7 PM ET "A Journey in Combinatorics with the Riordan Group"

A Riordan array is an infinite lower triangular array of numbers that is determined by a pair of generating functions meeting certain conditions. Riordan arrays can be effectively used to study many types of combinatorial problems, including the enumeration of lattice paths, plane trees and partitions. Moreover, under the right conditions Riordan arrays form a group, creating an algebraic structure out of which new combinatorial insights can be drawn. This talk will explore how the Riordan group can be used to discover interesting combinatorial identities and will also relate some well-known combinatorial problems to algebraic structure in the Riordan group.

7 PM ET "Arithmetic and Geometric Means from Jellyfish Swarms of Curves"

Classical work of Euler and Gauss on "Arithmetic and Geometric Means” (AGM) produces wonderful rapidly convergent sequences with common limit. Euler’s famous formula for pi is an extraordinary example. Here the speaker will introduce the AGM for finite fields, where finite directed graphs are spawned in lieu of infinite sequences. The collection of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e., tentacles connected to a bell head). These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves, and this interpretation gives a striking description of Gauss's class numbers as counts of spots on these jellyfish.

7 PM ET "Every Paper Tells a Story: Mathematics at the Monthly"

Over its 128-year history, the American Mathematical Monthly has not only featured a wide array of mathematics on its pages but also a host of other insights related to the discipline. From Nobel prize-winning ideas to careers inspired by a local drugstore to mathematical menus created by students, this talk showcases the riches of the Monthly and what they teach us about the profession. This talk also includes tips for publishing in the journal today.