Applied Mathematics Honours (B.Sc.) (63 credits)

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.

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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.

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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.

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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.

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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.

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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.

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Honours Applied Linear Algebra.

Terms offered: this course is not currently offered.

Matrix algebra, determinants, systems of linear equations. Abstract vector spaces, inner product spaces, Fourier series. Linear transformations and their matrix representations. Eigenvalues and eigenvectors, diagonalizable and defective matrices, positive definite and semidefinite matrices. Quadratic and Hermitian forms, generalized eigenvalue problems, simultaneous reduction of quadratic forms. Applications.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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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.

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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.

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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.

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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.

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Honours Complex Variables.

Terms offered: this course is not currently offered.

Functions of a complex variable; Cauchy-Riemann equations; Cauchy's theorem and consequences. Taylor and Laurent expansions. Residue calculus; evaluation of real integrals; integral representation of special functions; the complex inversion integral. Conformal mapping; Schwarz-Christoffel transformation; Poisson's integral formulas; applications. Additional topics if time permits: homotopy of paths and simple connectivity, Riemann sphere, rudiments of analytic continuation.

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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.

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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.

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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.

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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.

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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.

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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.

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Theory of Interest.

Terms offered: this course is not currently offered.

Simple and compound interest, annuities certain, amortization schedules, bonds, depreciation.

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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.

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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.

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Introduction to General Topology.

Terms offered: this course is not currently offered.

This course is an introduction to point set topology. Topics include basic set theory and logic, topological spaces, separation axioms, continuity, connectedness, compactness, Tychonoff Theorem, metric spaces, and Baire spaces.

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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.

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Computational Methods in Applied Mathematics .

Terms offered: this course is not currently offered.

Solution to initial value problems: Linear, Nonlinear Finite Difference Methods: accuracy and stability, Lax equivalence theorem, CFL and von Neumann conditions, Fourier analysis: diffusion, dissipation, dispersion, and spectral methods. Solution of large sparse linear systems: iterative methods, preconditioning, incomplete LU, multigrid, Krylov subspaces, conjugate gradient method. Applications to, e.g., weighted least squares, duality, constrained minimization, calculus of variation, inverse problems, regularization, level set methods, Navier-Stokes equations

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Honours Algorithm Design.

Terms offered: this course is not currently offered.

Basic algorithmic techniques, their applications and limitations. Problem complexity, how to deal with problems for which no efficient solutions are known.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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