Research

When it comes to mathematics, I am  generally interested in any problems that have their origin in Calculus, such as function spaces, derivative formulas, and anything to do with integration (which is my absolute favorite math topic). I also have a moderate interest in the fields of Algebra and Partial Differential Equations; the former for the beautiful theory  and categorization it provides and the latter as it contains applications for my research.

My main research focus is in extension theorems for functions in particular function spaces defined on a domain. In the case of my dissertation, the function space is matrix weighted Sobolev space and the domain has Lipschitz boundary. An example of this can be understood with the following visual:

If one is given an “n-dimensional snapshot” of a piece of landscape, how does one “fill-in” the rest of the landscape in a way that is both consistent, and whose size is less than (or equal to), the original piece? In this way we “extend” the landscape in all directions.

finite landscape
How could we fill in the rest of the above landscape in a way that looks natural?

The answer depends on what is meant by “size” as well as with which characteristics of the landscape we want to duplicate on the outside. For example, Sobolev space defines size with an integral expression, which is roughly the volume of the landscape. The only properties we want preserved are that the resulting landscape matches up perfectly at the edges of the original snapshot while also keeping the same level of smoothness of that original piece.

Some questions to think about:

  • What would the extension look like if the snapshot has size greater than or equal to the filled in landscape?
  • Why might you want such a size restriction in the first place?
  • How does one guarantee that the edges paste together in a smooth way?

 

My dissertation is currently in print (and I get royalties from it): Get in on Amazon

You can get a free copy here: Download it from Trace