Joshua Lee Padgett

I am currently an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas, an Affiliated Faculty member at the Center for Astrophysics, Space Physics, and Engineering Research, and a Research Partner/Honorary Adjunct Faculty in the Department of Mathematics and Statistics at Texas Tech University. I received my Ph.D. in Mathematics from Baylor University in August 2017 under the advisement of Qin "Tim" Sheng and completed my postdoctoral training in the Department of Mathematics and Statistics at Texas Tech University. Before that I received my B.S. in Mathematics from Gardner-Webb University, where I also was a member of the Track and Field team (I competed in the javelin and hammer throw). My undergraduate studies also included cellular biology, which led to several semesters of research into various aspects of cancer metabolism (in particular, the Warburg effect).

Here is a copy of my most recent Curriculum Vitae [updated January 2022].

Joshua Lee Padgett's Google Scholar profile.

Joshua Lee Padgett's ResearchGate profile.

A link with the information regarding the XVIII Red Raider Minisymposium I am co-organizing, may be found here.

A link with the information regarding the JMM Special Session I co-organized in 2020, may be found here.

With the support of Erica Graham, Candice Price, and Shelby Wilson, an amazing (former) colleague of mine, Raegan Higgins, maintains the website Mathematically Gifted and Black (which can be found by following the link). Please visit this webpage in order to learn more about the issue that black scholars face in academia and mathematics. The page also contains details regarding how one can support and nominate black scholars.

Note: I am currently looking for graduate students. Interested students should email me or fill out the form here. While I am interested in a wide range of topics (and, in general, am willing to learn about others), potential research students should take a look at my research descriptions below or my recent publications to have a better idea of the topics I am likely to consider or propose.

Recent Updates

Below you can find some (what I believe to be) important recent updates regarding myself or my research.

NSF-1903450

Onset of Turbulence in Dusty Plasma Liquids

I am co-PI on an NSF collaborative research grant with an interdisciplinary research team composed of myself and members from Baylor University and the University of Deleware. This exciting new project explores a novel method for studying turbulent behavior in dusty plasmas while also exploring the use of non-local operators. This project combines theoretical mathematical analysis, computational mathematics, high-performance computing, mathematical modeling, theoretical physics, and experimental physics.

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NSF-1956396

Modeling in a Heterogeneous World (XVIII Red Raider MiniSymposium)

I am co-PI on an NSF conference grant with an interdisciplinary team composed of myself and members from Texas Tech University's Departments of Mathematics and Statistics and Biological Sciences. This grant allows us to continue the long tradition of hosting a Red Raider Minisymposium and also allows for the provision of travel funds to interested early-career researchers. This year's focus in on the inclusion of heterogeneity in biologically relevant mathematical models.

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New Publication #1

My paper studying novel methods for approximating certain high-dimensional partial differential (PDEs) equations has been recently submitted for publication and may be found on arXiv (see link below). This paper provides a novel method for approximating high-dimensional PDEs while avoiding suffering from the so-called curse of dimensionality (CoD). This result provides the foundation needed to prove that deep neural networks possess the expressive power to provide CoD-free approximate solutions to high-dimensional PDEs (an important result which validates the use of deep learning techniques when approximating PDEs).

M. Hutzenthaler, A. Jentzen, B. Kuckuck, and J. L. Padgett, Strong L^p-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

arXiv version: arXiv:2110.08297

New Publication #2

My paper studying properties of objects which generalize classical strongly continuous semigroups has been accepted for publication in the recent book From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory. These generalizations include solution operators arising from certain singular integral equations, as well as those from the classical setting. Moreover, the analysis is carried out in abstract Banach spaces which guarantees that the results hold for numerical approximations, as well. This article appears in a book which has been published in honor of Lance Littlejohn's 70^th birthday.

T. F. Jones, J. L. Padgett, and Q. Sheng, Intrinsic properties of strongly continuous fractional semigroups in normed vector spaces

arXiv version: arXiv:21012.11092

Official book version: https://doi.org/10.1007/978-3-030-75425-9_14

New Publication #3

My paper studying discrete non-local operators and their explicit series representations has been accepted for publication in the Journal of Mathematical Analysis and Applications. This exciting publication is the result of a long standing collaboration across several fields with the ultimate goal of improving our ability to model realistic physical phenomena with a high level of accuracy.

T. F. Jones, E. G. Kostadinova, J. L. Padgett, and Q. Sheng, A series representation of the discrete fractional Laplace operator of arbitrary order

arXiv version: arXiv:2101.03629

Official JMAA version: https://doi.org/10.1016/j.jmaa.2021.125323

Research Interests

My primary research interests lie in the areas of numerical analysis, applied mathematics, and computational mathematics. I am particularly interested in problems arising in biology and physics whichexhibit nonstandard computational challenges—such as problems with singular, nonlocal, or stochastic influences. My work has employed a variety of mathematical techniques, with a particular focus on combining computational techniques with those from operator theory, spectral theory, and Lie group theory. My most significant contributions have been the development of the abstract numerical analysis for such problems, which allows for the obtained results to have a wider range of applications. Such efforts allow for the construction of qualitatively and quantitatively superior computational algorithms. Moreover, it allows for the results to be applied to numerous physical problems of interest, such as those arising in mathematical biology, combustion theory, and plasma physics.

My recent research efforts have been focused on machine learning and deep artificial networks, with a particular emphasis on how these tools may be employed to efficiently approximate high-dimensional partial differential equations. This area is of great importance in the scientific community and garners interest from a wide array of academic disciplines. My current focus is on the theoretical and mathematical considerations of deep learning; i.e., my focus is on proving theorems regarding deep artificial networks. The mathematics for this field is still in its infancy, and as such, there are a great deal of exciting problems to be pursued in this direction.

Topics of interest:

Applied mathematics
Numerical analysis
Computational mathematics
Geometric and Lie group integration methods
Operator splitting methods
Quenching-combustion differential equations
Singular integral equations
Stochastic differential equations
Machine learning
Mathematics for deep learning

Grants and Funding

The following is a list of grant proposals that have either been funded.

R&D Project: As part of a research and development effort, we developed a three-dimensional mathematical model for application to a manufacturing process for a Fortune 500 Consumer Products Company (2021) Amount: $105,039
Investigators: Edmund Harriss • Jeremy Van Horn-Morris • Joshua Lee Padgett
NSF-1903450: Onset of Turbulence in Dusty Plasma Liquids (2019 - 2022) Amount: $257,840
Investigators: Evdokiya Kostadinova • Constanze Liaw • Lorin Matthews • Joshua Lee Padgett
NSF-1956396: Modeling in a Heterogeneous World (XVIII Red Raider Mini-Symposium) (2020 - 2022) Amount: $17,750
Investigators: Linda J. S. Allen • Joshua Lee Padgett • Angela Peace • Kenneth Schmidt • Wenjing Zhang

My Dogs

Below are some pictures of my two dogs: Murphy (an Australian Shepherd) and Memphis (a Goldendoodle). Murphy is now approximately five years old and Memphis is now approximately two and a half years old.