Scientific

We develop a general framework for identifying phase reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett 123, 164101 (2019)] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use diffusively coupled complex Ginzburg-Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for non-weak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories can not.

Speaker Biography

Youngmin Park, Ph.D., is currently a PIMS Postdoc at the University of Manitoba under the supervision of Prof. Stéphanie Portet. He received his PhD in Mathematics from the University of Pittsburgh in 2018, where he applied dynamical systems methods to problems in neuroscience. His first postdoc involved auditory neuroscience research at the University of Pennsylvania in the Department of Otorhinolaryngology, before moving on to his next postdoc researching molecular motor dynamics in the Department of Mathematics at Brandeis University. He is now at Manitoba, continuing to apply dynamical systems methods to biological questions related to molecular motor transport and neural oscillators.

Large systems of interacting particles (or agents) are widely used to investigate self-organization and collective behavior. They frequently appear in modeling phenomena such as biological swarms, crowd dynamics, self-assembly of nanoparticles and opinion formation. Similar particle models are also used in metaheuristics, which provide empirically robust solutions to tackle hard optimization problems with fast algorithms. In this talk I will start with introducing some generic particle models and their underlying mean-field equations. Then we will focus on a specific particle model that belongs to the class of Consensus-Based Optimization (CBO) methods, and we show that it is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning.

Speaker Biography

Hui Huang, Ph.D., is currently a PIMS Postdoc at the University of Calgary under the supervision of Prof. Jinniao Qiu. Before moving to Calgary, he worked as a postdoctoral researcher in the Chair for Applied Numerical Analysis at the Technical University of Munich, Germany. Prior to being at TUM he was an Alan Mekler Postdoctoral Fellow in the Department of Mathematics at Simon Fraser University. In 2017, he received his PhD in Mathematics from Tsinghua University. His doctoral dissertation was conducted in consultations with Prof. Jian¬-Guo Liu from Duke University, where he studied as a joint PhD student from 2014 to 2016. His research has been focused on complex dynamical systems and their related kinetic equations.

Read more about Hui Huang on the PIMS Medium blog.

One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. A new approach to data assimilation known as the Azouani-Olson-Titi algorithm (AOT) introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, we present our tests on the robustness, improvements, and implementation of the AOT algorithm, as well as generate new ideas based off of these investigations. First, we discuss the application of the AOT algorithm to the 2D NSE with an incorrect parameter and prove it still converges to the correct solution up to an error determined by the error in the parameters. This led to the development of a simple parameter recovery algorithm, whose convergence we recently proved in the setting of the Lorenz equations. The implementation of this algorithm led us to provide rigorous proofs that solutions to the corresponding sensitivity equations are in fact the Fréchet derivative of the solutions to the original equations. Next, we present a proof of the convergence of a nonlinear version of the AOT algorithm in the setting of the 2D NSE, where for a portion of time the convergence rate is proven to be double exponential. Finally, we implement the AOT algorithm in the large scale Model for Prediction Across Scales - Ocean model, a real-world climate model, and investigate the effectiveness of the AOT algorithm in recovering subgrid scale properties.

Speaker Biography

Elizabeth Carlson, is a homeschooler turned math PhD! She grew up in Helena, MT, USA, where she also graduated from Carroll College with a Bachelor's in mathematics and minor in physics. She became interested in fluid dynamics as an undergraduate, and followed this interest through her graduate work at the University of Nebraska - Lincoln in Lincoln, NE, USA, where she just earned my PhD in May 2021. Her research focus is in fluid dynamics, focusing on the well-posedness of systems of partial differential equations and numerical computations and analysis in fluid dynamics. In her free time, she enjoys hiking, playing piano, reading, and martial arts.

Read more about Elizabeth Carlson on our PIMS Medium blog here.

Richard Guy's book "Unsolved Problems in Number Theory" was one of the first mathematical books I owned. I will discuss a selection of my favorite problems from the book, together with some of the progress that has been made on them in the 30 years since I acquired my copy.

Speaker Biography

Ben Green was born and grew up in Bristol, England. He was educated at Trinity College, Cambridge and has been the Waynflete Professor of Pure Mathematics at Oxford since 2013.

About the Series

The Richard & Louise Guy Lecture Series, presented from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world, celebrates the joy of discovery and wonder in mathematics for everyone.

This series of lectures covers a brief introduction into some algebro-geometric techniques used in the construction of BPS algebras. The starting point of our construction is a physical picture of D0-branes bound to D-branes of higher dimension. Using methods of the derived category of coherent sheaves, we are going to derive a framed quiver with potential describing supersymmetric quantum mechanics capturing the low-energy behavior of such D0-branes. For a large class of quivers, we are going to identify the space of BPS states with different melted-crystal configurations. Finally, by employing correspondences, we are going to construct an action of a BPS algebra known as the affine Yangian on the space of BPS states. The action of the affine Yangian factors through the action of various vertex operator algebras, Cherednik algebras, and more. This construction leads to an enormously rich interplay between physics, geometry and representation theory.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4