Title: On the faces of the Kunz cone and the numerical semigroups within them
| Speaker: | Harper Niergarth |
| Affiliation: | University of Waterloo |
| Location: | MC 5479 |
There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.
Abstract: A numerical semigroup is a subset of the natural numbers that is closed under addition, contains 0, and has finite complement. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a polyhedral cone $C_m \subset \mathbb{R}^{m-1}$ called the Kunz cone. Moreover, numerical semigroups corresponding to points on the same face $F$ of $C_m$ are known to share many properties, such as the number of minimal generators. But not all faces of the Kunz cone contain integer points corresponding to numerical semigroups. In this talk, we will classify all the faces that do contain such points. Additionally, we will present sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $C_m$ containing the point corresponding to $S$.
This is joint work with Levi Borevitz, Tara Gomes, Jiajie Ma, Christopher O'Neill, Daniel Pocklington, Rosa Stolk, Jessica Wang, and Shuhang Xue.