Mathematics

The basic purpose of this course is to deeply understand mathematical thought and values behind classical knowledge in mathematics.

This course provides and understanding of group theory and some of its applications. In this course, students work with cyclic groups, permutation groups, dihedral groups, equivalence classes, cosets, Lagrange's theorem, and direct product groups; are introduced to quotient groups, construct the groups of low order, learn about the conjugation map, and construct conjugacy classes; meet the classical matrix groups, which are examples of continuous (or Lie) groups; work with group homomorphisms, isomorphisms, automorphisms, normal subgroups, kernels of homomorphisms, and prove and make extensive use of the group homomorphism theorem (also known as the first isomorphism theorem); learn about the semi-direct product and semi-direct product groups; construct and investigate the Euclidean group; investigate the geometric structure of some of the classical matrix groups, in particular SU(2)and SO(3); work with group actions on sets, stabilisers and orbits; and prove the Sylow theorems.

This course provides an introduction to sets and functions: defining sets, subsets, intersections and unions; injections, surjections, bijections.; compositions and inverses of functions; an introduction to mathematical logic and proof: logical operations, implication, equivalence, quantifiers, converse and contrapositive; proof by induction and contradiction, examples of proofs. These ideas are then applied in the context of the real numbers to make rigorous arguments with sequences and series and develop the notions of convergence and limits. 

This course explores the quantitative instruments needed to pose and analyze economic problems with the aid of a formal model. Topics include: concepts of matrices and algebra of matrices; analysis of dynamic economic models; differential and difference equations and systems; examination of the qualitative behavior of solutions. 

This course explores the concepts of heuristics and optimization as two means of problem-solving and analysis. Topics include: dynamic programming; linear programming; constrained Boolean satisfiability; constraints programming; search. Pre-requisites: Programming; Algorithms and Data Structures; Discrete Mathematics; Artificial Intelligence. 

This course introduces the key principles of mathematical modelling, which then are explored through several real-world examples in disease modelling, environmental planning, and population dynamics. The techniques of calculus are essential, although core concepts such as differential equations are revisited. Students also touch on the mathematical models used in data science, particularly the techniques of principal component analysis and clustering, both essential tools in machine learning models.  

Graph theory provides a basis for computational sciences, financial engineering, chemistry, and epidemiology, among many others. This course covers graph theory including graphs, paths, cycles, trees, connectivity, Eulerian and Hamiltonian graphs, and planar graphs, as well as some important algorithms. 

This course provides a study of the fundamentals of statistical inference such as probability distribution, point estimation, interval estimation, and hypothesis testing. Other topics include: sampling distributions; parameter estimation; confidence intervals; parametric hypothesis testing; ANOVA and nonparametric contrasts; Bayesian inference.

This is a first course in real analysis and a concrete introduction to group theory and the mathematics of symmetry. Students study fundamental concepts of Analysis (completeness, epsilon-N, continuity, epsilon-delta) and Group Theory (groups, group actions, symmetries).

Students learn to identify and remove simple trends and seasonalities from time series data; describe the properties of stationary time series and their autocorrelations; define various time series probability models (ARMA, ARIMA, GARCH); construct time series probability models from data and verify model fit; define the spectral density function and understand it as a distribution of energy in the frequency domain; compute the periodogram and smoothed versions; and analyze multivariate time series.