- Brendan Harcourt

# Andrew Rosevear ‘22

Updated: Feb 23

This article is part of a series on the __Joint Mathematics Meeting__.

Andrew Rosevear explored a branch of graph theory known as algebraic graph theory. In graph theory, “graphs” refer to several points known as “vertices” connected by lines known as “edges.” Social media networks are a real-life example of graph theory: Facebook users (vertices) connect with other users by becoming friends (edges). Algebraic graph theory specifically concerns the relationships between graphs and complex structures known as “algebraic objects”. Rosevear studied three types of these algebraic objects: two “cohomology groups” and the graph’s “automorphism group”. Groups simply refer to a type of algebraic object that represents concrete structures in an abstract way. As an example, a Rubik’s cube group represents a Rubik’s cube puzzle in mathematical terms. Similarly, cohomology groups are vector spaces – abstract representations of normal *n*-dimensional spaces – in which every point in space is represented using *n* coordinates. For the 0th cohomology group, *n* equals the number of connected components (maximal sets of vertices that are connected by edges) of the graph. For the 1st cohomology group, *n *equals the number of edge cycles (pathways of edges in which the only repeated vertices are the initial and ending vertices) on the graph. The automorphism group measures the graph’s symmetries – the transformations of the graph that preserve its shape.

In his study, Rosevear derived two new algebraic objects by combining each cohomology group with the automorphism group. The 0th “cohomology symmetry group” determines the symmetry of the connected components of the graph. The 1st cohomology symmetry group roughly determines the amount of the graph’s symmetry located in its edge cycles as opposed to the rest of the graph. Rosevear acknowledges that his research has no immediate applications, but believes these findings will eventually prove useful in certain fields, such as image processing or social media networking.

Rosevear initiated his study at last summer’s SURF program. He thoroughly enjoyed his research experience, describing it as an opportunity to freely explore and play around with a self-assigned puzzle. “I love trying to make new connections and gain intuition for new, unexplored areas,” he notes. “Gaining intuition is the first crucial step towards proving theorems, and is one of the most fun parts of math.”

Rosevear is the president and co-founder of the Math Club and is a member of the Archery Club board. He is also involved with theater, devoting time to Theater and Dance department shows and student-run Green Room shows. He is currently pursuing a career in academia, likely in some division of pure mathematics.