#
Mathematics and Statistics

### Ottawa-Carleton Joint Program

The University of Ottawa offers a rich academic environment to study mathematics and statistics under the supervision of professors who have gained an international reputation for their research. Most major fields of research in mathematics and statistics are represented within the Department of mathematics and Statistics. Moreover, the Department is a participating unit in the master's level collaborative programs in bioinformatics and in biostatistics. Additional information about the Department and its programs is posted on the departmental website at www.mathstat.uottawa.ca.

The Department offers a PhD program and an MSc program in mathematics and statistics. There are three options for the MSc program: MSc with thesis, MSc with project or MSc by coursework (i.e. with courses only). The MSc by coursework in the field of probability and statistics and the MSc with project in all fields can be completed in one year by taking courses over three consecutive terms (sessions).

Since 1984, the graduate programs in mathematics and statistics have been under the umbrella of the Ottawa-Carleton Institute of Mathematics and Statistics (OCIMS). The OCIMS consists of the School of Mathematics and Statistics at Carleton University and the Department of Mathematics and Statistics at the University of Ottawa. The two units have pooled together their resources to offer each year a large selection of graduate courses.

The programs are governed by the graduate studies regulations of the two universities. Graduate courses are generally offered in English. However, research activities can be conducted in English or French or both depending on the language used by the professor and the members of the research group.

In accordance with the University of Ottawa regulation, students have a right to submit their work, thesis, and exams in French or in English.

## Programs

- Master of Science Mathematics and Statistics Concentration in Mathematics
- Master of Science Mathematics and Statistics Concentration in Statistics
- Master of Science Mathematics and Statistics Specialization in Bioinformatics
- Master of Science Mathematics and Statistics Specialization in Biostatistics
- Doctorate in Philosophy Mathematics and Statistics

## Professors

- Alvo, Mayer, Full Professor
*Nonparametric statistics; sequential analysis* - Aris-Brosou, Stéphane, Cross-appointment
*Computational molecular evolution; population genetics* - Balan, Raluca, Full Professor
*Bayesian nonparametric statistics; generalized estimation equations; stochastic partial differential equations, multi-parameter (or set-indexed) processes; Markov processes; processes with independent increments limit theorems for dependent sequences; strong approximations; self-normalized sequences* - Bergeron, Pierre-Jérôme, Adjunct Professor
*Survival analysis, truncated and censored data, biased sampling, computational statistics, recurrent event data, clinical trials* - Bickel, David, Cross-appointment
*Statistical bioinformatics: gene expression data analysis; molecular network reconstruction; model validation methodology; Bayesian and empirical Bayes inference; machine learning algorithms; and Monte Carlo simulation* - Blute, Richard, Full Professor
*Logic; category theory* - Bourgault, Yves, Full Professor
*Computational fluid dynamics; numerical methods; finite element; continuum mechanics modelling* - Boyd, Sylvia, Cross-appointment
*Combinatorial optimization; network design; integer programming; polyhedral combinatorics, travelling salesman problem; analysis and design of algorithms* - Broadbent, Anne, Assistant Professor
*Quantum information science; Cryptography* - Burkett, Kelly, Assistant Professor
*Statistical Genetics, Genetic Epidemiology, Biostatistics* - Collins, Benoit, Associate Professor
*Random matrices; free probability; non-commutative probability theory; operator algebras; asymptotic representation of groups; matrix integrals* - Daigle, Daniel, Full Professor
*Algebraic geometry; commutative algebra; geometry of affine spaces* - Dionne, Benoit, Associate Professor
*Singularity & groups in bifurcation theory* - Felty, Amy, Cross-appointment
*Theorem proving; automated deduction; formal methods in software engineering; computational logic* - Fiorilli, Daniel, Assistant Professor
- Fraser, Maia, Assistant Professor
- Giordano, Thierry, Full Professor
*Operator algebras and dynamical systems* - Guo, Hongbin, Adjunct Professor
- Handelman, David, Full Professor
*K-theory; operator algebras; ergodic theory; ring theory; etc* - Hofstra, Pieter, Associate Professor
*Logic and foundations of computing* - Ivanoff, B. Gail, Emeritus Professor
*Theory and applications of set-indexed and multiparameter stochastic processes; including martingale theory; Markov processes; renewal theory; survival analysis; stochastic orders* - Jessup, Barry, Full Professor
*Rational homotopy; Lie algebra cohomology* - Kaimanovich, Vadim, Full Professor
*Ergodic theory; geometry; probability theory and operator algebras* - Kousha, Termeh, Replacement Professor
*Application of statistics in environmental health sciences, Random matrix theory and Applied probability theory* - Kulik, Rafal, Associate Professor
*Limit theorems for weakly and strongly dependent random variables (empirical and quantile processes, trimmed sums, extremes, nonparametric statistics); with applications to time series (linear, GARCH and LARCH processes); wavelet methods* - Leblanc, Victor, Full Professor
*Differential equations; dynamical systems; bifurcation theory* - Levy, Jason, Assistant Professor
*Systems biology, cell dynamics, representation theory, automorphic forms* - Longtin, Andre, Cross-appointment
*Nonlinear dynamics, stochastic dynamics, biological physics and mathematical biology.* - Lutscher, Frithjof, Associate Professor
*Differential equations; dynamical systems; mathematical biology* - McDonald, David, Emeritus Professor
*Applied probability* - Mesfioui, Mhamed, Adjunct Professor
- Moura, Lucia, Associate Professor
*Combinatorial algorithms, combinatorial designs and their applications, combinatorial optimization* - Neher, Erhard, Emeritus Professor
*Lie algebras, Jordan algebras and groups* - Nevins, Monica, Full Professor
*Representation theory of p-adic Lie groups / coding theory* - Newman, Michael, Associate Professor
*Graph theory* - Novruzi, Arian, Associate Professor
*Partial differential equations; shape optimization; numerical analysis; modeling* - Parent, Paul-Eugène, Associate Professor
*Algebraic topology; homotopy theory* - Pestov, Vladimir, Full Professor
*Topological transformation groups; geometry of large dimensions* - Roy, Damien, Full Professor
*Transcendental number theory* - Sajna, Mateja, Associate Professor
*Graph theory* - Salmasian, Hadi, Associate Professor
*Unitary representations of Lie groups, quantization, orbit method.; combinatorial problems related to representation theory; infinite-dimensional Lie algebras and their representations* - Sankoff, David, Full Professor
*Mathematical genomics* - Savage, Alistair, Associate Professor
*Geometric representation theory; Lie algebras; quantum groups; crystal bases* - Schiopu-Kratina, Ioana, Adjunct Professor
*Statistics and Probability* - Schmah, Tanya, Assistant Professor
- Scott, Philip, Full Professor
*Logic; category theory; theoretical computer science* - Sebbar, Abdellah, Full Professor
*Number theory; quantum groups* - Selinger, Peter, Adjunct Professor
*Mathematical logic; category theory* - Smith, Aaron, Assistant Professor
*Statistics and Probability* - Smith, Robert
- Théberge, François, Adjunct Professor
*Applied probability & statistics* - Walsh, Gary, Adjunct Professor
*Number theory; diophantine equations* - Xu, Chen, Assistant Professor
- Zarepour, Mahmoud, Full Professor
*Resampling; nonparametric bayesian inference; infinite variance random variables* - Zaynullin, Kirill, Full Professor
*Motives and algebraic cycles on homogeneous varieties; invariant theory of the Weyl group and Algebraic cobordism; the Grothendieck-Serre conjecture, the Purity and the Gersten conjecture*

## AdmissionSpecific requirements

### Master's

Admission to the graduate programs in mathematics and statistics is governed by the general regulations of the Ottawa-Carleton Institute of Mathematics and Statistics (OCIMS) and by the graduate studies regulations of the University of Ottawa.

Applicants for admission must:

- Hold a bachelor's degree with a specialization or a major in mathematics and statistics (or equivalent) with a minimum average of 75% (B+).
- Demonstrate a good academic performance in previous studies as shown by official transcripts, research reports, abstracts or any other documents demonstrating research skills.
- Provide at least two confidential letters of recommendation from professors who have known the applicant and are familiar with the student work.
- Provide a statement of purpose indicating the career goals and the interests in the proposed research area.
- Identify at least one professor who is willing and available to act as thesis supervisor.

NOTE: The choice of supervisor will determine the primary campus location of the student. It will also determine which university awards the degree.

#### Collaborative Programs

The Department of Mathematics and Statistics is a participating unit in the collaborative programs in Bioinformatics and in Biostatistics. Students should indicate in their initial application for admission that they wish to be accepted into the collaborative program. For further details, see the description of these programs posted on the FGPS website.

### Doctorate

Admission to the graduate program in mathematics and statistics is governed by the general regulations of the Ottawa-Carleton Institute of Mathematics and Statistics (OCIMS) and by the graduate studies regulations of the University of Ottawa.

Applicants for admission must:

- Hold a master's in mathematics and statistics (or equivalent) with a minimum average of 75% (B+).
- Demonstrate a good academic performance in as shown by official transcripts, research reports, abstracts or any other documents demonstrating research skills.
- Provide at least two confidential letters of recommendation from professors who have known the applicant and are familiar with the student work.
- Provide a statement of purpose indicating the career goals and the interests in the proposed research area.
- Identify at least one professor who is willing and available to act as thesis supervisor.

NOTE: The choice of supervisor will determine the primary campus location of the student. It will also determine which university awards the degree.

#### Transfer from Master’s to PhD Program

Outstanding students enrolled in the MSc program may be allowed to transfer to the PhD program without being required to write a master’s thesis provided they meet the following conditions:

- Completion of two graduate courses (six units) with a grade of A- or better in each.
- Satisfactory progress in the research program.
- Written recommendation by the supervisor and the advisory committee;
- Approval by the graduate studies committee.

The transfer must take place within sixteen months of initial enrollment in the master’s. Following the transfer, all of the requirements of the doctoral program must be met: a total of 18 course units (including the units completed at the master's); the comprehensive exam; and a thesis.

## Program Requirements

### Master's

#### MSc in Mathematics and Statistics

At least 50% of the units must be from the student's home university. The project counts for 6 units and the thesis counts for 12 units.

The student must enroll for one of the following two concentrations.

**Concentration in mathematics**

More than 50% of the units must come from the list of courses in mathematics. See list below.

**Concentration in statistics**

More than 50% of the units must come from the list of courses in statistics. See list below.

#### MSc with thesis

- 12 units at the 5000 level or more in mathematics or in other related disciplines approved by the Department of Mathematics and Statistics.
- Presentation and successful defense of a thesis (MAT7999) based on an original research carried out under the direct supervision of a faculty member of the Department.

#### MSc with project

- 18 units at the 5000 level or more in mathematics or in other related disciplines approved by the Department of Mathematics and Statistics.
- Project (MAT6997).

#### MSc by coursework

- 24 units at the 5000 level or above in mathematics or in related disciplines approved by the Department of Mathematics and Statistics.

**Mathematics courses**

MAT5105 (MATH5818) DISCRETE APPLIED MATHEMATICS I: GRAPH THEORY (3 units)

MAT5107 (MATH 5819) DISCRETE APPLIED MATHEMATICS II: COMBINATORIAL ENUMERATION (3 units)

MAT5121 (MATH 5009) INTRODUCTION TO HILBERT SPACE (3 units)

MAT5122 (MATH 5003) BANACH ALGEBRAS (3 units)

MAT5125 (MATH 5007) REAL ANALYSIS I (Measure theory and integration) (3 units)

MAT5126 (MATH 5008) REAL ANALYSIS II (Functional analysis) (3 units)

MAT5127 (MATH 5005) COMPLEX ANALYSIS (3 units)

MAT5131 (MATH 5405) ORDINARY DIFFERENTIAL EQUATIONS (3 units)

MAT5133 (MATH 5406) PARTIAL DIFFERENTIAL EQUATIONS (3 units)

MAT5134 (MATH 5407) TOPICS IN DIFFERENTIAL EQUATIONS (3 units)

MAT5141 (MATH 5107) ALGEBRA I (3 units)

MAT5142 (MATH 5109) ALGEBRA II (3 units)

MAT5143 (MATH 5104) LIE ALGEBRAS (3 units)

MAT5144 (MATH 5001) COMMUTATIVE ALGEBRA (3 units)

MAT5145 (MATH 5106) GROUP THEORY (3 units)

MAT5146 (MATH 5103) RINGS AND MODULES (3 units)

MAT5147 (MATH 5108) HOMOLOGICAL ALGEBRA AND CATEGORY THEORY (3 units)

MAT5148 (MATH 5102) GROUP REPRESENTATIONS AND APPLICATIONS (3 units)

MAT5149 (MATH 5002) ALGEBRAIC GEOMETRY (3 units)

MAT5150 (MATH 5201) TOPICS IN GEOMETRY (3 units)

MAT5151 (MATH 5205) TOPOLOGY I (3 units)

MAT5152 (MATH 5206) TOPOLOGY II (3 units)

MAT5155 (MATH 5208) DIFFERENTIABLE MANIFOLDS (3 units)

MAT5158 (MATH 6104) LIE GROUPS (3 units)

MAT5160 (MATH 5300) MATHEMATICAL CRYPTOGRAPHY (3 units)

MAT5161 (MATH 5301) MATHEMATICAL LOGIC (3 units)

MAT5162 (MATH 6807) MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (3 units)

MAT5163 (MATH 5305) ANALYTIC NUMBER THEORY (3 units)

MAT5164 (MATH 5306) ALGEBRAIC NUMBER THEORY (3 units)

MAT5165 (MATH 5605) THEORY OF AUTOMATA (3 units)

MAT5167 (MATH/COMP 5807) FORMAL LANGUAGE AND SYNTAX ANALYSIS (3 units)

MAT5168 (MATH 5202) HOMOLOGY THEORY (3 units)

MAT5169 (MATH 5207) FOUNDATIONS OF GEOMETRY (3 units)

MAT5185 (MATH 5408) ASYMPTOTIC METHODS OF APPLIED MATHEMATICS (3 units)

MAT5187 (MATH 5403) TOPICS IN APPLIED MATHEMATICS (3 units)

MAT5301 (MATH 5609) TOPICS IN COMBINATORIAL MATHEMATICS (3 units)

MAT5303 (MATH 5801) LINEAR OPTIMIZATION (3 units)

MAT5304 (MATH 5803) NONLINEAR OPTIMIZATION (3 units)

MAT5106 (MATH5808) COMBINATORIAL OPTIMIZATION (3 units)

MAT5307 (MATH 5804) TOPICS IN OPERATIONS RESEARCH (3 units)

MAT5308 (MATH 5805) TOPICS IN ALGORITHM DESIGN (3 units)

MAT5309 (MATH 6002) HARMONIC ANALYSIS ON GROUPS (3 units)

MAT5312 (MATH 6201) TOPICS IN TOPOLOGY (3 units)

MAT5319 (MATH 6507) TOPICS IN PROBABILITY AND STATISTICS (3 units)

MAT5324 (MATH 5607) GAME THEORY (3 units)

MAT5325 (MATH 5802) TOPICS IN INFORMATION AND SYSTEMS SCIENCE (3 units)

MAT5326 (MATH 6008) TOPICS IN ANALYSIS (3 units)

MAT5327 (MATH 6101) TOPICS IN ALGEBRA (3 units)

MAT5328 (MATH 6008) TOPICS IN ANALYSIS (3 units)

MAT5329 (MATH 6009) TOPICS IN ANALYSIS (3 units)

MAT5330 (MATH 6102) TOPICS IN ALGEBRA (3 units)

MAT5331 (MATH 6103) TOPICS IN ALGEBRA (3 units)

MAT5341 (MATH5821) QUANTUM COMPUTING (3 units)

MAT5343 MATHEMATICAL ASPECTS OF WAVELETS AND DIGITAL SIGNAL PROCESSING (3 units)

MAT5361 (MATH 6806) TOPICS IN MATHEMATICAL LOGIC (3 units)

MAT5180 (MATH 5806) NUMERICAL ANALYSIS FOR DIFFERENTIAL EQUATIONS (3 units)

**Statistics courses**

MAT5175 (STAT 5506) ROBUST STATISTICAL INFERENCE (3 units)

MAT5181 (STAT 5703) DATA MINING I (3 units)

MAT5182 (STAT 5702) MODERN APPLIED / COMPUTATIONAL STATISTICS (3 units)

MAT5192 (STAT 5502) SAMPLING THEORY AND METHODS (3 units)

MAT5193 (STAT 5503) LINEAR MODELS (3 units)

MAT5195 (STAT 5505) DESIGN OF EXPERIMENTS (3 units)

MAT5196 (STAT 5509) MULTIVARIATE ANALYSIS (3 units)

MAT5313 (MATH 6507) TOPICS IN PROBABILITY AND STATISTICS (3 units)

MAT5314 (MATH 6508) TOPICS IN PROBABILITY AND STATISTICS (3 units)

MAT5315 ADVANCED DESIGN OF SURVEYS (3 units)

MAT5317 (STAT 5602) ANALYSIS OF CATEGORICAL DATA (3 units)

MAT5318 (STAT 5603) RELIABILITY AND SURVIVAL ANALYSIS (3 units)

MAT5375 (STAT 5610) MATHEMATICAL STATISTICS (3 units)

MAT5176 (STAT 5507) ADVANCED STATISTICAL INFERENCE (3 units)

MAT5177 (STAT 5500) MULTIVARIATE NORMAL THEORY (3 units)

MAT5992 (STAT 5902) SEMINAR IN BIOSTATISTICS (3 units)

**Mathematics and statistics courses**

MAT5170 (STAT 5708) PROBABILITY THEORY I (3 units)

MAT5171 (MATH 5709) PROBABILITY THEORY II (3 units)

MAT5172 (STAT 5508) TOPICS IN STOCHASTIC PROCESSES (3 units)

MAT5173 (STAT 5604) STOCHASTIC ANALYSIS (3 units)

MAT5174 (STAT 5704) NETWORK PERFORMANCE (3 units)

MAT5190 (STAT 5600) MATHEMATICAL STATISTICS I (3 units)

MAT5191 (STAT 5501) MATHEMATICAL STATISTICS II (3 units)

MAT5194 (STAT 5504) STOCHASTIC PROCESSES AND TIME SERIES ANALYSIS (3 units)

MAT5197 (STAT 5601) STOCHASTIC OPTIMIZATION (3 units)

MAT5198 (MATH 5701) STOCHASTIC MODELS (3 units)

MAT5990 (MATH 5900) SÉMINAIRE / SEMINAR (3 units)

MAT5991 (MATH 5901) TRAVAUX DIRIGÉS / DIRECTED STUDIES (3 units)

MAT5996 (MATH 5906) STAGE DE RECHERCHE / RESEARCH INTERNSHIP (3 units)

MAT6990 (MATH 6900) SÉMINAIRE / SEMINAR (3 units)

MAT6991 (MATH 6901) TRAVAUX DIRIGÉS / DIRECTED STUDIES (3 units)

MAT6997 (MATH5910) PROJET EN MATHÉMATIQUES ET STATISTIQUE / PROJECT IN MATHEMATICS AND STATISTICS (6 units)

MAT7999 THÈSE DE MAÎTRISE / MSc THESIS

MAT9998 EXAMEN DE SYNTHÈSE / COMPREHENSIVE EXAMINATION

MAT9999 (MATH 6909) THÈSE DE DOCTORAT / PhD THESIS

#### Collaborative program in Bioinformatics

The student is responsible for fulfilling both the participating unit requirements for the primary program and the requirements for the collaborative program.

The requirements specific to the collaborative program are as follows:

- 3 compulsory units in bioinformatics (BNF5106 / BIO5106).
- Enrollment in the seminar course in bioinformatics (BNF6100), which involves a written report, the presentation of a seminar, and regular attendance at departmental seminars.
- Presentation and defence of a research thesis on a topic in bioinformatics based on original research carried out under the supervision of a faculty member participating in the bioinformatics collaborative program.

The primary program may require students to take additional courses, depending on their backgrounds.

#### Collaborative program in Biostatistics

The student is responsible for fulfilling both the participating unit requirements for the primary program and the requirements for the collaborative program.

The following requirements must be met:

- 21 units including EPI5240, EPI5241, EPI6178, EPI6278, MAT5190, MAT5191 and another 3 units of graduate course in mathematics and statistics.
- Enrollment in the seminar course in biostatistics MAT5992 (STAT5902), which involves the presentation of a seminar, and regular attendance at the seminars presented by the Department.
- Presentation and defence of a thesis in biostatistics (MAT7999) based on an original research carried out under the supervision of a faculty member participating in the biostatistics collaborative program.

#### Collaborative program in Biostatistics (by coursework)

The student is responsible for fulfilling both the participating unit requirements for the primary program and the requirements for the collaborative program.

The following requirements must be met:

- 27 units including EPI5240, EPI5241, EPI6178, EPI6278, MAT5190, MAT5191 and three graduate courses of 3 units each in mathematics and statistics.
- Enrollment in the seminar course in biostatistics MAT5992 (STAT5902), which involves the presentation of a seminar, and regular attendance at the seminars presented by the Department of Mathematics and Statistics.

#### Transfer from master’s to PhD

Students enrolled in the MSc program may be allowed to transfer to the PhD program without being required to write a master’s thesis. For additional information, please consult the “Admission” section of the PhD program.

#### Duration of the program

The requirements of the program are usually fulfilled within two years. The maximum time permitted is four years from the date of initial enrollment.

#### Residence

All students admitted full-time must complete a minimum of three terms (sessions) of full-time enrollment.

#### Minimum standards

The passing grade in all courses is B. Students who fail two courses (equivalent to 6 units), or whose progress is deemed unsatisfactory are required to withdraw.

### Doctorate

**The following requirements must be met:**

- 18 units at the 5000 level or above in mathematics and statistics or in related disciplines approved by the Department of Mathematics and Statistics.
- Successful completion of a comprehensive examination (MAT9998) within eighteen months of the initial admission into the program.
- Presentation and successful defense of a thesis (MAT9999) based on an original research carried out under the direct supervision of a faculty member of the Institute.

The department may require students to take additional courses depending on their backgrounds.

#### Duration of the program

The requirements of the program are usually fulfilled within four years. The maximum time permitted is six years from the date of initial enrollment in the program, or seven years in the case of the students transferring from the master’s to the doctorate.

#### Residence

All students must complete a minimum of six terms (sessions) of full-time enrollment. In the case of transfer students, the residency period is nine full-time terms (sessions) from the initial enrollment in the program.

#### Minimum standards

The passing grade in all courses is B. Students who fail two courses (equivalent to 6 units), or the thesis proposal, or the comprehensive exam, or whose research progress is deemed unsatisfactory are required to withdraw.

#### Thesis Advisory Committee

During the first term (session) of the program, a thesis advisory committee (TAC) is formed for the candidate. The Committee’s membership will be determined by the specific interests of the candidate. It will be composed of the supervisor and 2-3 additional professors. At least one member of the thesis committee, in addition to the supervisor, must be from the Faculty of Science. The TAC is responsible for guiding the student throughout the program, including course selection, the comprehensive examination, and thesis defense.

A meeting between the student and the Thesis Advisory Committee will take place at least once per term (session). The thesis examining board may include members who are not part of the TAC.

## Courses

Course codes in parentheses are for Carleton University. A 3-unit course at the University of Ottawa is equivalent to a 0.5-unit course at Carleton University.

**MAT5105 (MATH5818) DISCRETE APPLIED MATHEMATICS I: GRAPH THEORY**(3 units)

Paths and cycles, trees, connectivity, Euler tours and Hamilton cycles, edge colouring, independent sets and cliques, vertex colouring, planar graphs, directed graphs. Selected topics from one or more of the following areas: algebraic graph theory, topological theory, random graphs.

**MAT5106 (MATH5808) COMBINATORIAL OPTIMIZATION**(3 units)

Network flow theory and related material. Topics will include shortest paths, minimum spanning trees, maximum flows, minimum cost flows. Optimal matching in bipartite graphs.

**MAT5107 (MATH 5819) DISCRETE APPLIED MATHEMATICS II: COMBINATORIAL ENUMERATION**(3 units)

Ordinary and exponential generating functions; product formulas; permutations; partitions; rooted trees; cycle index; WZ method. Lagrange Inversions; singularity analysis of generating functions and asymptotics. Selected topics from one or more of the following areas: random graphs, random combinatorial structures, hypergeometric functions.

**MAT5121 (MATH 5009) INTRODUCTION TO HILBERT SPACE**(3 units)

**MAT5122 (MATH 5003) BANACH ALGEBRAS**(3 units)

**MAT5125 (MATH 5007) REAL ANALYSIS I (Measure theory and integration)**(3 units)

General measure and integral, Lebesgue measure and integration on R, Fubini's theorem, Lebesgue-Radon-Nikodym theorem, absolute continuity and differentiation, Lp-Spaces. Selected topics such as: Daniell-Stone theory. Prerequisite(s): Permission of the Program Director.

*Prerequisites: MAT3125 (MATH 3001 and MATH 3002).*

**MAT5126 (MATH 5008) REAL ANALYSIS II (Functional analysis)**(3 units)

Banach and Hilbert spaces, bounded linear operators, dual spaces. Topics selected from: weak- and weak-topologies, Alaoglu's theorem, compact operators, differential calculus in Banach spaces, Riesz representation theorems.

*Prerequisite: MAT 5125 (MATH 5007).*

**MAT5127 (MATH 5005) COMPLEX ANALYSIS**(3 units)

**MAT5131 (MATH 5405) ORDINARY DIFFERENTIAL EQUATIONS**(3 units)

**MAT5133 (MATH 5406) PARTIAL DIFFERENTIAL EQUATIONS**(3 units)

First-order equations, characteristics method, classification of second-order equations, separation of variables, Green's functions. Lp and Soboloev spaces, distributions, variational formulation and weak solutions, Lax-Milgram theorem, Galerkin approximation. Parabolic PDes. Wave equations, hyperbolic systems, nonlinear PDes, reaction diffusion equations, infinite-dimensional dynamical systems, regularity.

*Prerequisite: An intermediate level course on Ordinary Differential Equations such as MAT3130 Dynamical Systems or equivalent, or the permission of the School or Department.*

**MAT5134 (MATH 5407) TOPICS IN DIFFERENTIAL EQUATIONS**(3 units)

**MAT5141 (MATH 5107) ALGEBRA I**(3 units)

Groups, Sylow subgroups, finitely generated abelian groups. Rings, field of fractions, principal ideal domains, modules. Polynomial algebra, Euclidean algorithm, unique factorization. Prerequisites: MAT 3141 and MAT 3143.

*Prerequisites: MAT3141 and MAT3143.*

**MAT5142 (MATH 5109) ALGEBRA II**(3 units)

Field theory, algebraic and transcendental extensions, finite fields, Galois groups. Modules over principal ideal domains, decomposition of a linear transformation, Jordan normal form.

*Prerequisite: MAT 5141 (MATH 5107).*

**MAT5143 (MATH 5104) LIE ALGEBRAS**(3 units)

**MAT5144 (MATH 5001) COMMUTATIVE ALGEBRA**(3 units)

Prime spectrum of a commutative ring (as a topological space); localization of rings and modules; tensor product of modules and algebras; Hilbert’s Nullstellensatz and consequences for finitely generated algebras; Krull dimension of a ring; integral dependence, going-up, going-down; Noether Normalization Lemma and dimension theory for finitely generated algebras over a field; noetherian rings and Hilbert Basis Theorem; introduction to affine algebraic varieties and their morphisms.

*Prerequisite: MAT3143*

**MAT5145 (MATH 5106) GROUP THEORY**(3 units)

**MAT5146 (MATH 5103) RINGS AND MODULES**(3 units)

**MAT5147 (MATH 5108) HOMOLOGICAL ALGEBRA AND CATEGORY THEORY**(3 units)

**MAT5148 (MATH 5102) GROUP REPRESENTATIONS AND APPLICATIONS**(3 units)

**MAT5149 (MATH 5002) ALGEBRAIC GEOMETRY**(3 units)

Brief overview of commutative algebra, Hilbert’s Nullstellensatz, algebraic sets, and Zariski topology. Affine and projective varieties over algebraically closed fields. Regular functions and rational maps. Additional topics chosen from: the relation of varieties over complex numbers to complex analytic manifolds, genus, divisors, line bundles, Riemann-Roch Theorem, Bézout’s Theorem.

*Prerequisite: MAT3143*

**MAT5150 (MATH 5201) TOPICS IN GEOMETRY**(3 units)

**MAT5151 (MATH 5205) TOPOLOGY I**(3 units)

Topological spaces, product and identification topologies, countability and separation axioms, compactness, connectedness, homotopy, fundamental group, net and filter convergence.

*Prerequisite: MAT 3153 (MATH 3001).*

**MAT5152 (MATH 5206) TOPOLOGY II**(3 units)

Covering spaces, homology via the Eilenberg-Steenrod axioms, applications, construction of a homology functor. Prerequisites: MAT 3143 and MAT 5151 (MATH 3100 and MATH 5205).

*Prerequisites: MAT3143 and MAT5151 (MATH 3100 and MATH 5205).*

**MAT5155 (MATH 5208) DIFFERENTIABLE MANIFOLDS**(3 units)

**MAT5158 (MATH 6104) LIE GROUPS**(3 units)

**MAT5160 (MATH 5300) MATHEMATICAL CRYPTOGRAPHY**(3 units)

Analysis of cryptographic methods used in authentication and data protection, with particular attention to the underlying mathematics, e.g. Algebraic Geometry, Number Theory, and Finite Fields. Advanced topics on Public-Key Cryptography: RSA and integer factorization, Diffie-Hellman, discrete logarithms, elliptic curves. Topics in current research. Prerequisites: undergraduate honours algebra, including group theory and finite fields.

*Prerequisite: undergraduate honours algebra, including group theory and finite fields.*

**MAT5161 (MATH 5301) MATHEMATICAL LOGIC**(3 units)

A basic graduate course in mathematical logic. Propositional and Predicate logic, Proof theory, Gentzen's Cut-Elimination, Completeness, Compactness, Henkin models, model theory, arithmetic and undecidability. Special Topics (time permitting) depending on interests of instructor and audience. Prerequisites: honours undergraduate algebra, analysis and topology (or permission of the instructor).

*Prerequisite: Honours undergraduate algebra, analysis and topology (or permission of the instructor).*

**MAT5162 (MATH 6807) MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE**(3 units)

Foundations of functional languages, lambda calculi (typed, polymorphically typed, untyped), Curry-Howard Isomorphism, proofs-as-programs, normalization and rewriting theory, operational semantics, type assignment, introduction to denotational semantics of programs, fixed-point programming. Topics chosen from: denotational semantics for lambda calculi, models of programming languages, complexity theory and logic of computation, models of concurrent and distributed systems, etc. Prerequisites: honours undergraduate algebra and either topology or analysis. Some acquaintance with Logic useful.

*Prerequisite: Honours undergraduate algebra and either topology or analysis. Some acquaintance with Logic useful.*

**MAT5163 (MATH 5305) ANALYTIC NUMBER THEORY**(3 units)

**MAT5164 (MATH 5306) ALGEBRAIC NUMBER THEORY**(3 units)

**MAT5165 (MATH 5605) THEORY OF AUTOMATA**(3 units)

**MAT5167 (MATH/COMP 5807) FORMAL LANGUAGE AND SYNTAX ANALYSIS**(3 units)

**MAT5168 (MATH 5202) HOMOLOGY THEORY**(3 units)

**MAT5169 (MATH 5207) FOUNDATIONS OF GEOMETRY**(3 units)

**MAT5170 (STAT 5708) PROBABILITY THEORY I**(3 units)

Probability spaces, random variables, expected values as integrals, joint distributions, independence and product measures, cumulative distribution functions and extensions of probability measures, Borel-Cantelli lemmas, convergence concepts, independent identically distributed sequences of random variables. Prerequisites: Permission of Program Director.

*Prerequisites: MAT3125 and MAT3172 (MATH 3001, MATH 3002 and MATH 3500).*

**MAT5171 (MATH 5709) PROBABILITY THEORY II**(3 units)

Laws of large numbers, characteristic functions, central limit theorem, conditional probabilities and expectation, basic properties and convergence theorems for martingales, introduction to Brownian motion.

*Prerequisite: MAT 5170 (STAT 5708).*

**MAT5172 (STAT 5508) TOPICS IN STOCHASTIC PROCESSES**(3 units)

**MAT5173 (STAT 5604) STOCHASTIC ANALYSIS**(3 units)

Brownian motion, continuous martingales and stochastic integration.

**MAT5174 (STAT 5704) NETWORK PERFORMANCE**(3 units)

The course will focus on advanced techniques in performance evaluation of large complex networks. Topic may include classical queueing theory and simulation analysis; models of packet networks; loss and delay systems; blocking probabilities. Prerequisites: Some familiarity with probability and stochastic processes and queueing, or permission of the instructor.

*Prerequisite: Some familiarity with probability and stochastic processes and queueing, or permission of the instructor.*

**MAT5175 (STAT 5506) ROBUST STATISTICAL INFERENCE**(3 units)

**MAT5181 (STAT 5703) DATA MINING I**(3 units)

Visualization and knowledge discovery in massive datasets; unsupervised learning: clustering algorithms; dimension reduction; supervised learning: pattern recognition, smoothing techniques, classification. Computer software will be used.

*Prerequisite: Permission of the Instructor.*

**MAT5182 (STAT 5702) MODERN APPLIED / COMPUTATIONAL STATISTICS**(3 units)

Resampling and computer intensive methods: bootstrap, jackknife with applications to bias estimation, variance estimation, confidence intervals, and regression analysis. Smoothing methods in curve estimation; Statistical classification and pattern recognition: error counting methods, optimal classifiers, bootstrap estimates of the bias of the misclassification error.

**MAT5185 (MATH 5408) ASYMPTOTIC METHODS OF APPLIED MATHEMATICS**(3 units)

Asymptotic series: properties, matching, application to linear and nonlinear differential equations. Asymptotic expansion of integrals: elementary methods, methods of Laplace, Stationary Phase and Steepest Descent, Watson's Lemma, Riemann-Lebesgue Lemma. Perturbation methods: regular and singular perturbation for differential equations, multiple scale analysis, boundary layer theory, WKB theory.

**MAT5187 (MATH 5403) TOPICS IN APPLIED MATHEMATICS**(3 units)

**MAT5190 (STAT 5600) MATHEMATICAL STATISTICS I**(3 units)

Statistical decision theory; likelihood functions; sufficiency; factorization theorem; exponential families; UMVU estimators; Fisher's information; Cramer-Rao lower bound; maximum likelihood and moment estimation; invariant and robust point estimation; asymptotic properties; Bayesian point estimation. Prerequisites: MAT 3172 and MAT 3375.

*Prerequisites: MAT3172 and MAT3375.*

**MAT5191 (STAT 5501) MATHEMATICAL STATISTICS II**(3 units)

Confidence intervals and pivotals; Bayesian intervals; optimal tests and Neyman-Pearson theory; likelihood ratio and score tests; significance tests; goodness-of-fit tests; large sample theory and applications to maximum likelihood and robust estimation.

*Prerequisite: MAT 5190.*

**MAT5192 (STAT 5502) SAMPLING THEORY AND METHODS**(3 units)

Unequal probability sampling with and without replacement; unified theory of standard errors; prediction approach; ratio and regression estimation; stratification and optimal designs; multistage cluster sampling; double sampling; domains of study; post-stratification; non-response; measurement errors. Related topics.

*Prerequisite: (MATH 4502).*

**MAT5193 (STAT 5503) LINEAR MODELS**(3 units)

Theory of non-full-rank linear models: estimable functions, best linear unbiased estimators, hypothesis testing, confidence regions; multi-way classification; analysis of covariance; variance component models: maximum likelihood estimation, MINQUE, ANOVA methods. Miscellaneous topics.

*Prerequisite: MAT 4175 (MATH 4500) or MAT 5190 (STAT 5600).*

**MAT5194 (STAT 5504) STOCHASTIC PROCESSES AND TIME SERIES ANALYSIS**(3 units)

**MAT5195 (STAT 5505) DESIGN OF EXPERIMENTS**(3 units)

Overview of linear model theory; orthogonality; randomized block and split plot designs; Latin square designs; randomization theory; incomplete block designs; factorial experiments; confounding and fractional replication; response surface methodology. Miscellaneous topics. Prerequisites: MAT 3375 and MAT 3376 or MAT 5190 (STAT 3505 and STAT 4500 or STAT 5600).

*Prerequisites: MAT3375 and MAT3376 or MAT5190 (STAT 3505 and STAT 4500 or STAT 5600).*

**MAT5196 (STAT 5509) MULTIVARIATE ANALYSIS**(3 units)

**MAT5197 (STAT 5601) STOCHASTIC OPTIMIZATION**(3 units)

Topics chosen from stochastic dynamic programming, Markov decision processes, search theory, optimal stopping.

*Prerequisite: STAT 3506 or MAT 4371.*

**MAT5198 (MATH 5701) STOCHASTIC MODELS**(3 units)

Markov systems, stochastic networks, queuing networks, spatial processes, approximation methods in stochastic processes and queuing theory. Applications to the modelling and analysis of computer-communications systems and other distributed networks.

**MAT5301 (MATH 5609) TOPICS IN COMBINATORIAL MATHEMATICS**(3 units)

**MAT5303 (MATH 5801) LINEAR OPTIMIZATION**(3 units)

**MAT5304 (MATH 5803) NONLINEAR OPTIMIZATION**(3 units)

**MAT5307 (MATH 5804) TOPICS IN OPERATIONS RESEARCH**(3 units)

**MAT5308 (MATH 5805) TOPICS IN ALGORITHM DESIGN**(3 units)

**MAT5309 (MATH 6002) HARMONIC ANALYSIS ON GROUPS**(3 units)

**MAT5312 (MATH 6201) TOPICS IN TOPOLOGY**(3 units)

**MAT5313 (MATH 6507) TOPICS IN PROBABILITY AND STATISTICS**(3 units)

**MAT5314 (MATH 6508) TOPICS IN PROBABILITY AND STATISTICS**(3 units)

**MAT5315 ADVANCED DESIGN OF SURVEYS**(3 units)

**MAT5317 (STAT 5602) ANALYSIS OF CATEGORICAL DATA**(3 units)

Analysis of one-way and two-way tables of nominal date; multi-dimensional contingency tables, log-linear models; tests of symmetry, marginal homogeneity in square tables; incomplete tables; tables with ordered categories; fixed margins, logistic models with binary response; measures of association and agreement; applications biological.

**MAT5318 (STAT 5603) RELIABILITY AND SURVIVAL ANALYSIS**(3 units)

**MAT5319 (MATH 6507) TOPICS IN PROBABILITY AND STATISTICS**(3 units)

**MAT5324 (MATH 5607) GAME THEORY**(3 units)

**MAT5325 (MATH 5802) TOPICS IN INFORMATION AND SYSTEMS SCIENCE**(3 units)

**MAT5326 (MATH 6008) TOPICS IN ANALYSIS**(3 units)

**MAT5327 (MATH 6101) TOPICS IN ALGEBRA**(3 units)

**MAT5328 (MATH 6008) TOPICS IN ANALYSIS**(3 units)

**MAT5329 (MATH 6009) TOPICS IN ANALYSIS**(3 units)

**MAT5330 (MATH 6102) TOPICS IN ALGEBRA**(3 units)

**MAT5331 (MATH 6103) TOPICS IN ALGEBRA**(3 units)

**MAT5341 (MATH5821) QUANTUM COMPUTING**(3 units)

Space of quantum bits; entanglement. Observables in quantum mechanics. Density matrix and Schmidt decomposition. Quantum cryptography. Classical and quantum logic gates. Quantum Fourier transform. Shor's quantum algorithm for factorization of integers.

*Prerequisite: Undergraduate honours linear algebra, or permission of the School or Department.*

**MAT5343 MATHEMATICAL ASPECTS OF WAVELETS AND DIGITAL SIGNAL PROCESSING**(3 units)

Lossless compression methods. Discrete Fourier transform and Fourier-based compression methods. JPEG and MPEG. Wavelet analysis. Digital filters and discrete wavelet transform. Daubechies wavelets. Wavelet compression. Prerequisites: Linear algebra and Fourier series, or permission of the School or Department.

**MAT5361 (MATH 6806) TOPICS IN MATHEMATICAL LOGIC**(3 units)

**MAT5375 (STAT 5610) MATHEMATICAL STATISTICS**(3 units)

Limit theorems; sampling distributions; parametric estimation; concepts of sufficiency and efficiency; Neyman-Pearson paradigm, likelihood ratio tests; parametric and non-parametric methods for two-sample comparisons; notions of experimental design, categorical data analysis, the general linear model, decision theory and Bayesian inference. Prerequisites: MAT2121, (MAT2141 or MAT2342), MAT2375.

*Exclusion: Students in the MSc program cannot combine this course with MAT5190 (STAT5600) for credit towards the master’s program.*

**MAT5176 (STAT 5507) ADVANCED STATISTICAL INFERENCE**(3 units)

Pure significance tests; uniformly most powerful unbiased and invariant tests; asymptotic comparison of tests; confidence intervals; large sample theory of likelihood ratio and chi-square tests; likelihood inference; Bayesian inference. Topics such as empirical Bayes inference, fiducial and structural inference, resampling methods. Prerequisites: MAT 4170 or equivalent and MAT 5191.

*Prerequisite: MAT4170 or equivalent and MAT5191.*

**MAT5177 (STAT 5500) MULTIVARIATE NORMAL THEORY**(3 units)

**MAT5180 (MATH 5806) NUMERICAL ANALYSIS FOR DIFFERENTIAL EQUATIONS**(3 units)

Floating pointing arithmetic; numerical solution of ordinary differential equations; finite difference methods for partial differential equations; stability, consistency and convergence: von Neumann analysis, Courant-Friedrichs-Lewy condition, Lax theorem; finite element methods: boundary value problems and elliptic partial differential equations; spectral and Pseudo-spectral methods. Prerequisites: MAT2324 and MAT3380.

**MAT5990 (MATH 5900) SÉMINAIRE / SEMINAR**(3 units)

**MAT5991 (MATH 5901) TRAVAUX DIRIGÉS / DIRECTED STUDIES**(3 units)

**MAT5996 (MATH 5906) STAGE DE RECHERCHE / RESEARCH INTERNSHIP**(3 units)

Cours visant à donner à l’étudiant la possibilité d’entreprendre de la recherche mathématique dans le contexte d’un projet en collaboration avec un organisme parrain des secteurs public ou privé. Inclut des séminaires sur des sujets pertinents au projet de l’étudiant. Note finale S (satisfaisant) ou NS (non satisfaisant), à décider par le professeur responsable du cours en consultation avec le superviseur du stage, fondée sur le contenu mathématique et sur la présentation orale et écrite des résultats.

*Préalable : Permission de l'Institut. / Project-oriented course affording students the opportunity to undertake research in applied mathematics as a cooperative project with governmental or industrial sponsors. Project work and seminars on related topics. Grade S (satisfactory) or NS (not satisfactory) to be assigned based upon the mathematical content as well as upon the oral and written presentation of results, and to be determined by the professor in charge of the course in consultation with the internship supervisor. Prerequisite: Permission of the Institute.*

**MAT6990 (MATH 6900) SÉMINAIRE / SEMINAR**(3 units)

**MAT6991 (MATH 6901) TRAVAUX DIRIGÉS / DIRECTED STUDIES**(3 units)

**MAT6997 (MATH5910) PROJET EN MATHÉMATIQUES ET STATISTIQUE / PROJECT IN MATHEMATICS AND STATISTICS**(6 units)

Projet en mathématiques et statistique dirigé par un professeur approuvé par le directeur des études supérieures et donnant lieu à la rédaction d'un rapport approfondi (30-40 pages approx). Noté S (satisfaisant) ou NS (non satisfaisant) par le directeur du projet et un autre professeur nommé par le directeur des études supérieures en mathématiques et statistique. Le projet est normalement complété en une session.

*Préalable : approbation du directeur des études supérieures en mathématiques et statistique. / Project in mathematics and statistics supervised by a professor approved by the director of graduate studies and leading to the writing of an in-depth report (approx. 30-40 pages). Graded S (satisfactory) or NS (not satisfactory) by the supervisor and by another professor appointed by the director of graduate studies in mathematics and statistics. The project will normally be completed in one session. Prerequisite: approval of director of graduate studies in mathematics and statistics.*

**MAT7999 THÈSE DE MAÎTRISE / MSc THESIS**

**MAT9998 EXAMEN DE SYNTHÈSE / COMPREHENSIVE EXAMINATION**

**MAT9999 (MATH 6909) THÈSE DE DOCTORAT / PhD THESIS**