebra of quasisymmetric functions. In this talk, we generalize shuffle-compatibility to objects such as words, parking functions, and set partitions using the Hopf monoids associated with these objects. We investigate statistics on these objects and then define various algebraic structures on the equivalence classes formed by these statistics that are, in many cases, quotients of (or isomorphic to) well-known combinatorial Hopf… Read more.