Probability and Dynamical Systems

David Wood

  •  28 Apr 2025 - 2 May 2025
     8:00 am - 5:00 pm

Organisers:
Sakshi Jain (Monash University)
Tim Garoni (Monash University)
Luchezar Stoyanov (University of Western Australia)
Dalia Terhesiu (Leiden University, Netherlands)

Program Description:

Probability and dynamical systems are two fundamental areas of mathematics that intersect in the study of systems evolving over time under uncertainty. The combination of probability and dynamical systems plays a critical role in analysing long-term behaviour, stability, and optimal control. Probabilistic methods allow researchers to study phenomena such as ergodicity, where time-averaged behaviour aligns with space-averaged behaviour, and to explore invariant measures describing the statistical properties of the system’s evolution.
This program aims at bringing together experts, students and early-career-researchers working in the areas of probability and dynamical systems from across the globe. The theme of the program is to focus on applications of probability in the study of dynamical systems through mini-courses, and brainstorming sessions. In the spirit of MATRIX and this program, the schedule is designed to facilitate collaborations among participants. This program will be of remarkable benefit to the community of mathematicians working in the areas of probability and dynamical systems.
Program Structure:

Mini Course by Caroline Wormell
Title: A tasting plate of Koopman theory
Abstract: Nonlinear dynamical systems, even with complex behaviour, can be encapsulated in a linear object known as a Koopman operator. The spectral information contained in this operator can be used to make long-term forecasts of the system’s behaviour, and identify emergent patterns in the dynamics.  In this minicourse, I will introduce some ideas common in the study of Koopman operators, and give a sense of their diverse mathematical underpinnings.
Lecture 1: Koopman operators, and the L^2 theory
Lecture 2: Extended Dynamical Mode Decomposition
Lecture 3: Dynamical Mode Decomposition with delay variables

Mini Course by Gary Froyland
Click here for the Lecture Notes
Title: Finite-time coherent dynamics and applications
Lecture 1: Diffusive transfer operators and finite-time mixing
Lecture 2: Dynamic isoperimetry, the dynamic Laplacian, and finite-time coherent sets
Lecture 3: An inflated dynamic Laplacian and the birth/death of coherent dynamics
Overview of lectures:

  1. I could start with an overview of transfer operator cocycles…which might already be covered by Dmitry’s 1st lecture, and at a high level relate them to mixing in time-dependent dynamics over an infinite time horizon.  Then move on to mixing in finite-time dynamics.  The lecture title could be something like “Diffusive transfer operators and finite-time mixing”.
  2. I could then move on to talking about advective (non-diffusive) mixing, and how finite-time coherent sets can be posed as a dynamic isoperimetric problem (which I would introduce). I would then introduce the dynamic Laplacian and connect it with the zero-noise limit of the diffusive transfer operators in the previous lecture.  And prove some general theory about the dynamic Laplacian spectrum and consequences for the dynamic isoperimetric problems. A lecture title could be “Dynamic isoperimetry, the dynamic Laplacian, and finite-time coherent sets”.
  3. In the third lecture I could talk about some recent work with Peter Koltai, namely a kind of time-expanded “inflated” dynamic Laplacian, which aims to characterise the appearance and disappearance of finite-time coherent sets.  A lecture title could be “An inflated dynamic Laplacian and the birth/death of coherent dynamics”.

Mini Course by Kasun Fernando:
Title: Transfer operator method in dynamical systems
Abstract: This minicourse will introduce the use of transfer operator method to establish limit theorems, including the Central Limit Theorem along with its higher-order corrections for exponentially mixing dynamical systems.
Lecture 1: Dynamical systems and the associated transfer operator
Lecture 2: Spectral gap and perturbation theory of transfer operators
Lecture 3: Limit Theorems for dynamical systems

Participant List:
Gary Froyland (University of New South Wales)
Natalia McAlister (Monash University)
Warwick Tucker (Monash University)
Caroline Wormell (University of Sydney)
Polina Vytnova (University of Surrey)
Andy Hammerlindl (Monash University)
Marisa Cantarino (Monash University)
Axel Péneau (University of Tours, Institut Denis Poisson)
Dalia Terhesiu (Leiden University)
Sakshi Jain (Monash University)
Cecilia Gonzalez-Tokman (University of Queensland)
Jack Mills (Monash University)
Maxence Phalempin (University of New South Wales)
Renee Oldfield (University of Queensland)
Josh Peters (University of Queensland)
Ziwen Zhong (Monash University)
Andrea Collevecchio (Monash University)
Kasun Fernando Akurugodage (Brunel University of London)
Miranda Xu (Monash University)
Bryan Bajar (Macquarie University)

Registration:

  • Registration is now closed
  • Arrival date: 27 April 2025

ASSOCIATED EVENTS
MATRIX Wine and Cheese Afternoon 29 April 2025
On the first Tuesday of each program, MATRIX provides a pre-dinner wine and cheese afternoon. Produce is locally-sourced to showcase delicacies from the region.