An Introduction to Categorification

Abstract

Many algebraic and combinatorial identities can be understood as shadows of deeper relationships between mathematical objects. Categorification provides a systematic framework for uncovering and explaining such relationships and has led to important developments in areas ranging from algebra and combinatorics to knot theory. In this talk, we will explore the basic ideas of categorification through elementary examples from linear algebra and representation theory. After introducing monoidal categories and Grothendieck groups, we will see how representations of general linear groups can be used to categorify symmetric polynomials. In particular, we will discover how this perspective gives a conceptual explanation for some interesting combinatorial positivity phenomena. No prior knowledge of category theory or representation theory will be assumed.

Biography

Jonas Antor is a postdoctoral researcher at the University of Bonn. He obtained his DPhil in Mathematics from the University of Oxford in 2024 under the supervision of Kevin McGerty. His research focuses on geometric representation theory, using geometry to study algebraic structures such as affine Hecke algebras and quantum groups. His work includes geometric realisations of Hecke algebras with unequal parameters, methods for computing canonical bases, exotic Springer correspondences, and geometric approaches to the Langlands correspondence.

AToM@MAS

This talk is part of a new regular seminar series called AToM@MAS organized by Andrew Kricker and Emile Okada. This stands for The Algebra, Topology, and Mathematical Physics Seminar. This series is intended to complement the regular MAS Seminar Series by providing speakers with an opportunity to give supplementary introductory talks aimed at a wider mathematical audience, including researchers not working in the discipline in question. It also offers researchers at all career stages within SPMS a platform to present their work and hosts in-depth learning seminars on topics of interest within algebra, topology, mathematical physics, and their interdisciplinary spaces.