Updated December, 2020

Welcome

James Madison University (JMU) has been chosen for a National Science Foundation Research Experiences for Undergraduates (REU) site in mathematics. All interested undergraduates (who are US citizens or permanent residents) are encouraged to apply for this eight week program, in which students will work in groups of 2-4 under the supervision of their faculty mentors. Due to the COVID-19 pandemic, our site will run online using remote work technologies.

A $4,500 stipend will be provided for participation in this remote program.

Students from groups underrepresented within their disciplines (e.g., women, underrepresented minorities, students with disabilities), veterans of the U.S. Armed Forces, first-generation college students and students from socioeconomically depressed regions (e.g., Appalachia) are especially encouraged to apply.

Here is a link to a flyer.

Dates

May 24 2021 - July 16 2021 (remotely).

Applications

Please visit mathprograms.org to apply. Application deadline is February 15, 2021.

Projects

Bias Mitigation for Machine Learning Estimators: As machine learning estimators are gaining more and more attentions, so is the study of the accuracy of these estimators. In this project, we will assess the accuracy of some most popular machine learning estimators, including random forest and boosting estimators. In many situations, the dominant component in the risk turns out to be the squared bias, which leads to the necessity of bias correction. We will focus on comparing the degree of biasedness of these estimators and exploring the effectiveness of adaptive bagging and bias correction bootstrap techniques in reducing bias.

The students will get familiar with the popular machine learning estimators and learn how to assess statistical risks. The students will also gain experiences with R programming.

Mentors: Lihua Chen and Prabhashi Gamage

Power Series and (highly!) nonlinear ODEs: Join us as we continue to develop, analyze, and exploit an exciting approach to generate numeric and symbolic series solutions of nearly all Ordinary Differential Equation (ODE) systems, even those which are highly nonlinear. The technique, discovered by JMU faculty Ed Parker and Jim Sochacki, was initially based on application of the Picard iterative method to ODE systems of a particular form -- those which can be written as systems of first order initial value problems with polynomial forcing. The approach has evolved into a more efficient method based on power series.

What is perhaps most exciting about the technique which we call PSM is that the theory is remarkably accessible, and that it can be applied in many ways. At the core, the fundamental PSM object is a Taylor series. There are many questions we wish to answer, and here are a few:

Explore adaptive methods. Both adaptive in time and adaptive in order are possible. We know that there is a sharp error bound for general functions, but can this be improved for specific problem classes? Is there an optimal choice, and can it be proven? And which ad-hoc heuristics might be effective and competitive?  How can computational complexity be minimized?

Investigate the generation and application of symbolic Taylor representations. Since our approximate solutions are constructible, the basic ideas from calculus are applicable. Use root-finding techniques to find critical events, for example, or calculate derivatives to study sensitivity of the system to changes in initial conditions and parameters. Can PSM be used to control systems?

Experiment with Pade approximants. These can be generated from Taylor series using a linear solver or with continued fractions. Pade will capture singularities in the solution and is therefore useful in regions where the polynomial form doesn't converge or doesn't converge quickly, and regions which are stiff. Is Pade provably more efficient, and computationally more effective?

Develop and analyze PSM based solvers for Hamiltonian systems. Symplectic systems, those without friction-like terms and which conserve energy-like Hamiltonian quantities, are important. Two examples, electrical and gravitational systems, are reversible and independent of path. Numerical drag is almost always introduced by numerical solvers, and unfortunately destroys reversibility and path independence. How and when can PSM avoid measurable numerical drag and conserve Hamiltonian dynamics?

We are interested in these questions about the theory and practice of PSM, and more. But we don't expect all of them to excite you. Our 8 week session will start with survey material to give you some background, and then together we will develop a fun and exciting project that will be a great fit for your talents and interest, and which will give you the opportunity to learn a ton.

Mentors:  Jim Sochacki and Roger Thelwell

Common reply date

This REU participates in the Mathematical Sciences REU Common Reply Date agreement. Any student who receives an offer from an REU that participates in the Common Reply Date agreement, including this REU, is not required to accept or decline the offer until March 8, 2021 (or later). The goal of the Common Reply Date agreement is to help students make informed decisions, particularly when faced with the potential of receiving offers from multiple REUs. 

Questions?

Please write to mathreu@jmu.edu with any questions.

This material is based upon work supported by the National Science Foundation under Grant Number 1950370. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Inclusivity Commitment

The Department of Mathematics and Statistics is committed to creating learning environments that support and are improved by a diversity of thought, perspective, and experience. We affirm that the lives and experiences of Black, Indigenous, and People of Color matter. We recognize that within the study and culture of mathematics and statistics there are deep-rooted and systemic inequalities, racism, and sexism that have disproportionately affected some members of our community. We strive to recognize and reverse these inequities. We embrace all backgrounds, identities, names, and pronouns. We see you, we hear you, and we stand with you. You are welcome in our department.

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