Many interesting phenomena are modelled by dynamical systems for which the state of the system evolves according to some deterministic rule — nevertheless, these phenomena often appear to behave randomly over long enough time scales. (The canonical example is the weather.) I will explain where this randomness comes from, how the tools of dynamical systems allow us to make meaningful predictions for such systems, and what some of the challenges are in extending the theory more broadly.

The presence of instabilities in models of physical processes often makes it practically impossible to simulate individual orbits for long periods of time. It is therefore natural to view dynamical systems probabilistically, an idea that can be traced back to Boltzmann and the development of statistical mechanics. Although dynamical systems defined by flows or maps are deterministic, they can nevertheless give rise to time series that look as if they were generated by stochastic processes with strong independence properties. In this talk we will introduce the probabilistic viewpoint and discuss one particular manifestation: Borel-Cantelli lemmas for dynamical systems.

The talk is about how randomness can arise in deterministic dynamics and how we can measure and quantify randomness. We indicate how in systems which are statistically indistinguishable from fair coin tossing we can nonetheless see structure.

Some systems that look chaotic can be described by a relatively simple symbolic dynamics (trajectories in a finite graph, with specified transition probabilities). These are the "uniformly hyperbolic" systems, considered in the 1960's by Dmitri Anosov and Stephen Smale. However, many chaotic systems exhibit only "non-uniformly hyperbolic" behavior. A model to describe these was introduced in the late 1990's by Lai-Sang Young. We will sketch these models, and discuss consequences one can derive from them.

Studying a simple ODE, Lorenz discovered in 1963 an object that is called today a strange attractor: nearby points are attracted to a set of fractal dimension, and move around this set chaotically.

Understanding this attractor was one of the 18 problems for the twenty-first century proposed by Field medalist Steven Smale.

Namely: Is the dynamics of the ordinary differential equations of Lorenz that of the geometric Lorenz attractor of Williams, Guckenheimer, and Yorke? Tucker answered this question in the affirmative in 2002. His technical proof makes use of a combination of normal form theory and validated interval arithmetic.

The goal of this talk is to explain what it means to understand systems that look chaotic. Link to Andrew Török's website for more information.

Attractors are topological structures which are useful in the study of dynamical systems. This talk will first sketch the development of expanding attractors from 1) the classical solenoid attractor of S. Smale to 2) generalized solenoid attractors by R. Williams to 3) the introduction of symmetry into Williams' structures by M. Field, I. Melbourne, and M. Nicol. The symmetry is by way of a finite group acting on 3-space. In the case of Field, Melbourne and Nicol, the group acts freely on the expanding attractor. I will then discuss recent developments to construct expanding attractors on which the finite group action is not free.

We briefly describe the mathematical theory of Brownian motion and give applications to PDES, chaotic dynamical systems and mathematical finance.

We will describe how seemingly unrelated fields, knot theory (low dimensional topology) and von Neumann algebras (a topic of Functional Analysis), were connected when Vaughan Jones discovered a new knot invariant. [In part for this work, Jones received the Fields Medal in 1990. A link about V. Jones is at here.]

It is now well known that deterministic dynamical systems can behave "chaotically". But what does this actually mean and how can we measure it? In this talk we explore what is meant by terms such as "random" and show how simple deterministic systems can have statistics indistinguishable from coin-tossing. The talk will include a visual component where we will show some of the intricate structure that can be embedded within chaotic systems.