National Research Experiences for Undergraduates Program

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. 

Housing in Harrisonburg, VA during the eight weeks of the program and a $4,750 stipend are supplied, as well as full use of university facilities and travel expenses

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 22 2023 - July 14 2023.

Applications

Please visit mathprograms.org to apply. Application deadline is Noon on February 20, 2023.

Projects

Effects of Riparian Buffers to Maintain a Healthy Environment in Shenandoah Valley, Virginia

Shenandoah Valley: the breathtaking, scenic valley which has the greatest agricultural activities and farming practices in Virginia. The heavy agronomic land use and farming has taken the attention of the authorities with some issues related to the eco-system health of Shenandoah Valley. To reduce the pollution due to these practices, the Conservation Reserve Enhancement Program (CREP) by U.S. Department of Agriculture has suggested the landowners to install a riparian buffer-an area between the pollutant source and the water stream which is often vegetated with grasses and some trees.

In this project, we focus on several farms in Shenandoah Valley selected via Soil Water Conservation District (SWCD). We will determine the water (soil) quality by measuring pH levels, chloride/ sulfate/ nitrate concentrations, calcium/ magnesium concentration, etc. The GIS software will be used to quantify some environmental/landscape factors to include but not limited to slope of the land, elevation, distance to the farm from the stream. Through some linear models, we will determine the impact of riparian buffers (age, size, species richness), and the influence of some other factors (to include environmental and geographical) in the water (soil) quality in Shenandoah Valley.

Join us this summer to get a wonderful outreach experience in serving the Shenandoah community while having fun by visiting the farms and kayaking in beautiful streams in Shenandoah Valley for sample collection.

Mentors: Dr. Prabhashi Withana Gamage and Dr. Dhanuska Wijesinghe

 

Some Invariants in Algebraic Graph Theory

Consider a matrix whose rows are indexed by the subsets of size r (r-subsets) of a finite set, and whose columns are indexed by the s-subsets. (Say r < s.) Place a 1 as the (X,Y)-entry of the matrix if X is a subset of Y, and put a 0 otherwise. This zero-one incidence matrix describes the fundamental relation of subset inclusion. Certain properties of the matrix that are independent of its construction become properties of the relation itself, and tools of linear and abstract algebra can be used to investigate and unravel these mysterious and often beautiful invariants.

Subsets are fundamental objects in mathematics and many other incidence relations besides inclusion can be considered; for example, disjointness, or more generally intersection in a fixed size. When the incidence matrix is the adjacency matrix of a graph (for example, of the Kneser or Johnson graphs on subsets) then the spectrum is an interesting invariant that gives much useful information about the graph. The ranks over various fields, or more generally the integer invariants (e.g., Smith normal form), of the matrix are also of great interest. Again in the case of a graph, the integer invariants of the Laplacian matrix describe the "sandpile group" of the graph. This finite abelian group goes by several other names and describes a sort of discrete flow across the edges; its order is equal to the number of spanning trees of the graph.

Most of these invariants remain undescribed. The spectrum of the Kneser graphs is known, but only in the case of 2-subsets have the integer invariants of the adjacency and Laplacian matrices been described. There is a long-standing problem on determining the sandpile group of the n-dimensional hypercube graph (whose vertices may be viewed as subsets); here the 2-primary component of the group is what remains to be found. Many of these questions also have analogous statements concerning subspaces of a vector space, where even less is known.

Participants in this REU will learn about and make progress on these problems and related topics. Various approaches will be employed, including matrix methods, chip-firing games, and applying the representation theory of the symmetric group.

Mentors: Dr. Josh Ducey and Dr. Brant Jones

 

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, 2022 (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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