## Javier Cueto García, University of Nebraska

A Whole New (Nonlocal) World

4:00 pm–4:50 pm

Avery Hall
Room: 115

1144 T St

Lincoln NE 68508

Lincoln NE 68508

Additional Info: AVH

Contact:

Mikil Foss

Can you imagine if we could have a ¾ derivative? -

- What? No, why would I do that? It’s Friday!

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.

- What? No, why would I do that? It’s Friday!

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.