Wolfgang and Luise Kappe Alumni Speaker
Title: From Groups to Semigroups
Abstract: Semigroups have connections to many areas of math, but are often understudied directly. I have found many interesting questions by taking questions for groups and asking them about semigroups. I will introduce semigroups (a set with an associative operation) and outline two such questions in this talk.
The covering numbers of a group G is the minimum number of proper subgroups whose union is G. Covering numbers of groups have been studied since the early 1900s and have many fascinating results. Luise-Charlotte Kappe and I investigated covering numbers of semigroups in joint work. Our results highlighted some surprising differences between groups and semigroups, specifically that the covering number of most semigroups is 2, whereas no group has covering number 2.
A group is 3/2-generated if every non-identity element is contained in a two-element generating set. It was recently shown that a finite group G is 3/2-generated if and only if G only has cyclic quotients. For semigroups (that may or may not have an identity), I am proposing multiple definitions of 3/2-generation, which leads to multiple classifications. I will detail the differences in the definitions and resulting semigroups.
Title: Representation Asymptotics in the Homology of Pure Graph Braid Groups
Abstract: The homology groups of the unordered configuration spaces of a graph form a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. Wishing to lift this result to an analogue of representation stability, one encounters the unavoidable fact that the corresponding algebraic structure at the level of ordered configuration spaces is non-Noetherian. Thus, finite generation results are likely to be neither useful nor within easy reach. Drawing on ideas from twisted algebra and factorization homology, we circumvent this obstacle to give a complete asymptotic calculation of the Betti numbers of pure graph braid groups over any field and, in characteristic zero, of the multiplicities of many irreducible representations of the symmetric groups. This talk represents joint work with Hainaut and Wawrykow, building on joint work with An and Drummond-Cole.
Title: Discrete Geometry Beyond Matroid
Abstract: Many classic results in discrete geometry have "colorful" generalizations. These include Helly's theorem, Caratheodory's theorem, the KKM theorem, the Borsuk–Ulam theorem among others. A new level of sophistication was found in the 2000's that replaced the coloring information with matroids. The purpose of this talk is to introduce selection structures, which allow us to generalize these results beyond matroids, and unify them under a common combinatorial umbrella. Selection structures allow us to specify the coloring conditions with non-matroidal structures, such as matching complexes and chessboard complexes.