Winter 2024 Term
The purpose of the AARMS Advanced Course program is to give graduate students and upper year undergraduates from AARMS Member Universities an opportunity to take courses not offered at their home institution free of charge. In Winter 2024, the courses listed below will be offered in an exclusively online format.
Students may be able to claim credit for AARMS Advanced Courses from their home institution using mechanisms similar to the AARMS Summer School. Students hoping to receive academic credit for courses in this program are strongly encouraged to consult with their home institution about this process before the start of lectures.
- November 17, 2023: Application form opens for regular AARMS Advanced courses (applications will be accepted on a rolling basis)
- December 17, 2023: Application form closes for regular AARMS Advanced courses (individual courses may close earlier without notice)
- Early January, 2024: Classes begin
Advanced graph theory
This course will cover a variety of topics in graph theory at the advanced undergraduate or graduate level. These are expected to be: matchings and factors (including 1-factorizations, cycle decompositions, Steiner triple systems and edge colourings); spectral graph theory (including graph eigenvalues, strongly regular graphs, cliques and cocliques, and the Erdős–Ko–Rado theorem); graph products (including Cartesian products, vertex- and edge-colourings, spectral properties).
Students should have taken a course in discrete mathematics, graph theory, or a related subject, but this is not an absolute requirement; however, a background in linear algebra (including eigenvalues, eigenvectors and orthogonal diagonalization) is essential. They should be comfortable writing proofs.
Lie groups describe continuous symmetries and are widely used in mathematics and physics. It was discovered by Sophus Lie that these – typically nonlinear – objects can be studied using linear objects called Lie algebras, whose properties capture much of the structure of the groups. This course will give an introduction to the theory of Lie algebras starting from the basics (solvability, nilpotency, simplicity, and semisimplicity) and progressing to cover the Killing-Cartan classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero and the associated concepts (Cartan subalgebra, Killing form, root systems, and Dynkin diagrams). Students will have an opportunity to explore additional topics in their course projects.
Prerequisites: undergraduate abstract algebra and advanced linear algebra (vector spaces and linear transformations). Knowledge of advanced group or ring theory is not required.
Topics covered in this course include an introduction to computing, likelihood methods and optimization (e.g., the Expectation Maximization and MM algorithms), subsampling methods (bootstrapping and cross validation), Bayesian methods and integration (e.g., Metropolis-Hastings algorithm, Gibbs sampling, MCMC), and Lasso if time permits.
The course will use R statistical software. Students taking this course should have some familiarity with programming in R and an intermediate level statistical theory course, equivalent to STAT 3460 offered at Dalhousie University.
Functional Analysis II
The aim of this course is to provide an overview of the rapidly developing Brunn-Minkowski theory for convex bodies and its analytic lifting to the log-concave functions. Topics covered in this course range from the basic theory of convex bodies and convex functions to some selected higher level results including the Brunn-Minkowski inequality, the Minkowski inequality, the Prékopa–Leindler inequality, the affine isoperimetric inequalities and the Minkowski problems. If time permits, we will also cover some fantastic topics such as the optimal mass transpose, the symmetrization and rearrangements, etc.
Students should have at least taken courses in Real Analysis and Linear Algebra. Ideally, students should have some knowledge in Functional Analysis and Lebesgue Measure Theory. Students completing this course should gain a good understanding of the geometric and analytic properties of convexity, along with the ability to apply these concepts to other areas. Timeslot: Mondays and Wednesdays, 10:30‐11:45, Newfoundland time.