Graph it up! Making abstract algebra less “abstract”

Faculty of Science
Mathematics and statistics
Master’s graduate Saima Samchuck-Schnarch wearing a blue hoodie and displaying Brauer diagrams illustrated on a green chalkboard.
Math may not be everyone’s cup of tea, but to Saima Samchuck-Schnarch — an MSc graduate under the supervision of Professor Alistair Savage — it is his passion.

Saima is not only eager to learn about the latest pure mathematical theories, but also enjoys sharing his knowledge. His enthusiasm for teaching earned him the Department Teaching Assistant (TA) of the Year award, the first of its kind in the Department of Mathematics. The recipient was chosen based on a combination of feedback from professors and testimonials from students who had direct contact with the TA.

In addition, Saima received an NSERC graduate scholarship to support his MSc research in abstract algebra, which is the study of algebraic structures using mainly expressions (e.g., x2 + 3y = 5) instead of numbers. Saima’s thesis links two existing topics, namely Brauer categories and Frobenius algebras, using a graphical method. Brauer categories are built from Brauer diagrams, which are comprised of strings that twist around in various shapes and can potentially cross each other. The strings can also be multiplied in various orders, meaning they can be stacked on top of each other in different ways, resulting in numerous possibilities. For example, imagine two strings crossing each other (✕) multiplied by two separate parallel strings ( | | ). Depending on the order that the strings are multiplied (i.e., | | or | | ), they will be stacked on top of each other in opposite manners.

Saima’s research generalizes Brauer categories by adding “tokens” to the linked strings, with each token labelled by an element of the Frobenius algebra. This unifies several existing theories and gives an intuitive graphical way of viewing a topic that was previously very abstract and only interpreted using complex equations. In other words, Saima has defined a diagrammatic language to illustrate Brauer categories and Frobenius algebras in a simpler and visual way. This has led to important new results in a field of mathematics known as representation theory. Saima is grateful for the encouragement and support of the late Professor Pieter Hofstra and Prof. Savage who played an important role in his decision to pursue graduate studies.

Saima is also involved in the Svitanok Ukrainian Dance Ensemble where he is an active dancer, choreographer, and instructor. He performed with his group during the 2022 Canada Day festivities. His advice for graduate students is to “keep working on your research even if you run into problems that seem insurmountable because you will eventually get over it and see the other side of the tunnel”.