Patterns and Invariants in Mathematics

Professor Nicole Snashall

Department of Mathematics

Lecture Summary

We start by using a variety of mathematical and non-mathematical examples (including pinecones and chocolates) to informally explore the concept of a mathematical invariant, that is, an innate property of a system which is independent of the particular presentation or situation in which that system arises. These examples naturally introduce us to algebras, representations, and homological invariants, and we include some illustrations and practical applications of representation theory. Hochschild cohomology is an important invariant which allows us to study the structural properties and representation theory of algebras. We give an overview of some of the recent work in this area which demonstrates how connections between algebraic and geometrical properties have led to a deeper understanding of this beautiful area of pure mathematics.