Mathematics Honours (B.Sc.) (63 credits)
Offered by: Mathematics and Statistics (Faculty of Science)
Degree: Bachelor of Arts
Program credit weight: 63
Program Description
The B.Sc.; Honours in Mathematics provides an in-depth training, at the honours level, in mathematics. It gives the foundations and tools needed to explore diverse areas of mathematics such as analysis, number theory, geometry, geometric group theory, and probability. This program may be completed with a minimum of 60 credits or a maximum of 63 credits.
Degree Requirements — B.A. students
To be eligible for a B.A. degree, a student must fulfil all Faculty and program requirements as indicated in Degree Requirements for the Faculty of Arts.
We recommend that students consult an Arts OASIS advisor for degree planning.
Note: For information about Fall 2025 and Winter 2026 course offerings, please check back on May 8, 2025. Until then, the "Terms offered" field will appear blank for most courses while the class schedule is being finalized.
Program Prerequisites
The minimum requirement for entry into the Honours program is that the student has completed with high standing the following courses below or their equivalents.
| Course | Title | Credits |
|---|---|---|
| MATH 133 | Linear Algebra and Geometry. | 3 |
Linear Algebra and Geometry. Terms offered: Summer 2025 Systems of linear equations, matrices, inverses, determinants; geometric vectors in three dimensions, dot product, cross product, lines and planes; introduction to vector spaces, linear dependence and independence, bases. Linear transformations. Eigenvalues and diagonalization. | ||
| MATH 150 | Calculus A. | 4 |
Calculus A. Terms offered: this course is not currently offered. Functions, limits and continuity, differentiation, L'Hospital's rule, applications, Taylor polynomials, parametric curves, functions of several variables. | ||
| MATH 151 | Calculus B. | 4 |
Calculus B. Terms offered: this course is not currently offered. Integration, methods and applications, infinite sequences and series, power series, arc length and curvature, multiple integration. | ||
In particular, MATH 150 Calculus A./MATH 151 Calculus B. and MATH 140 Calculus 1./MATH 141 Calculus 2./MATH 222 Calculus 3. are considered equivalent.
Students who have not completed an equivalent of MATH 222 Calculus 3. on entering the program must consult an academic adviser and take MATH 222 Calculus 3. as a required course in the first semester, increasing the total number of program credits from 60 to 63. Students who have successfully completed MATH 150 Calculus A./MATH 151 Calculus B. are not required to take MATH 222 Calculus 3..
Students who transfer to Honours in Mathematics from other programs will have credits for previous courses assigned, as appropriate, by the Department.
To be awarded the Honours degree, the student must have, at time of graduation, a CGPA of at least 3.00 in the required and complementary Mathematics courses of the program, as well as an overall CGPA of at least 3.00.
Required Courses (45 credits)
| Course | Title | Credits |
|---|---|---|
| MATH 222 | Calculus 3. 1 | 3 |
Calculus 3. Terms offered: Summer 2025 Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals. | ||
| MATH 251 | Honours Algebra 2. | 3 |
Honours Algebra 2. Terms offered: this course is not currently offered. Linear equations over a field. Introduction to vector spaces. Linear maps and their matrix representation. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Real and complex inner product spaces. Diagonalization of self-adjoint operators. | ||
| MATH 255 | Honours Analysis 2. | 3 |
Honours Analysis 2. Terms offered: this course is not currently offered. Basic point-set topology, metric spaces: open and closed sets, normed and Banach spaces, Hölder and Minkowski inequalities, sequential compactness, Heine-Borel, Banach Fixed Point theorem. Riemann-(Stieltjes) integral, Fundamental Theorem of Calculus, Taylor's theorem. Uniform convergence. Infinite series, convergence tests, power series. Elementary functions. | ||
| MATH 325 | Honours Ordinary Differential Equations. | 3 |
Honours Ordinary Differential Equations. Terms offered: this course is not currently offered. First and second order equations, linear equations, series solutions, Frobenius method, introduction to numerical methods and to linear systems, Laplace transforms, applications. | ||
| MATH 356 | Honours Probability. | 3 |
Honours Probability. Terms offered: this course is not currently offered. Sample space, probability axioms, combinatorial probability. Conditional probability, Bayes' Theorem. Distribution theory with special reference to the Binomial, Poisson, and Normal distributions. Expectations, moments, moment generating functions, uni-variate transformations. Random vectors, independence, correlation, multivariate transformations. Conditional distributions, conditional expectation.Modes of stochastic convergence, laws of large numbers, Central Limit Theorem. | ||
| MATH 357 | Honours Statistics. | 3 |
Honours Statistics. Terms offered: this course is not currently offered. Sampling distributions. Point estimation. Minimum variance unbiased estimators, sufficiency, and completeness. Confidence intervals. Hypothesis tests, Neyman-Pearson Lemma, uniformly most powerful tests. Likelihood ratio tests for normal samples. Asymptotic sampling distributions and inference. | ||
| MATH 358 | Honours Advanced Calculus. | 3 |
Honours Advanced Calculus. Terms offered: this course is not currently offered. Point-set topology in Euclidean space; continuity and differentiability of functions in several variables. Implicit and inverse function theorems. Vector fields, divergent and curl operations. Rigorous treatment of multiple integrals: volume and surface area; and Fubini’s theorem. Line and surface integrals, conservative vector fields. Green's theorem, Stokes’ theorem and the divergence theorem. | ||
| MATH 454 | Honours Analysis 3. 2 | 3 |
Honours Analysis 3. Terms offered: this course is not currently offered. Measure theory: sigma-algebras, Lebesgue measure in R^n and integration, L^1 functions, Fatou's lemma, monotone and dominated convergence theorem, Egorov’s theorem, Lusin's theorem, Fubini-Tonelli theorem, differentiation of the integral, differentiability of functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus. | ||
| MATH 455 | Honours Analysis 4. | 3 |
Honours Analysis 4. Terms offered: this course is not currently offered. Review of point-set topology: topological spaces, dense sets, completeness, compactness, connectedness and path-connectedness, separability, Baire category theorem, Arzela-Ascoli theorem, Stone-Weierstrass theorem..Functional analysis: L^p spaces, linear functionals and dual spaces, Hilbert spaces, Riesz representation theorems. Fourier series and transform, Riemann-Lebesgue Lemma,Fourier inversion formula, Plancherel theorem, Parseval’s identity, Poisson summation formula. | ||
| MATH 456 | Honours Algebra 3. | 3 |
Honours Algebra 3. Terms offered: this course is not currently offered. Groups, quotient groups and the isomorphism theorems. Group actions. Groups of prime order and the class equation. Sylow's theorems. Simplicity of the alternating group. Semidirect products. Principal ideal domains and unique factorization domains. Modules over a ring. Finitely generated modules over a principal ideal domain with applications to canonical forms. | ||
| MATH 457 | Honours Algebra 4. | 3 |
Honours Algebra 4. Terms offered: this course is not currently offered. Representations of finite groups. Maschke's theorem. Schur's lemma. Characters, their orthogonality, and character tables. Field extensions. Finite and cyclotomic fields. Galois extensions and Galois groups. The fundamental theorem of Galois theory. Solvability by radicals. | ||
| MATH 458 | Honours Differential Geometry. | 3 |
Honours Differential Geometry. Terms offered: this course is not currently offered. In addition to the topics of MATH 320, topics in the global theory of plane and space curves, and in the global theory of surfaces are presented. These include: total curvature and the Fary-Milnor theorem on knotted curves, abstract surfaces as 2-d manifolds, the Euler characteristic, the Gauss-Bonnet theorem for surfaces. | ||
| MATH 466 | Honours Complex Analysis. | 3 |
Honours Complex Analysis. Terms offered: this course is not currently offered. Functions of a complex variable, Cauchy-Riemann equations, Cauchy's theorem and its consequences. Uniform convergence on compacta. Taylor and Laurent series, open mapping theorem, Rouché's theorem and the argument principle. Calculus of residues. Fractional linear transformations and conformal mappings. | ||
| MATH 470 | Honours Research Project. | 3 |
Honours Research Project. Terms offered: this course is not currently offered. The project will contain a significant research component that requires substantial independent work consisting of a written report and oral examination or presentation. | ||
| MATH 475 | Honours Partial Differential Equations. | 3 |
Honours Partial Differential Equations. Terms offered: this course is not currently offered. First order partial differential equations, geometric theory, classification of second order linear equations, Sturm-Liouville problems, orthogonal functions and Fourier series, eigenfunction expansions, separation of variables for heat, wave and Laplace equations, Green's function methods, uniqueness theorems. | ||
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Students who have successfully completed MATH 150 Calculus A./MATH 151 Calculus B. or an equivalent of MATH 222 Calculus 3. on entering the program are not required to take MATH 222 Calculus 3..
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Not open to students who have taken MATH 354 .
Complementary Courses (15 credits)
3 credits selected from:
| Course | Title | Credits |
|---|---|---|
| MATH 242 | Analysis 1. | 3 |
Analysis 1. Terms offered: this course is not currently offered. A rigorous presentation of sequences and of real numbers and basic properties of continuous and differentiable functions on the real line. | ||
| MATH 254 | Honours Analysis 1. 1 | 3 |
Honours Analysis 1. Terms offered: this course is not currently offered. Properties of R. Cauchy and monotone sequences, Bolzano- Weierstrass theorem. Limits, limsup, liminf of functions. Pointwise, uniform continuity: Intermediate Value theorem. Inverse and monotone functions. Differentiation: Mean Value theorem, L'Hospital's rule, Taylor's Theorem. | ||
- 1
It is strongly recommended that students take MATH 254 Honours Analysis 1..
3 credits selected from:
| Course | Title | Credits |
|---|---|---|
| MATH 235 | Algebra 1. | 3 |
Algebra 1. Terms offered: this course is not currently offered. Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; homomorphisms and quotient groups. | ||
| MATH 245 | Honours Algebra 1. 1 | 3 |
Honours Algebra 1. Terms offered: this course is not currently offered. Honours level: Sets, functions, and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. In-depth study of rings, ideals, and quotient rings; fields and construction of fields from polynomial rings; groups, subgroups, and cosets, homomorphisms, and quotient groups. | ||
- 1
It is strongly recommended that students take both MATH 245 Honours Algebra 1. and MATH 254 Honours Analysis 1..
0-6 credits from the following courses for which no Honours equivalent exists:
| Course | Title | Credits |
|---|---|---|
| MATH 204 | Principles of Statistics 2. | 3 |
Principles of Statistics 2. Terms offered: this course is not currently offered. The concept of degrees of freedom and the analysis of variability. Planning of experiments. Experimental designs. Polynomial and multiple regressions. Statistical computer packages (no previous computing experience is needed). General statistical procedures requiring few assumptions about the probability model. | ||
| MATH 208 | Introduction to Statistical Computing. | 3 |
Introduction to Statistical Computing. Terms offered: this course is not currently offered. Basic data management. Data visualization. Exploratory data analysis and descriptive statistics. Writing functions. Simulation and parallel computing. Communication data and documenting code for reproducible research. | ||
| MATH 308 | Fundamentals of Statistical Learning. | 3 |
Fundamentals of Statistical Learning. Terms offered: this course is not currently offered. Theory and application of various techniques for the exploration and analysis of multivariate data: principal component analysis, correspondence analysis, and other visualization and dimensionality reduction techniques; supervised and unsupervised learning; linear discriminant analysis, and clustering techniques. Data applications using appropriate software. | ||
| MATH 329 | Theory of Interest. | 3 |
Theory of Interest. Terms offered: this course is not currently offered. Simple and compound interest, annuities certain, amortization schedules, bonds, depreciation. | ||
| MATH 338 | History and Philosophy of Mathematics. | 3 |
History and Philosophy of Mathematics. Terms offered: this course is not currently offered. Egyptian, Babylonian, Greek, Indian and Arab contributions to mathematics are studied together with some modern developments they give rise to, for example, the problem of trisecting the angle. European mathematics from the Renaissance to the 18th century is discussed, culminating in the discovery of the infinitesimal and integral calculus by Newton and Leibnitz. Demonstration of how mathematics was done in past centuries, and involves the practice of mathematics, including detailed calculations, arguments based on geometric reasoning, and proofs. | ||
| MATH 378 | Nonlinear Optimization . | 3 |
Nonlinear Optimization . Terms offered: this course is not currently offered. Optimization terminology. Convexity. First- and second-order optimality conditions for unconstrained problems. Numerical methods for unconstrained optimization: Gradient methods, Newton-type methods, conjugate gradient methods, trust-region methods. Least squares problems (linear + nonlinear). Optimality conditions for smooth constrained optimization problems (KKT theory). Lagrangian duality. Augmented Lagrangian methods. Active-set method for quadratic programming. SQP methods. | ||
| MATH 430 | Mathematical Finance. | 3 |
Mathematical Finance. Terms offered: this course is not currently offered. Introduction to concepts of price and hedge derivative securities. The following concepts will be studied in both concrete and continuous time: filtrations, martingales, the change of measure technique, hedging, pricing, absence of arbitrage opportunities and the Fundamental Theorem of Asset Pricing. | ||
| MATH 462 | Machine Learning . | 3 |
Machine Learning . Terms offered: this course is not currently offered. Introduction to supervised learning: decision trees, nearest neighbors, linear models, neural networks. Probabilistic learning: logistic regression, Bayesian methods, naive Bayes. Classification with linear models and convex losses. Unsupervised learning: PCA, k-means, encoders, and decoders. Statistical learning theory: PAC learning and VC dimension. Training models with gradient descent and stochastic gradient descent. Deep neural networks. Selected topics chosen from: generative models, feature representation learning, computer vision. | ||
| MATH 463 | Convex Optimization. | 3 |
Convex Optimization. Terms offered: this course is not currently offered. Introduction to convex analysis and convex optimization: Convex sets and functions, subdifferential calculus, conjugate functions, Fenchel duality, proximal calculus. Subgradient methods, proximal-based methods. Conditional gradient method, ADMM. Applications including data classification, network-flow problems, image processing, convex feasibility problems, DC optimization, sparse optimization, and compressed sensing. | ||
6-12 credits selected from:
| Course | Title | Credits |
|---|---|---|
| COMP 250 | Introduction to Computer Science. 1 | 3 |
Introduction to Computer Science. Terms offered: this course is not currently offered. Mathematical tools (binary numbers, induction,recurrence relations, asymptotic complexity,establishing correctness of programs). Datastructures (arrays, stacks, queues, linked lists,trees, binary trees, binary search trees, heaps,hash tables). Recursive and non-recursivealgorithms (searching and sorting, tree andgraph traversal). Abstract data types. Objectoriented programming in Java (classes andobjects, interfaces, inheritance). Selected topics. | ||
| COMP 252 | Honours Algorithms and Data Structures. | 3 |
Honours Algorithms and Data Structures. Terms offered: this course is not currently offered. The design and analysis of data structures and algorithms. The description of various computational problems and the algorithms that can be used to solve them, along with their associated data structures. Proving the correctness of algorithms and determining their computational complexity. | ||
| MATH 350 | Honours Discrete Mathematics . | 3 |
Honours Discrete Mathematics . Terms offered: this course is not currently offered. Discrete mathematics. Graph Theory: matching theory, connectivity, planarity, and colouring; graph minors and extremal graph theory. Combinatorics: combinatorial methods, enumerative and algebraic combinatorics, discrete probability. | ||
| MATH 352 | Problem Seminar. | 1 |
Problem Seminar. Terms offered: this course is not currently offered. Seminar in Mathematical Problem Solving. The problems considered will be of the type that occur in the Putnam competition and in other similar mathematical competitions. | ||
| MATH 365 | Honours Groups, Tilings and Algorithms. | 3 |
Honours Groups, Tilings and Algorithms. Terms offered: this course is not currently offered. Transformation groups of the plane. Inversions and Moebius transformations. The hyperbolic plane. Tilings in dimension 2 and 3. Group presentations and Cayley graphs. Free groups and Schreier's theorem. Coxeter groups. Dehn's Word and Conjugacy Problems. Undecidability of the Word Problem for semigroups. Regular languages and automatic groups. Automaticity of Coxeter groups. | ||
| MATH 376 | Honours Nonlinear Dynamics. | 3 |
Honours Nonlinear Dynamics. Terms offered: this course is not currently offered. This course consists of the lectures of MATH 326, but will be assessed at the honours level. | ||
| MATH 377 | Honours Number Theory. | 3 |
Honours Number Theory. Terms offered: this course is not currently offered. This course consists of the lectures of MATH 346, but will be assessed at the honours level. | ||
| MATH 387 | Honours Numerical Analysis. | 3 |
Honours Numerical Analysis. Terms offered: this course is not currently offered. Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations. | ||
| MATH 397 | Honours Matrix Numerical Analysis. | 3 |
Honours Matrix Numerical Analysis. Terms offered: this course is not currently offered. The course consists of the lectures of MATH 327 plus additional work involving theoretical assignments and/or a project. The final examination for this course may be different from that of MATH 327. | ||
| MATH 398 | Honours Euclidean Geometry . | 3 |
Honours Euclidean Geometry . Terms offered: this course is not currently offered. Honours level: points and lines in a triangle. Quadrilaterals. Angles in a circle. Circumscribed and inscribed circles. Congruent and similar triangles. Area. Power of a point with respect to a circle. Ceva’s theorem. Isometries. Homothety. Inversion. | ||
| MATH 462 | Machine Learning . | 3 |
Machine Learning . Terms offered: this course is not currently offered. Introduction to supervised learning: decision trees, nearest neighbors, linear models, neural networks. Probabilistic learning: logistic regression, Bayesian methods, naive Bayes. Classification with linear models and convex losses. Unsupervised learning: PCA, k-means, encoders, and decoders. Statistical learning theory: PAC learning and VC dimension. Training models with gradient descent and stochastic gradient descent. Deep neural networks. Selected topics chosen from: generative models, feature representation learning, computer vision. | ||
| MATH 480 | Honours Independent Study. | 3 |
Honours Independent Study. Terms offered: this course is not currently offered. Reading projects permitting independent study under the guidance of a staff member specializing in a subject where no appropriate course is available. Arrangements must be made with an instructor and the Chair before registration. | ||
| MATH 488 | Honours Set Theory. | 3 |
Honours Set Theory. Terms offered: this course is not currently offered. Axioms of set theory, ordinal and cardinal arithmetic, consequences of the axiom of choice, models of set theory, constructible sets and the continuum hypothesis, introduction to independence proofs. | ||
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Students with limited programming experience should take COMP 202 Foundations of Programming. or COMP 204 Computer Programming for Life Sciences. or COMP 208 Computer Programming for Physical Sciences and Engineering . or equivalent before COMP 250 Introduction to Computer Science..
all MATH 500-level courses
Students may select other courses with the permission of the Department.