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Research Assistant Positions

A research assistant position is your chance as an undergraduate student to explore the exciting world of research in mathematics and statistics. You have the opportunity to do research with one of the researchers in the Department, to learn beautiful mathematics and statistics, and make some money.

The projects listed below are available under one of the following funding programs:

	Other funding programs

Available Undergraduate Research Positions

There is no available position at this time.

To apply for a position, please fill in the following and submit it to

Potential Undergraduate Research Projects:

If you are interested in one of these projects, please contact the supervisor of the project to express your interest and to discuss funding for the project.

Supervisor	Title	Summary
	The Integers	Let A be an n x n matrix with integer entries (likely n will be ten or more). We look at a reduced form for such matrices modulo the action of GL(n,Z) acting on the left and the permutation matrices acting on the right. (+) It is known that for reasonably large n, roughly 84% will have a reduced form exactly one of whose columns is not that of the identity matrix. The question is, what is the corresponding density for matrices A such that both it and its transpose have such a reduced form. [The properties are not independent–in fact, they seem to anti-correlate, so we cannot simply square 0.84.] This problem comes from an equivalence relation on integer matrices intermediate between Hermite equivalence and Smith equivalence; it comes up in the classification of lattice simplices and of commutative codes. The proposal would involve taking a large sample (thousands) of say size ten matrices with bounds on their entries, and finding the number of matrices which satisfy the conditions. It will involve integer linear algebra, number theory, and statistics. The prerequisites would include some programming ability (in particular, the algorithms to do the reductions would have to be efficient), and some knowledge of number theory and linear algebra.
	Introduction à la théorie des algèbres de Lie	Les algèbres de Lie sont des espaces vectoriels avec une structure additionnelle, à savoir une multiplication. (+) Un exemple important est l’espace vectoriel des matrices de format n x n avec la multiplication donnée par le commutateur. Les algèbres de Lie sont un sujet important de la recherche en mathématiques. Dans le projet, l’étudiant(e) va d’abord apprendre les rudiments de la théorie pour ensuite travailler sur les représentations des matrices, vues comme algèbres de Lie. La langue de travail peut être le français ou l’anglais, mais la plupart des ressources sur les algèbres de Lie sont en anglais. Préalables: MAT2541, MAT2543 recommandés (ou équivalents)
	Simple groups
	Théorèmes de Frobenius et Wedderburn sur les algèbres de division
	Representations of n x n - matrices
	Théorie de Galois
	p-adic Numbers and Representation Theory	The p-adic numbers are a collection of fields which contain the rationals, are complete like the reals, but correspond to a different metric (corresponding to the prime number p). (+) They are useful in number theory and representation theory. In this project, you will delve into the p-adic numbers and consider some questions arising from representation theory.
	calculus of variations and shape optimization; application to the shape of drops, minimal surfaces, hydrogen fuel cells and artficial hearts
	From Hyperbolic Geometry to Expander Networks	Expander networks are a class of networks with high connectivity. Their study was first motivated by problems in wireless communications, neural networks, internet, etc. (+) But the most influential results on expanders and their constructions were first obtained by mathematicians! Somewhat surprisingly, the study of expander graphs is connected with objects in mathematics called symmetric spaces. The simplest example of such an object is the hyperbolic plane, which is where the non-Euclidean geometry lives. (This is the geometry in which the Parellel Axiom of Euclid fails!) If you are interested in learning more about this connection, you should come and work with me on this project! Students who have taken MAT2355 (Introduction to Geometry)
	Category Theory and Supergeometry	Supergeometry is a mathematical theory that was introduced in the 1970's in connection with advancements in particle physics (e.g. supersymmetry, string theory, the standard model, the Higgs boson, etc. (+) Which are currently being studied by physicists using the Large Hadron Collider). The main idea of supergeometry is to replace geometric spaces with their graded analogues. Sometimes the objects are as simple as vector spaces and linear maps, but usually one works with more sophisticated and therefore more interesting objects. It turns out that one can use categories and functors to construct objects of supergeometry. The goal of this project is to implement such a construction to obtain reproducing-kernel vector spaces in the super setting. Students with a strong background in Algebraic Structures (MAT2143) are encouraged to apply. Familiarity with category theory is not necessary, but will be useful.
	The Heisenberg group, the uncertainty principle in physics, and applications
	Plant Genomes	A project using discrete algorithms and data analysis to study the evolution of plant genomes.
	Category theoretic interpretations of algebraic structures
	Crystals and two-dimensional ice
	Heisenberg doubles
	Weyl groups as groups of symmetries	Study various examples of the Weyl groups as groups of symmetries of certain polyhedra in R3 and describe their combinatorial properties and geometry. Prerequisites: MAT2143, MAT2141
	Quadratic and bilinear forms over real numbers	This project is devoted to elementary classification of quadratic forms over real numbers using the signature and the Witt-decomposition theorem. Prerequisites: MAT2143, MAT2141
	Matrices and their twisted forms	This project is essentially an introduction to the theory of central simple algebras. Examples and various properties (e.g. the reduced norm) are discussed. Prerequisites: MAT2143, MAT2141
	Linear transformations and projective spaces	Study the geometry of linear subspaces in a projective space (Grassmannians and their properties). Prerequisites: MAT2143, MAT2141
	Geometry of quaternions/octonions	Study the quaternions/octonions and the associated quadratic forms. Prerequisites: MAT2143, MAT2141
	Bilinear forms, orthogonal groups and reflections	The Cartan-Dieudonne Theorem (rotations generated by reflections). Prerequisites: MAT2143, MAT2141