Research


Overview

In mathematics, a common goal is to understand how to use established techniques and theory to help solve more complicated questions. Representation theory is the study of understanding abstract algebraic objects through the lens of linear algebra by representing abstract elements as matrices. In effect, representation theory allows one to use techniques of linear algebra to understand far more complicated objects.

Quantum algebra is the area of mathematics that in some way relates to the fact that noncommutativity (that is, when $a \cdot b \neq b \cdot a$) is found in quantum mechanics. Examples of noncommutative operations are division (e.g., $2 \div 5 \neq 5 \div 2$), subtraction (e.g., $8-3 \neq 3-8$), and matrix multiplication.

In the area of quantum algebra are special noncommutative objects known as quantum groups, whose representation theories are known to have particular importance in quantum physics. One such family of quantum groups are called Yangians, which come equipped with the foundational fact that their representation theory produces rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself, with such research expanding to super Yangians: a formulation of Yangians in terms of supersymmetry. However, the theory of super Yangians is comparatively less developed to its non-super counterpart, which is the cause of further exploration.

Conferences Attended


  • Workshop on Symmetric Spaces, Their Generalizations, and Special Functions

    • August 2022 | University of Ottawa
  • Quantum Field Theories and Quantum Topology Beyond Semisimplicity

    • November 2021 | Banff International Research Station
  • QRST: Quantum Groups, Representation Theory, Superalgebras, and Tensor Categories (Virtual)

    • August 2020 | University of Ottawa
  • Hopf Algebras, Algebraic Groups, and Related Structures

    • June 2016 | Memorial University of Newfoundland
  • Algebraic Groups and Lie Algebras

    • August 2015 | Bonne Bay Marine Station of Memorial University of Newfoundland

Presentations


  • Super Yangians: Where We Are Today

    • May 2022 | PIMS-USask Geometry, Algebra and Physics Seminar | University of Saskatchewan
    • Abstract: Given any finite-dimensional simple Lie algebra $\mathfrak{a}$ over $\mathbb{C}$, the Yangian $\mathbf{Y}(\mathfrak{a})$ is a certain unital associative $\mathbb{C}$-algebra. In particular, Yangians form a family of so-called quantum groups. The main property these algebras is the foundational fact that their representations produce rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself and has expanded to the study of super Yangians based on Lie superalgebras; however, the theory of super Yangians is comparatively less developed than its non-super counterpart. In this talk, we will survey what recent advancements have been made in the study of super Yangians and view what else remains to do.
  • A Friendly Introduction to Representation Theory

    • October 2021 | Mathematical and Statistical Sciences Graduate Colloquium | University of Alberta
    • Abstract: Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. When that abstract algebraic object is being represented on a finite-dimensional vector space, its elements are described by matrices and its algebraic operations are described by matrix multiplication and matrix addition. The power of such methodology allows one to reduce abstract algebra problems to linear algebra problems. In this talk, we will introduce the basic notions of representation theory with a focus on group representations.
    • Slides
  • A Friendly Introduction to Representation Theory

    • February 2021 | Mathematical and Statistical Sciences Graduate Colloquium (Virtual) | University of Alberta
    • Abstract: Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. When that abstract algebraic object is being represented on a finite-dimensional vector space, its elements are described by matrices and its algebraic operations are described by matrix multiplication and matrix addition. The power of such methodology allows one to reduce abstract algebra problems to linear algebra problems. In this talk, we will introduce the basic notions of representation theory with a focus on group representations.
    • Slides
  • Dualizing Algebras

    • January 2020 | Graduate Algebra Seminar | University of Alberta
    • Abstract: An algebra is traditionally defined as a certain object in the category of modules over some ring or field. We shall realize that such definition can be generalized to work in any monoidal category and can moreover be dualized, introducing the concept of a coalgebra.
  • Introduction to Hopf Algebras

    • December 2017 | University of Western Ontario
    • Abstract: By dualizing the definition of an algebra, we obtain a notion of a coalgebra. Combining these notions with some extra compatibility conditions, one arrives at the definition of a Hopf algebra. We shall discuss why Hopf algebras are important and provide some examples.
  • Yetter-Drinfel’d Modules and the Radford Projection Theorem

    • April 2017 | Bachelor Thesis Public Presentation | Memorial University of Newfoundland
    • Abstract: In the theory of groups, the semidirect product is an approach to decompose a group into a product of two subgroups with one being normal. We shall examine a Hopf algebra analogue of the semidirect product, called the Radford biproduct.
  • Olshansky’s Modification of Golod’s Example of Non-Nilpotent Finitely Generated Nil-Algebras

    • October 2016 | Algebra Seminar | Memorial University of Newfoundland
    • Abstract: Burnside’s problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. A counter-example was provided by Golod in the 1960’s and in this talk we shall provide Olshanky’s version of such counter-example.