Titles and Abstracts

Scott Lewis - One Dimensional Dynamics

Dynamical Systems has many applications to science and engineering. Behind these applications there lies a rich mathematical subject that can be divided into two basic areas which can be roughly described by the terms “discrete”, centered on orbits of iteration of a function, and “continuous”, meaning solutions of ordinary differential equations. In this introduction we will consider the discrete side of dynamics. From iterations of simple functions on an interval we get an astounding amount of interesting areas to investigate. Some topics of interest include fixed and periodic points, families of maps, symbolic dynamics, topological conjugacy, and more. Join with me as we consider some of these ideas in the amazing world of one-dimensional dynamics.

 

Abigail Linscott - Geodesic flow on compact hyperbolic surfaces: density of periodic orbits

With the goal of constructing a compact hyperbolic surface, we will choose an appropriate isometry group on the hyperbolic disc to quotient by. After constructing this surface, we will study the properties of the geodesic flow, focusing specifically on periodic orbits, which we will identify with hyperbolic isometries. We will then see that such periodic orbits form a dense set on the tangent bundle of the surface.

 

Yibo Zhai - Self-joinings and Ratner's property for special flows

The study of self-joinings is a branch of ergodic theory that has many applications to the classification of dynamical systems. In the 1980s, Ratner used self-joinings to classify factors of the horocycle flow. A key idea in the proof is now called Ratner's property. It's natural to ask whether some measure-preserving systems have Ratner's property or not.  In this talk, I will introduce self-joinings, and discuss Ratner's property for special flows.

 

Quinlan Leishman - Intrinsic Stability for classes of Delay Differential Equations

Introducing time delays into a dynamical system often changes the dynamics drastically in ways that are difficult to determine. This has implications for modeling, since time delays arise naturally in many real-world scenarios due to nonzero processing time and separation of components. Since directly modeling and analyzing time delays tends to be difficult, the usual approach is to try to determine properties of the time-delayed system using only properties of the undelayed system. A particularly powerful result in this direction is the idea of intrinsic stability. This criterion ensures that the fixed point will be asymptotically stable for any choice of time delays, so long as a matrix related to the system has spectral radius less than one. We discuss an extension of this method to apply to stable fixed points of continuous-time dynamical systems, and some of its implications.

 

Davi Obata - Stable ergodicity in smooth dynamics

I will present a survey on the stable ergodicity problem in smooth dynamics. I will present the state of the art and some open questions.