Joe Kraisler (Amherst College)
Dispersive estimates for a Dirac equation with a domain wall

• In this talk we will introduce a one dimensional Dirac equation which interpolates between two phase shifted constant coefficient equations at $x=\pm\infty$. Such equations arise in the study of topologically protected states for one dimensional quantum materials. This Dirac operator depends continuously on a parameter $\tau$ which measures the relative shift between the limiting operators. When $\tau = 0$ the operator has an edge resonance which leads to slow dispersive decay of the solutions, while for $\tau \neq 0$ no such resonance exists and solutions decay more rapidly. We derive a $\tau$ dependent bound which smoothly interpolates between these two different rates of decay.

Presley Kimball (Brown University)
Optimizing RSV Vaccination Policy to Account for Regional Dynamics

• Respiratory Syncytial Virus (RSV) is a common viral pathogen that is the leading cause of lower respiratory tract infection in young infants. As of the Summer of 2024, two immunizations have been approved to help mitigate the risk of severe RSV in infants: a monoclonal antibody infusion for infants under 19 months and a vaccine for pregnant people to pass passive immunity to newborns. However, the effectiveness of these immunizations is suboptimal and only lasts for one RSV season. Since RSV is a seasonal infection, climate has been shown to be an important driver in RSV dynamics, and we have also found evidence that the RSV age distribution depends on urbanization factors. Therefore, we develop an age-structured compartmental model of RSV and calibrate it to county-level, time-series RSV hospitalization data. We utilize this model to evaluate and optimize vaccination strategies to better inform public health policy while accounting for the temporal, spatial, and urbanization factors that affect RSV dynamics.

KEYNOTE: Govind Menon (Brown University)
The geometry of the deep linear network

• The deep linear network is a phenomenological model of deep learning introduced by Arora, Cohen and Hazan. It captures the effect of overparametrization despite being a description of learning for linear functions. We describe an explicit analysis of the DLN using the tools of dynamical systems theory along with some insights for deep learning.

Juliana Londono Alvarez (Brown University)
Attractor-based models for sequences and pattern generation in neural circuits

• Neural circuits in the brain perform a variety of essential functions, including input classification, pattern completion, and the generation of rhythms and oscillations that support functions like breathing and locomotion. There is also substantial evidence that the brain encodes memories and processes information via sequences of neural activity. Traditionally, rhythmic activity and pattern generation have been modeled using coupled oscillators, whereas input classification and pattern completion have been modeled using attractor neural networks. In this talk, I will present models for several different neural functions using threshold-linear networks. Our goal is to develop a unified modeling framework around attractor-based models. The models presented include: a “counter” network that can count the number of external inputs it receives, encoded as a sequence of fixed points; a model for locomotion that encodes five different quadruped gaits as limit cycles; and a model that connects the sequence of fixed points in the counter network with the attractors of the locomotion network to obtain a new network that steps through a sequence of locomotive gaits. I will also introduce a general architecture for layering networks which produces “fusion” attractors by minimizing interference between the attractors of individual layers.

Shrohan Mohapatra (UMass Amherst)
PainleveBacklundCheck: A Sympy-powered Kivy app for the Painlevé property of nonlinear dispersive PDEs and auto-Bäcklund transformations

• In the present work, we revisit the Painlev\’e property for partial differential equations. We consider the PDE variant of the relevant algorithm on the basis of the fundamental work of Weiss, Tabor and Carnevale and explore a number of relevant examples. Subsequently, we present an implementation of the relevant algorithm in an open-source platform in Python and discuss the details of a Sympy-powered Kivy app that enables checking of the property and the derivation of associated auto-B{\”a}ck{u}nd transform when the property is present. Examples of the relevant code and its implementation are also provided, as well as details of its open access for interested potential users.

John Ivanhoe (Boston University)
Global Solutions in a Constrained Potential for Stochastic Reaction-Diffusion Equations