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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Introduction to Algebraic Geometry.

Terms offered: this course is not currently offered.

Affine varieties. Radical ideals and Hilbert's Nullstellensatz. The Zariski topology. Irreducible decomposition. Dimension. Tangent spaces, smoothness and singularities. Projective spaces and projective varieties. Regular functions and morphisms. Rational maps and indeterminacy. Blowing up. Divisors and linear systems. Projective curves.

Combinatorics.

Terms offered: this course is not currently offered.

Enumerative combinatorics: inclusion-exclusion, generating functions, partitions, lattices and Moebius inversion. Extremal combinatorics: Ramsey theory, Turan's theorem, Dilworth's theorem and extremal set theory. Graph theory: planarity and colouring. Applications of combinatorics.

Combinatorial Optimization.

Terms offered: this course is not currently offered.

Algorithmic and structural approaches in combinatorial optimization with a focus upon theory and applications. Topics include: polyhedral methods, network optimization, the ellipsoid method, graph algorithms, matroid theory and submodular functions.

Algorithmic Game Theory.

Terms offered: this course is not currently offered.

Foundations of game theory. Computation aspects of equilibria. Theory of auctions and modern auction design. General equilibrium theory and welfare economics. Algorithmic mechanism design. Dynamic games.

Real Analysis and Measure Theory.

Terms offered: this course is not currently offered.

Abstract theory of measure and integration: Borel-Cantelli lemmas, regularity of measures, product measures, Fubini-Tonelli theorem, signed measures, Hahn and Jordan decompositions, Radon-Nikodym theorem, differentiation in R^n.

Functional Analysis.

Terms offered: this course is not currently offered.

Review of the basic theory of Banach and Hilbert spaces, L^p spaces, open mapping theorem,closed graph theorem, Banach-Steinhaus theorem, Hahn-Banach theorem, weak and weak-* convergence, weak convergence of measures, Riesz representation theorems, spectral theorem for compact self-adjoint operators, Fredholm theory, spectral theorem for bounded self-adjoint operators, Fourier series and integrals, additional topics.

Advanced Complex Analysis and Riemann Surfaces

Terms offered: this course is not currently offered.

Brief review of holomorphicity and contour integration. Analytic continuation and the monodromy theorem, normal families. Riemann surfaces; elliptic functions; Picard theorem. Riemann zeta function and prime number theorem. Relations with harmonic analysis; the uniformization theorem. Time permitting, additional material such as: Hodge decomposition, divisors, the Riemann Roch formula, or rudiments of moduli spaces.

Higher Algebra 1.

Terms offered: this course is not currently offered.

Free groups and free products of groups. Categories and functors. Universal properties and adjoint functors. Limits. Introduction to commutative algebra. Tensor products. Localization. Noetherian rings and Hilbert’s basis theorem. Hilbert's Nullstellensatz.

Higher Algebra 2.

Terms offered: this course is not currently offered.

Flat, projective, and injective modules. Introduction to homological algebra. Ext and Tor. Group cohomology. Semi-simple rings and modules. The Artin-Wedderburn Theorem.

Geometry and Topology 1.

Terms offered: this course is not currently offered.

Basic point-set topology, including connectedness, compactness, product spaces, separation axioms, metric spaces. The fundamental group and covering spaces. Simplicial complexes. Singular and simplicial homology. Part of the material of MATH 577 may be covered as well.

Geometry and Topology 2.

Terms offered: this course is not currently offered.

Basic properties of differentiable manifolds, tangent and cotangent bundles, differential forms, de Rham cohomology, integration of forms, Riemannian metrics, geodesics, Riemann curvature.

Advanced Partial Differential Equations 1 .

Terms offered: this course is not currently offered.

Classification and wellposedness of linear and nonlinear partial differential equations; energy methods; Dirichlet principle. Brief introduction to distributions; weak derivatives. Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle. Representation formulae for solutions of heat and wave equations, Duhamel's principle. Method of Characteristics, scalar conservation laws, shocks.

Advanced Partial Differential Equations 2 .

Terms offered: this course is not currently offered.

Systems of conservation laws and Riemann invariants. Cauchy- Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.

Advanced Probability Theory 1.

Terms offered: this course is not currently offered.

Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.

Advanced Probability Theory 2.

Terms offered: this course is not currently offered.

Characteristic functions: elementary properties, inversion formula, uniqueness, convolution and continuity theorems. Weak convergence. Central limit theorem. Additional topic(s) chosen (at discretion of instructor) from: Martingale Theory; Brownian motion, stochastic calculus.

Advanced Set Theory.

Terms offered: this course is not currently offered.

Students will attend the lectures and fulfill all the requirements of MATH 488. In addition, they will complete a project on an advanced topic agreed on with the instructor. Topics may be chosen from combinatorial set theory, Goedel's constructible sets, forcing, large cardinals and descriptive set theory.

Model Theory.

Terms offered: this course is not currently offered.

Structures, theories, and definable sets; elementary equivalence and elementary embeddings; compactness and Löwenheim– Skolem theorems; types, the omitting types theorem, and saturation; categoricity and the Ryll-Nardzewski theorem; quantifier elimination and applications to algebra; ultraproducts; homogeneous structures and Fraïssé theory; infinitary logics. Optional topics: indiscernibles and Morley's theorem; stability; o-minimality; elimination of imaginaries.

Descriptive Set Theory.

Terms offered: this course is not currently offered.

Polish spaces; universality of the Hilbert cube, the Cantor space, and the Baire space; the Cantor–Bendixson theorem; Baire spaces; the Borel hierarchy; change of topology techniques; infinite games; analytic and co-analytic sets; analytic separation; the Luzin–Souslin theorem; the Borel and measure isomorphism theorems; regularity properties of analytic sets; uniformization; the projective hierarchy. Optional topics: Polish groups and their actions; definable equivalence relations and graphs; effective descriptive set theory.

Mathematical Logic.

Terms offered: this course is not currently offered.

Propositional logic: truth-tables, formal proof systems, completeness and compactness theorems, Boolean algebras; first-order logic: formal proofs, Gödel's completeness theorem; axiomatic theories; set theory; Cantor's theorem, axiom of choice and Zorn's lemma, Peano arithmetic; Gödel's incompleteness theorem.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.