Drs. Kelly Yancey and Matt Yancey of the Center for Computing Sciences will be speaking at 2pm and at 3pm in BAB 130.
Title: A Tour of Rigidity in Ergodic Theory
Speaker: Kelly B. Yancey
Abstract: A dynamical system can be thought of as a probability space equipped with a self map that is invariant with respect to the measure. In the field of ergodic theory one of the things that we study is the long term behavior of such dynamical systems. In this talk I will begin with some basic constructions in ergodic theory and explain how the study began. I will then move on to describe a generic class of maps called rank-one transformations and a method to construct them. We will discuss the rigidity property of dynamical systems and explore the partial rigidity of Chacon's transformation, a classic map that was the first example of a map that is weakly mixing but not strongly mixing.
With these examples under our belt, I will describe some new results by Jon Fickenscher and myself on characterizing rank-one rigid maps. For rank-one systems we will examine partial rigidity and bound the partial rigidity constant away from one for a subclass of transformations that are canonically bounded (a term coined by Gao and Hill). We will also discuss rigidity in the unbounded case.
Title: Vertex Partitions into an Independent Set and a Forest with Each Component Small
Speaker: Matthew P. Yancey
Abstract: The maximum average degree of a graph $G$, denoted $mad(G)$, is the maximum across all subgraphs of twice the number of edges divided by the number of vertices. This parameter is a special case of graph sparsity, which is connected to graph topology, rigidity theory, and has computational applications. At the Barbados Graph Theory Workshop in 2019, Hendrey, Norin, and Wood asked for the maximum function $g(a, b)$ such that any graph $G$ with $mad(G) < g(a,b)$ can have its vertex set partitioned into sets $A,B$ such $mad(G[A]) < a$ and $mad(G[B]) < b$. In joint work with Dan Cranston, we answer the question when $a=1$ and $b < 2$. The only known values previously were $g(1,1)$, $g(1,4/3)$, and $g(1, 2)$.
Our argument frames the problem as a task in graph coloring, where we apply the "Potential Method." The Potential Method is an enhancement to the technique of "Discharging," which has been used to solve many problems including the Four Color Theorem. We will give an introduction to this style of proof.