Joshua Lee Padgett's Research


1. Introduction and Overview

My research interests lie in the areas of numerical analysis, operator theory, differential equations, and nonlinear analysis. Within these fields, I also have interests in approximation methods, spectral theory, fractional and stochastic differential operators, Lie group methods, and Hopf algebras. However, the majority of my contributions have been in the fields of operator theory and numerical analysis.

I am currently engaged in several areas of research. In order to help any interested readers, I have split this page into sections which contain my projects falling into the same general category. These categories are constantly expanding, so I make no claim that these descriptions are in any way complete. Readers may contact me via email if they would like more details regarding one (or all) of the listed areas. Each section may be expanded by clicking on the title of said section. For the sake of clarity, within each section there is a tabbed “Results” addendum. This tabbed section briefly outlines my personal contributions to each subject.

2. Operator Splitting and Approximation

Operator theory, operator splitting, and approximation of operators have been my primary research areas, throughout my academic career. My endeavors have considered both deep theoretical issues and computational issues. Using techniques from each of these fields, I analyze various numerical methods via appropriate abstract techniques. Operator splitting methods enjoy numerous useful properties (such as stability, explicitness, efficiency, sub-problems may often be solved exactly, preserves geometric invariants, etc) and have been well-studied in numerous classical settings. However, there are still a wealth of open problems in regards to the abstract analysis and applications of these methods to non-classical settings (stochastic, fractional, optimization problems, and data analysis). Thus, many of my goals center around the exciting prospect of extending the notions of operator splitting to these novel settings.

2.1.

2.2.

2.3.

3. Fractional Differential Equations

Fractional differential equations are a newly blossoming field of interest that have applications in numerous scientific fields. My primary interest in such problems revolves around the study of the highly non-local solution operators which arise when solving equations involving fractional derivatives. This direction yields numerous interesting mathematical problems which have important open questions that deserve both theoretical and computational attention.

I have considered two distinct problems involving fractional operators. I briefly outline the projects below.

3.1.

3.2.

3.3.

4. Spectral Theory for Anderson-type Hamiltonians

Some of, what I believe to be, my most interesting work has been done in conjunction with the phenomenal analyst Constanze Liaw, from the University of Deleware, and physicist Eva Kostadinova, Lorin Matthews, and Truell Hyde, from Baylor University. Conni pioneered a novel approach for studying spectral properties of operators, which allows one to mathematically study the Anderson localization phenomenon independent of artificially imposed boundary conditions or contruction of eigenfunctions. Continuing in this direction, we have now extended this approach to consider fractional operator models, as well. Most importantly, this work has demonstrated results that better match observed experimental data due to the chosen approach and our improved model.

4.1.

4.2.

4.3.

5. Algebraic Theory of Numerical Integration

This section is still under construction...Feel free to contact me with questions...