Candidates will be selected on the basis of their past academic record, letters of recommendation and the relevance of the proposed area of research to the areas of specialization of the Department. The normal requirement for admission to the program is a MSc degree, with high standing in Mathematics or an allied discipline from a recognized university. Exceptional candidates who have successfully completed one-year's study at the Master's level may, upon approval by the Graduate Studies Committee, be exempted from the required completion of the Master's degree and admitted directly into the PhD program.
Proficiency in English. Applicants whose primary language is not English must demonstrate that their knowledge of English is sufficient to pursue graduate studies in their chosen field. Please refer to the Graduate Admission page for further information on the Language Proficiency requirements and exemptions.
Requirements for the Degree
- Credits. Students must complete a program of 90 credits, consisting of the following components:
- Comprehensive examinations (12 credits);
- Six courses or seminars (18 credits);
- Thesis (60 credits).
- Comprehensive Examination. The comprehensive examination is composed of the following two parts:
Part A (6 credits)Part B (6 credits)
This is a written examination, consisting of two parts. The first part of the Comprehensive A examination is to test the candidate's general knowledge of fundamental mathematical concepts. It will normally be completed within one year (3 terms) of the candidate's entry into the program or the equivalent of part-time study. The second part of the Comprehensive A examination tests the candidate's knowledge of topics in his or her area of specialization. The material will be chosen from the list of course descriptions given by the Graduate Studies Committee in consultation with the candidate's research supervisor and the student's Advisory Committee. Candidates are allowed at most one failure in the Part A examination.
The Comprehensive B examination is an oral presentation of the candidate's plan of his or her doctoral thesis in front of the student's Advisory Committee. It is normally taken within two-three years of the candidate's entry into the program (or the equivalent of part-time study) and at least one year before the expected completion of the thesis.
- Thesis. Concurrently with the preparation for the Part B exam, the students will be engaging in their research work towards the dissertation. After submitting the doctoral thesis, the candidate is required to pass an oral defence of the thesis. The doctoral thesis must make an original contribution to mathematical knowledge, at a level suitable for publication in a reputable professional journal in the relevant area.
- Average Time to Completion. Normally a student completes all requirements for the degree, except for the thesis, within two years of entering the program. The normal period for completion of the program, for a student already holding the equivalent of an MA/MSc degree, is three to four years.
- Academic Standing. Please refer to the Academic Standing section of the Calendar for a detailed review of the Academic Regulations.
- Residence. The minimum period of residence is two years of full-time graduate study, beyond the MA/MSc, or the equivalent in part-time study. (A minimum of one year of full-time study is normally expected).
- Time Limit. Please refer to the Academic Regulation page for further details regarding the Time Limit requirements.
- Graduation Requirement. In order to graduate, students must have a cumulative GPA of at least 3.00.
Number Theory and Computational Algebra
MAST 830 Cyclotomic Fields (3 credits)
L-series, Dirichlet theorem, Gauss sums, Stickelberger theorem, class groups and class number, circular units, analytic formulae.
MAST 831 Class Field Theory (3 credits)
Local and global class field theory, ideles and adeles, reciprocity laws, existence theorem.
MAST 832 Elliptic Curves (3 credits)
Introduction to elliptic curves over finite fields, local and global fields, rational points, Mordell-Weil theorem, formal groups.
MAST 833 Selected Topics in Number Theory (3 credits)
MAST 834 Selected Topics in Computational Algebra (3 credits)
MAST 837 Selected Topics in Analysis (3 credits)
MAST 838 Selected Topics in Pure Mathematics (3 credits)
Mathematical Physics and Differential Geometry
MAST 840 Lie Groups (3 credits)
The mathematical theory of Lie groups and introduction to their representation theory with applications to mathematical physics. Topics will include classical Lie groups, one-parameter subgroups, Lie algebras and the exponential mapping, adjoint and coadjoint representations, roots and weights, the Killing form, semi-direct products, Haar measure and decompositions such as those of Cartan and Iwasawa. The theory of unitary representations on Hilbert spaces. Physical applications of compact Lie groups (such as SU(2) and SU(3)) and non-compact groups (such as the Lorentz and Poincaré groups).
MAST 841 Partial Differential Equations (P.D.E.’s) (3 credits)
Introduction to the mathematical theory of P.D.E.’s, including applications to mathematical physics. Topics will include Sturm-Liouville systems, boundary value and eigenvalue problems, Green’s functions for time-independent and time-dependent equations, Laplace and Fourier transform methods. Additional topics will be selected from the theory of elliptic equations (e.g. Laplace and Poisson equations), hyperbolic equations (e.g., the Cauchy problem for the wave equation) and parabolic equations (e.g., the Cauchy problem for the heat equation). Links will be made with the theory of differential operators and with analysis on manifolds.
MAST 851 Differential Geometric Methods in Physics (3 credits)
Manifolds, differential systems, Riemannian, Kahlerian and symplectic geometry, bundles, supermanifolds with applications to relativity, quantization, gauge field theory and Hamiltonian systems.
MAST 852 Algebro-Geometric Methods in Physics (3 credits)
Algebraic curves, Jacobi varieties, theta functions, moduli spaces of holomorphic bundles and algebraic curves, rational maps, sheaves and cohomology with applications to gauge theory, relativity and integrable systems.
MAST 853 Gauge Theory and Relativity (3 credits)
Yang-Mills theory, connections of fibre bundles, spinors, twistors, classical solutions, invariance groups, instantons, monopoles, topological invariants, Einstein equations, equations of motion, Kaluza-Klein, cosmological models, gravitational singularities.
MAST 854 Quantization Methods (3 credits)
Geometric quantization, Borel quantization, Mackey quantization, stochastic and phase space quantization, the problems of prequantization and polarization, deformation theory, dequantization.
MAST 855 Spectral Geometry (3 credits)
Schrödinger operators; min-max characterization of eigenvalues, geometry of the spectrum in parameter space, kinetic potentials, spectral approximation theory, linear combinations and smooth transformations of potentials, applications to the N-body problem.
MAST 856 Selected Topics in Mathematical Physics (3 credits)
MAST 857 Selected Topics in Differential Geometry (3 credits)
MAST 860 Differentiable Dynamical Systems (3 credits)
The study of dynamical properties of diffeomorphisms or of one-parameter groups of diffeomorphisms (flows) defined on differentiable manifolds. Periodic points, the non-wandering set, and more general invariant sets. Smale’s horseshoe, Anosov, and Morse-Smale systems, general hyperbolic systems, the stable manifold theorem, various forms of stability, Markov partitions and symbolic dynamics.
MAST 861 Absolutely Continuous Invariant Measures (3 credits)
Review of functional analysis, Frobenius-Perron operator and its properties, existence of absolutely continuous invariant measures for piecewise expanding transformations, properties of invariant densities, compactness of invariant densities, spectral decomposition of the Frobenius-Perron operator, bounds on the number of absolutely continuous invariant measures, perturbations of absolutely continuous invariant measures.
MAST 862 Numerical Analysis of Nonlinear Problems (3 credits)
Continuation of solutions, homotopy methods, asymptotic stability, bifurcations, branch switching, limit points and higher order singularities, Hopf bifurcation, control of nonlinear phenomena, ODE with boundary and integral constraints, discretization, numerical stability and multiplicity, periodic solutions, Floquet multipliers, period doubling, tori, control of Hopf bifurcation and periodic solutions, travelling waves, rotations, bifurcation phenomena in partial differential equations, degenerate systems.
MAST 863 Bifurcation Theory of Vector Fields (3 credits)
Local and global bifurcations. Generalized Hopf bifurcation and generalized homoclinic bifurcation. Hamiltonian systems and systems close to Hamiltonian systems, local codimension two bifurcations of flows.
MAST 865 Selected Topics in Dynamical Systems (3 credits)
Statistics and Actuarial Mathematics
MAST 871 Advanced Probability Theory (3 credits)
Definition of probability spaces, review of convergence concepts, conditioning and the Markov property, introduction to stochastic processes and martingales.
MAST 872 Stochastic Processes (3 credits)
Stochastic sequences, martingales and semi-martingales, Gaussian processes, processes with independent increments, Markov processes, limit theorems for stochastic processes.
MAST 873 Advanced Statistical Inference (3 credits)
Decision functions, randomization, optimal decision rules, the form of Bayes’ rule for estimation problems, admissibility and completeness, minimax, rules, invariant statistical decisions, admissible and minimax decision rules, uniformly most powerful tests, unbiased tests, locally best tests, general linear hypothesis, multiple decision problems.
MAST 874 Advanced Multivariate Inference (3 credits)
Wishart distribution, analysis of dispersion , tests of linear hypotheses, Rao’s test for additional information, test for dimensionality, principal component analysis, discriminant analysis, Mahalanobis distance, cluster analysis, relations with sets of variates.
MAST 875 Advanced Sampling (3 credits)
Unequal probability sampling, multistage sampling, super population models, Bayes and empirical Bayes estimation, estimation of variance from complex surveys, non-response errors and multivariate auxiliary information.
MAST 876 Survival Analysis (3 credits)
Failure time models, inference in parametric models, proportional hazards, non-parametric inference, multivariate failure time data, competing risks.
MAST 877 Reliability Theory (3 credits)
Reliability performance measures, unrepairable systems, repairable systems, load-strength reliability models, distributions with monotone failure rates, analysis of performance effectiveness, optimal redundancy, heuristic methods in reliability.
MAST 878 Advanced Risk Theory (3 credits)
Generalizations of the classical risk model, renewal processes, Cox processes, diffusion models, ruin theory and optimal surplus control.
MAST 881 Selected Topics in Probability, Statistics and Actuarial Mathematics (3 credits)
MAST 858 Seminar in Mathematical Physics (3 credits)
MAST 859 Seminar in Differential Geometry (3 credits)
MAST 868 Seminar in Dynamical Systems (3 credits)
MAST 889 Seminar in Probability, Statistics and Actuarial Mathematics (3 credits)
MAST 898 Seminar in Number Theory (3 credits)
MAST 899 Seminar in Computational Algebra (3 credits)
Thesis and Comprehensive Examinations
MAST 890 Comprehensive Examination A (6 credits)
MAST 891 Comprehensive Examination B (6 credits)
MAST 892 Doctoral Thesis (60 credits)