MAA NCS Spring 2025 meeting

Invited Speaker: Opel Jones (MAA – NAM Speaker)

Talk Title: Pattern Avoidance in Restricted Permutations

Talk Abstract:  In 1974 Dumont found two types of permutations are counted by the same sequence. The first type is a permutation in which each even entry is followed by a smaller entry, and each odd entry is followed by a larger entry, or ends the permutation. The second type is a permutation wherein if an entry is a deficiency, it must be even, and if an entry is an exceedance or a fixed point, it must be odd. These are now known as Dumont permutations of the first and second kinds. There are two other types of permutations which are also counted by the same sequence, known as Dumont permutations of the third and fourth kinds. In this talk we will discuss several enumerations of restricted Dumont permutations, that is Dumont permutations avoiding certain patterns. We will also briefly discuss their proofs which involve methods using induction, block decomposition, Dyck paths, and generating functions. We will conclude with a conjecture that the patterns 2143 and 3421 are indeed Wilf-equivalent on Dumont permutations of the first kind. 

Invited Speaker: Claudio Gómez-Gonzáles (MAA NCS Section NExT Speaker, Carleton College)

Talk Title: How hard could it be? An invitation to resolvent degree

Talk Abstract:  Solving algebraic equations are among the oldest problems in mathematics. In this talk, we offer a concrete, visual, and historical introduction to resolvent degree (RD), an invariant that aspires to quantify just how hard these problems are. This overview makes contact with the origins of topology, miracles of classical algebraic geometry, Klein’s “hypergalois” program, and centuries-old exploits in reducing coefficients, which dare us to push beyond the solvable/unsolvable dichotomy. We will build towards the notion of versality central to Klein’s vision and, time permitting, allude to a general framework implemented in joint work with Alexander Sutherland and Jesse Wolfson that addresses resolvent questions via classical invariant theory. Throughout the talk, we will reflect on the past and future of resolvent problems, the value in grappling with hard questions even if we might not solve them, and what we do and do not know about RD. No background knowledge will be expected or required beyond calculus and linear algebra.