Combinatorics/Algebra Seminar
Spring 2023 Archive:

Wednesday, May 3

Justin Stevenson: Covering finite groups with quasigroups (preliminary exam).

Monday, April 24

Mitchell Ashburn: Vector-matrix algebras, and heterogeneous algebras over symmetric monoidal categories (preliminary exam).

Wednesday, April 19

Connor Depies: Octonions as Clifford-like algebras (preliminary exam).

April 10

Jonathan Smith: The quatedral loop.

Abstract: The quatedral loop, of order 8, interpolates between the quaternion group and dihedral group of order 8. Although the quatedral loop has lurked on the GAP loop package for several years, as NilpotentLoop(8,116), its independent existence and surprising properties have only recently become apparent. A combinatorial version of the supersymmetry technique ia used in the identification and study of the loop.

March 27, April 3

Cody Fleming (Mechanical Engineering, ISU):
Categorical semantics of cyber-physical systems theory. [8MB]

Abstract: Cyber-physical systems (CPS) require the construction and management of various models to assure their correct, safe, and secure operation. I will first describe CPS applications and the various formalisms present in current CPS research. Unfortunately, to date the different model views of cyber-physical systems are disparate and largely related informally, which raises issues with the degree of formal consistency between those various models of requirements, system behavior, and system architecture. I will present a category-theoretic framework that makes different types of composition explicit in the modeling and analysis of cyber-physical systems, which assist in verifying the system as a whole. In particular, I will focus on modeling system behavior via algebras on the monoidal category of wiring diagrams, system requirements (e.g. on safety) again as an algebra on certain objects in this monoidal category, and the CPS design process using the notion of a slice category.

March 20: no seminar.

March 13: Spring break.

February 27, March 6

Connor Depies: Generalizing the octonions with trilinear forms.

Abstract: The quaternions can be created using a symmetric bilinear form on the real vector space spanned by i and j. We then generalize by changing the bilinear form to get other structures; e.g., the split quaternions and the exterior algebra of 4 dimensions. Such generalizations give us Clifford algebras. We generalize the octonions in a similar manner, but non-associativity requires that in addition to a symmetric bilinear form, we need an antisymmetric trilinear form. We study these generalizations using a vector space of three dimensions, obtaining eight-dimensional algebras, and present some particular examples.

February 20: no seminar.

February 6, 13

Justin Stevenson: An introduction to covering quasigroups.

Abstract: In the first talk I will introduce the idea of a covering quasigroup, and present a proof of their existence which is due to Hilton and Wojciechowski. We will show how these covering quasigroups can be used to get an analogue of the character group that works in the non-abelian group case. In the second talk I will discuss the metacyclic groups, their characters, and what their covering quasigroups look like.

January 23, 30

Jonathan Smith: Catalan numbers, convex polytopes, and Fermat curves.

Abstract: Catalan numbers count the number of possible ways to parse a non-associative product. In these talks (reporting joint work with A. Romanowska, A. Zamojska-Dzienio, and L. Long), we discuss two of the many places where Catalan numbers show up: in the decomposition of convex polytopes, and in the number-theoretical study of Fermat curves.

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