This is a brief introduction to the 2024 Polymath Jr project that will be run by Mikil Foss and Petronela Radu. 

 

Nonlocal systems have been used to predict cracks and damage in dynamic fracture, to denoise in image processing, to provide more accurate models in binary fluids and phase separation, and to study many other physical or social phenomena. The project aims are largely to advance understanding of nonlocal models and to develop mathematical tools for prediction and analysis of the solutions for nonlocal systems. More precisely, we will focus on nonlocal versions of initial value problems that are encountered in ordinary differential equations. The outcomes of this project will answer questions such as:

• Formulate well-posed IVPs for nonlocal models.

• Prove existence and uniqueness theorems of solutions, depending on assumptions for the interaction kernel.

• Study dependence of solutions based on initial data and kernels.

• Produce numerical schemes to obtain approximations to these systems.

 

If time allows, we will consider different types of nonlinear forcing terms, and possibly systems. ​

 

Pre-requisites for the project would be Calculus and Differential Equations. Additional coursework in Analysis, numerical methods, some programming experience could be valuable.