For our Friday seminar this week, we have a guest speaker Dr. Javier Cueto. He is a postdoc currently at University of Nebraska-Lincoln. He studies nonlocal differential operator theory and is going to be giving us a talk on the basics of his research. This talk should prepare you for his afternoon talk at the Applied Math Seminar. I’ve attached below his abstract. See you guys at 11am on Friday for Javi’s talk!

Title: A Whole New (Nonlocal) World.

Abstract: Well, as we all know, the use of mathematics has been quite effective in describing natural phenomena. Popularly through the use of (partial) differential equations to model systems in physics, biology or economics for example. The function that describes the system is ‘hidden’ in these differential equations which stablish a relation between a function and its derivatives (related to how the function changes). But if we stretch something, for example this wooden beam…. (crack!) a fracture appears!  Thus, sometimes singularity phenomena may arise, and that implies functions with discontinuities which do not fit very well in these classical models.

There is something that can tackle this. A new fantastic point of view! What is nonlocal?  

We will try to understand that. Basically, we will consider a relaxed notion of gradient, typically made of an integral of a difference quotient. As a consequence, less regularity is needed and long range interactions can be taken into account (nonlocal: points at a finite distance may exert an interaction upon each other).

This means we have new horizons to pursue! In particular, we will need to obtain several tools, to the extent possible, similar to those of the classical case, so that we can study these new models. Fortunately, we have already been able to obtain quite a few, where a key ingredient has been a nonlocal version of the fundamental theorem of calculus.