Geordie Richards

Education & Employment Background

Prof. Geordie Richards received his PhD in Mathematics from the University of Toronto in 2012. From 2012-2022, Richards held academic positions at the Institute for Mathematics and its Applications at the University of Minnesota (Postdoctoral Research Fellow, 2012-2013), the Department of Mathematics at the University of Rochester (Visiting Assistant Professor, 2013-2016), the Department of Mechanical & Aerospace Engineering at Utah State University (Assistant Professor, 2016-2021), and the Department of Mathematical & Computational Sciences at the University of Toronto Mississauga (Teaching Stream Assistant Professor, 2021-2022). He joined the Department of Mathematics & Statistics at the University of Guelph in 2022.

Research Themes

General Description

Richards’ research focuses on the analysis of nonlinear partial differential equations (PDEs) coming from physics and engineering applications, specifically equations modeling fluids and other dispersive (i.e., wave-like) phenomena. He is interested in new directions in the analysis of PDEs which incorporate ideas from probability theory and the theory of dynamical systems.

Current Research Themes

1. Random data for dispersive PDEs – In studying the well-posedness of a dispersive PDE, one often encounters a critical regularity, below which the PDE is ill-posed. That is, below the critical regularity, there is no hope to establish well-posedness for all initial data. However, in some cases, one can equip the initial data function space with a probability measure, and prove well-posedness, below the critical threshold, almost surely with respect to this measure. Richards made early contributions to this type of random data Cauchy theory for dispersive PDEs, and his current work expands on the breadth of the underlying techniques.
2. Ergodic theory of stochastic PDEs – Ergodic hypotheses are a foundational assumption of statistical theories of turbulence, but the modern theory of deterministic PDEs, whether for fluids or other physical phenomena, does not appear capable of proving ergodicity directly from the equations of motion. Strikingly, however, it is possible to prove uniqueness and ergodicity of the invariant measure for some stochastic PDEs. Richards’ work contributes to this emergent theory of ergodicity for stochastic PDEs modeling fluids and dispersive phenomena.
3. Singular stochastic dispersive PDEs – In the last 10-15 years, there has been tremendous progress in the well-posedness theory of singular parabolic stochastic PDEs, including the theory of regularity structures (for which Prof. M. Hairer won a Fields medal in 2014) and the theory of paracontrolled distributions (for which Prof. M. Gubinelli spoke at the ICM in 2018). Richards’ current work seeks to expand these important new approaches toward the analysis of dispersive stochastic PDEs, equations which are poorly understood by comparison.
4. Uncertainty quantification in engineering applications – Richards collaborates with scientists and engineers on applied research projects that seek to quantify the interaction of nonlinear dynamics with randomness, in topics ranging from measurement uncertainty in turbulent flows to sensitivity analysis of models for small modular reactor designs.

Highlights

• Nuclear Regulatory Commission (NRC) Faculty Development Grant, $431,628 USD, 2019-2022 (Co-PI with PI Barton Smith and Co-PI Hailei Wang).
• Mechanical & Aerospace Engineering Teacher of the Year, Utah State University, 2019.
• National Science Foundation (NSF) Conference Grant, $17,875 USD, 2017 (Co-PI with PI Jared Whitehead, and Co-PIs Mark Allen and Blake Barker).
• Research Membership at the Simons Laufer Mathematical Sciences Institute at UC Berkeley (SLMath, formerly MSRI) for thematic program on “New Challenges in PDE: Deterministic Dynamics and Randomness in High and Infinite Dimensional Systems”, September-October 2015.
• Natural Sciences and Engineering Research Council (NSERC) Graduate Scholarship, 2006.