# Research

The faculty in the Department of Mathematics at Western Michigan University have a well-funded and active research program that addresses a wide range of research topics in mathematics. See faculty research area groups.

## Active grants

**On quantitative aspects of Ramsey theory**

Funded by Simons Foundation ($42,000; September 1, 2017- August 31, 2022). The main goal of the project is to study problems in Ramsey theory, currently one of the most active areas in combinatorics. Ramsey theory can loosely be described as the study of structure which is preserved under finite decomposition. The classical Ramsey theorem states that for a given integer t there is an integer n = n(t) such that any 2-coloring of the edges of the complete graph on n vertices yields a monochromatic copy of a complete graph on t vertices. The smallest such integer n is called the Ramsey number. Determining the order of magnitude of the Ramsey numbers, as well as that of some generalized Ramsey numbers is one of the major (and still wide open) problems in combinatorics. The project aims to demonstrate how some geometric ideas in conjunction with probabilistic methods can be used in solving such problems. This approach provides a new method for determining asymptotic behaviors of certain Ramsey- type numbers, many of which have withstood decades of active attempts. The award provides support for travel and visitors. Andrzej Dudek is the principal investigator.

**Tracking ****the Longitudinal Development of STEM Majors’ Autonomy and Agency in Mathematical Proof and Proving**

Funded by National Science Foundation ($299,000, September 2019 - March 2023). Dr. Mariana Levin (WMU), together with Dr. Shiv Karunakaran and Dr. Jack Smith of Michigan State University have a National Science Foundation IUSE (Improving Undergraduate STEM Education) grant to explore the longitudinal experiences of cohorts of STEM students across their proof-intensive upper- division coursework. A core question of interest for the project concerns the way upper-division STEM students respond when they face challenges in their mathematical work. In particular, the project has focused on the interrelations between students’ mathematical agency (their felt capacity to take action in response to challenges) and their mathematical autonomy (qualities of action that reflect active resistance to a priori endorsing or replicating the reasoning of mathematical authorities). The team has presented their work at the RUME (Research in Undergraduate Mathematics Education) Conference.

**Probabilistic Combinatorics and Constrained Random Processes**

Funded by Simons Foundation ($42,000; September 1, 2021 - August 31, 2026). A (p, q)-coloring on n vertices is a coloring of all possible edges with the property that every set of p vertices has at least q different colors among its edges. Erdos and Shelah first defined and studied f (n, p, q), the smallest number of colors needed for a (p, q)-coloring on n vertices. The project is focused on the asymptotics of f (n, p, q) based on the analysis of randomized coloring algorithms using the differential equation method developed and popularized by Wormald. Patrick Bennett is the principal investigator.

**The Building on MOSTs (Mathematical Opportunities in Student Thinking)**

Funded by National Science Foundation ($4,953,821 total; $1,990,705 for WMU). The project is a collaboration among researchers at Western Michigan University, Michigan Technological University, and Brigham Young University that focuses on improving the teaching of secondary school mathematics by exploring the teaching practices that allow teachers to elicit and take advantage of MOSTs. This work has received funding from NSF for the past ten years. Faculty member Dr. Laura Van Zoest is the Principal Investigator for WMU and WMU alum Dr. Shari Stockero (PhD, 2006) is the Principal Investigator for MTU. The project has over 75 publications and presentations and has provided research opportunities for numerous graduate students, post-docs, visiting scholars, and secondary school mathematics teachers.

**Using Student Feedback to Improve the Value of Desmos Activities for Students**

Funded by Faculty Scholars Award (SFSA) through WMU Office of Research and Innovation. Drs. Melinda Koelling and Tabitha Mingus were awarded the grant ($1990). The goal of the project was to work with calculus students who had used activities in Desmos and develop a rubric that can be used to evaluate and improve the content and accessibility of the activities.