# Mathematics and Statistics

Graduate Co-ordinator: Karen Meagher, PhD

#### Department Description

The Department is engaged in a broad range of research in pure and applied mathematics and statistics, and offers programs leading to the MSc, and PhD degrees in mathematics or statistics. The MSc degrees normally requires two years of study, and students may chose either a course-based option or a thesis option for the research component of the degree. The PhD program is an advanced, research-oriented course of study for which an original contribution to knowledge in the form of a dissertation is to be written and defended. The PhD program involves course work, comprehensive examinations, seminars, original research, and the defence of the dissertation.

The entrance requirement for the MSc program is a BSc degree in mathematics or statistics, with a grade average of at least 75%. Applicants to the PhD program should have an MSc degree or its equivalent, and show strong evidence of research potential.

#### Degree Requirements

**Master of Science (MSc) in Mathematics (thesis) **

Course work minimum (chosen from Mathematics graduate course offerings) * |
15 credit hours |

MATH 900 | 1 credit hour |

MATH 900 | 1 credit hour |

MATH 902 | 0 credit hours |

MATH 901 | 13 credit hour |

Total |
30 credit hours |

Master of Science (MSc) in Mathematics (course)

Course work minimum (chosen from Mathematics graduate course offerings)** |
27 credit hours |

MATH 802 | 3 credit hours |

Total |
30 credit hours |

**Master of Science (MSc) in Statistics (thesis)**

Course work minimum (chosen from Statistics graduate course offerings)* |
15 credit hours |

STAT 900 | 1 credit hour |

STAT 900 | 1 credit hour |

STAT 902 | 0 credit hours |

STAT 901 | 13 credit hour |

Total |
30 credit hours |

**Master of Science (MSc) in Statistics (Course) **

Course work minimum (chosen from Statistics graduate course offerings)** |
27 credit hours |

STAT 802 | 3 credit hours |

Total |
30 credit hours |

*Up to two courses may be from a related discipline. Thesis based students may take at most three credits in Math 890AA-ZZ and STAT 890AA-ZZ, except where permission has been granted by Department Head.

**Up to four courses may be from a related discipline. Course based students may take at most six credits in Math 890AA-ZZ and STAT 890AA-ZZ, except where permission has been granted by Department Head.

#### Degree Requirements

After a Master's degree, the PhD program in Mathematics or Statistics consists of course work, seminar, thesis proposal and original research resulting in the presentation of a thesis. All doctoral candidates are required to pass two comprehensive examinations (MATH 903/904 or STAT 903/904) that are normally taken after the completion of the course work. Doctoral candidates must also complete a seminar and a thesis proposal.

**Doctor of Philosophy (PhD) in Mathematics (after MSc)**

Course work minimum* | 15 credit hours |

MATH 900 | 1 credit hour |

MATH 902 | 0 credit hours |

MATH 903 | 1 credit hour |

MATH 904 | 1 credit hour |

MATH 905 | 1 credit hour |

MATH 901 | 41 credit hours |

Total |
60 credit hours |

*Up to two courses may be from a related discipline. PhD students may take at most 3 credits in MATH 890AA-ZZ and STAT 890AA-ZZ, except where permission has been granted by Department Head.

**Doctor of Philosophy (PhD) in Statistics (after MSc) **

Course work minimum* | 15 credit hours |

STAT 900 | 1 credit hour |

STAT 902 | 0 credit hours |

STAT 903 | 1 credit hour |

STAT 904 | 1 credit hour |

STAT 905 | 1 credit hour |

STAT 901 | 41 credit hours |

Total |
60 credit hours |

*Up to two courses may be from a related discipline. PhD students may take at most 3 credits in MATH 890AA-ZZ and STAT 890AA-ZZ, except where permission has been granted by Department Head.

#### Courses

Please note that the frequency of the offering of any particular course in the list below depends on students' needs and the availability of suitable instructors.

**MATH 800 Comprehensive Review of a Selected Topic in Mathematics (3)**

The student will conduct an in-depth literature review of a selected area in Mathematics and prepare a report pertaining to the selected topic. The topic will be chosen in consultation with the supervisor and the Department Head. A final examination (written, oral or both) will be conducted by a committee of the department.

**MATH 801 Report (3-6)**

Every Master's candidate, Report Option, will register in this course either for 6 credit hours in one semester or for 3 credit hours in each of two consecutive semesters. In this course the student, with the supervisor, prepares an expository report on some approved topic in mathematics. The candidate must present the report and be examined on it by a departmental committee chaired by the graduate program coordinator or by his or her designate.

**MATH 802 Major Essay in Mathematics (3) ** Essay on a selected topic for students in the course-based MSc program in Mathematics.

**MATH 803 Approved Summer School (3 or 6) **

This course is available to full-time Mathematics graduate students in good standing. Students will participate in a summer school offered by an approved institute. The school and credit award must be approved by the Graduate Coordinator for Mathematics and Statistics (or designee). Students may only take MATH 803 once.

**MATH 809 Foundations of Mathematics (3)**

Development of the real numbers, axioms for set theory, Gödel’s theorem.

**MATH 810 Measure of Integration (3) (Cross-listed with MATH 411)**

Integration and measure theory, spaces of continuous functions, and LP spaces.

**MATH 812 Complex Analysis (3) (Cross-listed with MATH 412)**

Riemann mapping theorem, analytic continuation, Riemann surfaces.

**MATH 813 Functional Analysis (3) (Cross-listed with MATH 485)**

Banach spaces, Banach algebras, and operator theory.

**MATH 814 Operator Algebras (3)**

C*-algebras and von Neumann algebras.

**MATH 816 Introduction to Quantum Information Theory (3)**

A first course in the mathematics of quantum information theory. Topics include information measures, quantum states and observables, qubits, entanglement, quantum channels, entropy, and measurements.

**MATH 818 Introduction to Lie Algebras and Representation Theory (3)**

The course is an introduction to the structure of finite dimensional complex semisimple Lie algebras, via root systems, as well as their finite dimensional irreducible representations, through highest weight modules.

**MATH 819 Topics in Analysis (3)**

Advanced study of selected areas of analysis.

**MATH 820 Introduction to Commutative Algebra (3) **

A first graduate course in commutative algebra. Topics include prime and maximal ideals, radicals, Nakayama's Lemma, exact sequences, tensor products, localization, Noetherian and Artinian rings and selected additional topics.

**MATH 821 Number Theory (3)**

Topics from analytic and algebraic number theory.

**MATH 822 Linear Algebra (3) (Cross-listed with MATH 422)**

Vector spaces, linear transformations and matrices, canonical forms, multilinear algebra.

**MATH 823 Algebra (3) (Cross-listed with MATH 423)**

Advanced study of group theory, Galois theory, and ring and module theory.

**MATH 824 Topics in Algebra (3)**

Advanced study of selected areas of algebra.

**MATH 825 Matrix Analysis (3)**

Matrix canonical forms, norms, spectral theory, perturbation theory, special classes of matrices.

**MATH 826 Combinatorial Matrix Theory (3)**

Matrices arising from directed and undirected graphs, and related connections between matrix theory and combinatorial mathematics.

**MATH 827 Graph Theory (3) (Cross-listed with MATH 427)**

Advanced study of selected areas of graph theory.

**MATH 828 Combinatorics (3)**

Advanced study of selected areas of combinatorics.

**MATH 831 Differential Geometry (3) (Cross-listed with MATH 431)**Differentiable manifolds, the tangent bundle, differential forms, and the general Stokes’ theorem.

**MATH 832 Topics in Differential Geometry and Topology (3)**

Advanced study of selected areas of differential geometry and topology.

**MATH 837 Introduction to Algebraic Number Theory (3) **A course on rings of integers of algebraic number fields, Dedekind rings, and Ideal Class Groups.

**MATH 838 Associative Algebras, Groups, and Representation Theory (3)**

An introductory course on the fundamental results concerning associative algebras, groups, and the representation theory of groups and algebras.

Cross-listd with MATH 438.

**MATH 839 Geometry (3)**

Advanced study of selected areas of geometry.

**MATH 841 General Topology (3) (Cross-listed with MATH 441)**

Separability of spaces, paracompactness, metrization theorems, function spaces.

**MATH 842 Algebraic Topology (3)**

Introduction to homotopy groups, and to the homology and cohomology of topological spaces.

**MATH 843 Homological Algebra (3)**A first course in homological algebra. Topics include modules over rings, chain complexes, homology, projective and injective resolutions, derived functors, abelian categories, derived categories, and selected additional topics.

**MATH 849 Topics in Topology (3)**

Advanced study of selected areas of topology.

**MATH 869 Numerical Analysis (3) (Cross-listed with MATH 461)**

Advanced study of selected areas of numerical analysis.

**MATH 881 Partial Differential Equations (3) (Cross-listed with MATH 481)**

The Cauchy problem, the Fredholm alternative in Banach spaces, the potential equation, the Dirichlet problem, the heat equation, Green’s functions, and the separation of variables.

**MATH 882 Topics in Applied Mathematics (3) (Cross-listed with MATH 482)**

Advanced study of selected topics in applied mathematics.

**MATH 885AA-ZZ Special Topics in Mathematics (3) **Lecture course in various specialized areas of Mathematics.

**MATH 890AA-ZZ Directed Readings (3)**

Directed readings in various areas of Mathematics.

**MATH 900 Seminar (1)**

Preparation and presentation of a one-hour lecture to graduate students and faculty.

**MATH 901 Research (1-15)**

Thesis research.

**MATH 902 Research Tools in Mathematics (0)**

Software tools (Maple, Mathematica, Matlab, GAP, LaTeX) and literature databases (Math Sci Net, Zentralblatt Math, Euler, preprint archives).

**MATH 903 Comprehensive Exam 1 (1)**Students must complete a comprehensive exam in one of the following topics: Matrix Theory and Linear Algebra, Commutative Algebra, Abstract Algebra, or Combinatorics and Graph Theory. It is evaluated on a pass/fail basis.

**MATH 904 Comprehensive Exam 2 (1)**Students must complete a comprehensive exam in one of the following topics: Topology, Algebraic Topology, Functional Analysis, Measure and Integration, Differential Geometry, or Probability Theory. It is evaluated on a pass/fail basis.

**MATH 905 Research Proposal (1)**Students are required to submit a written research proposal for their PhD thesis research project during its early stages. The candidate will give a seminar before the department to defend their proposal. The topic must be approved by the research supervisor and the candidates PhD committee. It is evaluated on a pass/fail basis. This course is required of all PhD students in Mathematics, and will usually be completed following the completion of MATH 903 and 904.

**STAT 800 Comprehensive Review of a Selected Topic in Statistics (3)**

The student will conduct an in-depth literature review of a selected area in Statistics and prepare a report pertaining to the selected topic. The topic will be chosen in consultation with the supervisor and the Department Head. A final examination (written, oral or both) will be conducted by a committee in the Department.

**STAT 801 Report (3-6)**

Every master’s candidate, Report Option, will register in this course either for 6 credit hours in one semester or for 3 credit hours in each of two consecutive semesters. In this course the student, with the supervisor, prepares an expository report on some approved topic in Statistics. The candidate must present the report and be examined on it by a departmental committee chaired by the graduate program coordinator or by his or her designate.

**STAT 802 Major Essay in Statistics (3) ** Essay on a selected topic for students in the course-based MSc program in Statistics.

**STAT 803 Approved Summer School (3 or 6) **

This course is available to full-time Statistics graduate students in good standing. Students will participate in a summer school offered by an approved institute. The school and credit award must be approved by the Graduate Coordinator for Mathematics and Statistics (or designee). Students may only take STAT 803 once.

**STAT 826 Advanced Survival Analysis (3) (Cross-listed with STAT 426)**

Life table, survival distributions, types of censoring, estimation of and inference for basic survival quantities, proportional hazards regression model, goodness of fit tests.

**STAT 851 Probability (3) (Cross-listed with STAT 451)**

Probability measures; distribution functions; sequences of random variables; characteristic functions; modes of convergence; convergence theorems; weak and strong laws of large numbers; Central Limit Theorem.

**STAT 852 Statistical Inference (3) (Cross-listed with STAT 452)**

Detailed theoretical development of statistical inference; statistical models; exponential families; sufficiency; completeness; properties of point estimation; testing hypothesis and confidence regions; asymptotic properties of estimators.

**STAT 853 Limit Theorems (3)**

Probability inequalities, weak limit theorems (central limit theorem, weak law of large numbers), strong limit theorems (strong law of large numbers, law of iterated logarithm).

**STAT 854 Regression Analysis (3)**

Inferences in simple and multiple regression models, model fitting, diagnostic checking and selection.

**STAT 855 Generalized Linear Models (3)**

Generalized linear models, exponential family, likelihood-based inference, analysis of contingency tables, estimation procedures.

**STAT 856 Stochastic Processes (3)**A first graduate course in stochastic processes. Topics include Markov chains, Poisson process, renewal theory, Brownian motions and selected additional topics.

**Note:**This class is cross-listed with STAT 456 and ACSC 456.

**STAT 857 Sampling Theory (3)**

Simple random sampling, systematic sampling, cluster sampling, stratified sampling, sampling with unequal probabilities, multistage and double sampling, ratio and regression estimators.

**STAT 858 Statistical Modeling of Dependence and Extremes (3)**

A first graduate course in extreme value theory and copula dependence modelling. Topics include copula models, dependence measures, order statistics, maximum domains of attraction, extreme value distribution, peak over threshold method, generalized Pareto distribution and selected additional topics. Prerequisite: STAT 851 or permission of the Department Head.

**STAT 859 Design of Experiments (3) (Cross-listed with STAT 485)**

Completely randomized designs, randomized block designs, factorial and fractional factorial designs, nested designs, fixed and random effects models.

**STAT 861 Applied Multivariate Analysis (3)**

Sample geometry, random sampling from multivariate normal distribution, inference about a mean vector, comparison of several mean vectors, multivariate linear regression models, principal components, factor analysis, discriminant analysis.

**STAT 862 Advanced Topics in Stochastic Processes (3)**

This is an advanced course in stochastic processes. Topics include: Measure theoretic probability theory, stopping theorems, Poisson process, renewal processes, Markov processes, Brownian motion, Gaussian processes, martingales, stochastic integration, and applications.

**STAT 870 Bootstrap Methods (3) **

A first course in Bootstrap techniques. Topics include bootstrap and jackknife procedures, confidence intervals, hypotheses testing, standard errors, regression models. Additional topics may vary.

Cross listed with STAT 470.

**STAT 871 Time Series Analysis (3) (Cross-listed with STAT 471)**

Stationarity, trend and seasonality, autoregressive and moving average processes, nonstationary processes, Box-Jenkins modeling method, spectrum and its estimation.

**STAT 872 Large Sample Methods (3)**

Asymptotic behavior of estimators and test statistics, asymptotic relative efficiency, large sample theory for regression models.

**STAT 881 Nonparametric Methods (3)**

Order statistics, rank statistics, sign test, Mann-Whitney test, Wilcoxon signed rank test, distribution of linear rank statistics, tests of randomness.

**STAT 882 Categorical Data Analysis (3)**

Univariate discrete responses, cross-classified responses, loglinear models, logistical regression, bayesian methods.

**STAT 885AA-ZZ Special Topics in Statistics (3) **Lecture course in various specialized areas of Statistics.

**STAT 890AA-ZZ Directed Readings (3)**

Directed readings in various areas of Statistics.

**STAT 900 Seminar (1)**

Preparation and presentation of a one-hour lecture to graduate students and faculty.

**STAT 901 Research (1-15)**

Thesis research.

**STAT 902 Research Tools in Statistics (0)**

Software tools (SAS, Maple, Mathematica, Matlab, LaTeX) and literature databases (Math Sci Net, STM-Z, preprint archives).

**STAT 903 Comprehensive Exam 1 - Probability Theory (1)**Students must complete a comprehensive exam in Probability Theory. The exam will also include one of the following elective topics: Stochastic Processes, Dependence and Extremes, Limit Theorems, or Measure and Integration. It is evaluated on a pass/fail basis.

**STAT 904 Comprehensive Exam 2 - Statistical Inference (1)**Students must complete a comprehensive exam in Statistical Inference. The exam will also include one of the following elective topics: Generalized Linear models, Survivial Analysis, Experimental Design, Time Series Analysis, Linear Models, or Sampling Theory. It is evaluated on a pass/fail basis.

**STAT 905 Research Proposal (1)**Students are required to submit a written research proposal for their PhD thesis research project during its early stages. The candidate will give a seminar before the department to defend their proposal. The topic must be approved by the research supervisor and the candidates PhD committee. It is evaluated on a pass/fail basis. This course is required of all PhD students in Statistics, and will usually be completed following the completion of STAT 903 and 904.