# Mathematics and Statistics

#### Staff

Department Head: Shaun Fallat, PhD

Graduate Co-ordinator: Karen Meagher, PhD

Faculty Listing: http://www.uregina.ca/science/mathstat/faculty-staff/index.html

#### 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's Program

**MSc in Mathematics (Thesis Option) ( effective 201930)**

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

Seminar (MATH 900) | 2 credit hours |

Thesis research (MATH 901) | 13 credit hours |

Research Methods (MATH 902) | 0 credit hour |

Total | 30 credit hours |

**MSc in Mathematics (Course-Based Option) ( effective 201930)**

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

Major Essay (MATH 802) | 3 credit hours |

Total | 30 credit hours |

**MSc in Statistics (Thesis Option) ( effective 201930)**

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

Seminar (STAT 900) | 2 credit hours |

Thesis research (STAT 901) | 13 credit hours |

Research Methods (STAT 902) | 0 credit hour |

Total | 30 credit hours |

**MSc in Statistics (Course-Based Option) ( effective 201930)**

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

Major Essay (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 3 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 6 credits in Math 890AA-ZZ and STAT 890AA-ZZ, except where permission has been granted by Department Head.

#### Degree Requirements

#### Doctoral Program (effective 202030)

After a Master's degree, the PhD program in Mathematics or Statistics consists of course work, seminars, and original research resulting in the presentation of a thesis. All doctoral candidates are required to pass a comprehensive examination that is normally taken after the completion of the course work.

**PhD in Mathematics after MSc **

Course work minimum* | 15 credit hours |

Seminar (MATH 900) | 2 credit hours |

Research Methods (MATH 902) | 0 credit hour |

Research (MATH 901) | 43 credit hours |

Total | 60 credit hours |

**PhD in Statistics after MSc **

Course work minimum* | 15 credit hours |

Seminar (STAT 900) | 2 credit hours |

Research Methods (STAT 902) | 0 credit hour |

Research (STAT 901) | 43 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.

#### Course Descriptions

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 (Variable credit 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 (411) Real Analysis (3)**

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

**MATH 812 (412) Complex Analysis (3)**

Riemann mapping theorem, analytic continuation, Riemann surfaces.

**MATH 813 (485) Functional Analysis (3)**

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) effective 202010**

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 (422) Linear Algebra (3)**

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

**MATH 823 (423) Algebra (3)**

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 (427) Graph Theory (3)**

Advanced study of selected areas of graph theory.

**MATH 828 Combinatorics (3)**

Advanced study of selected areas of combinatorics.

**MATH 831 (431) Differential Geometry (3)**

Differentiable manifolds, Lie groups, differential forms.

**MATH 832 (432) 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) effective 201930**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 (441) General Topology (3)**

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 849 Topics in Topology (3)**

Advanced study of selected areas of topology.

**MATH 869 (461) Numerical Analysis (3)**

Advanced study of selected areas of numerical analysis.

**MATH 881 (481) Partial Differential Equations (3)**

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 (482) Topics in Applied Mathematics (3)**

Advanced study of selected topics in applied mathematics.

**MATH 885AA-ZZ Special Topics in Mathematics (3) effective 201930**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 (Variable credit 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).

**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 (Variable credit 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 (426) Advanced Survival Analysis (3)**

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 (451) Probability (3)**

Measure spaces, independence, expectations, convergence theorems, distribution functions and characteristic functions.

**STAT 852 (452) Statistical Inference (3) effective 202010**

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 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 (485) Design of Experiments (3)**

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 Stochastic Processes (3)**

Poisson process, renewal processes, Markov processes, Brownian motion, Gaussian processes, random walks and martingales.

**STAT 870 – Bootstrap Methods (3) effective 201930**

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 (471) Time Series Analysis (3)**

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) effective 201930**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 (Variable credit 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).