##### Overview

Students accepted into our National Science Foundation (NSF) funded Research Experiences for Undergraduates (REU) program will receive a stipend to live and work on the JMU campus. In this eight week program, participants will be divided into groups of 2-4 and conduct research projects under the supervision and mentorship of our Mathematics & Statistics faculty.

The application process for the 2023 program is now closed.

##### 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.

##### 2023 Projects

##### Riparian Buffers and the Environment in Shenandoah Valley, VA

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**

##### 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**