This talk is about a question in arithmetic dynamics. Namely, the question of which quadratic polynomial maps over a fixed finite field yield isodynamic dynamical systems. In this talk, we will explain what we know (which is very little) as well as what we don't know (which is very much). This talk will be accessible to math students.

Distributive lattices appear in many different mathematical contexts. Drawing from familiar examples and assuming no prior knowledge, I will introduce the concept of a lattice and characterize an important class of distributive lattices. Finally, I will state (and prove if there is time) Garrett Birkhoff's representation theorem for finite distributive lattices which provides a beautiful correspondence between the category of finite distributive lattices and the category finite partially ordered sets.

Some optimization problems can be solved more easily by answering equivalent questions in the dual space. This motivates the research and development of criteria to guarantee that the solutions are equivalent (called "strong duality"). This talk will discuss a particular optimization problem of the Fenchel form and the development of a particular set of criteria guaranteeing strong duality. The talk will be accessible to graduate students.

A group is left-orderable if there exists a strict total ordering such that multiplication on the left preserves order. We will consider some basic properties of left-orderable groups (they are torsion free!) along with examples of manifolds whose fundamental groups are or aren't left-orderable; in particular, the circle (S¹), the torus, and Lens spaces. Left-orderability is related to existence of nontrivial homomorphisms of a certain type in the case of fundamental groups of 3-manifolds. We will discuss this result along with other recent developments relating left-orderability to topology.

In 2014, Flynn and Garton bounded the average number of components of the functional graphs of polynomials of fixed degree over a finite field. In many cases, their lower bound matched Kruskal's asymptotic for random functional graphs. Their upper bound was far less satisfactory, as it was far from Kruskal's bound. In our work, we introduce a heuristic for counting the average number of long cycles in such functional graphs. We provide numerical data comparing cycle counts to our heuristic.