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MAT Mathematics Courses


MAT123H1
Calculus and Linear Algebra for Commerce (A) 26L, 10T

First term of MAT133Y. Students in academic difficulty in MAT133Y who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT133Y and enrol in MAT123H in the Spring Term. These students are informed of this option by the beginning of the Spring Term. Classes begin in the second week of the Spring Term; late enrolment is not permitted. Students not enrolled in MAT133Y in the Fall Term are not allowed to enrol in MAT123H. MAT123H together with MAT124H is equivalent for program and prerequisite purposes to MAT133Y.
Exclusion: MAT125H, 126H, 133Y, 135Y, 137Y, 157Y
NOTE: students who enrol in MAT133Y after completing MAT123H but not MAT124H do not receive degree credit for MAT133Y; it is counted ONLY as an "Extra Course."
Prerequisite: Enrolment in MAT133Y, and withdrawal from MAT133Y after two midterms, with a mark of at least 20% in the second midterm.


MAT124H1
Calculus and Linear Algebra for Commerce (B) 26L, 13T

Second Term content of MAT133Y; the final examination includes topics covered in MAT123H. Offered in the Summer Session only; students not enrolled in MAT123H in the preceding Spring Term will NOT be allowed to enrol in MAT124H. MAT123H together with MAT124H is equivalent for program and prerequisite purposes to MAT133Y.
Exclusion: MAT125H, 126H, 133Y, 135Y, 137Y, 157Y
Prerequisite: MAT123H successfully completed in the preceding Spring Term


MAT125H1
Calculus I (A) 26L

First term of MAT135Y. Students in academic difficulty in MAT135Y who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT135Y and enrol in MAT125H in the Spring Term. These students are informed of this option by the beginning of the Spring Term. Classes begin in the second week of the Spring Term; late enrolment is not permitted. Students not enrolled in MAT135Y in the Fall Term will not be allowed to enrol in MAT125H. MAT125H together with MAT126H is equivalent for program and prerequisite purposes to MAT135Y.
Exclusion: MAT123H, 124H, 133Y, 135Y, 137Y, 157Y
NOTE: students who enrol in MAT135Y after completing MAT125H but not MAT126H do not receive degree credit for MAT135Y; it is counted ONLY as an "Extra Course."
Prerequisite: Enrolment in MAT135Y, and withdrawal from MAT135Y after two midterms, with a mark of at least 20% in the second midterm.


MAT126H1
Calculus I (B) 26L, 13T

Second Term content of MAT135Y; the final examination includes topics covered in MAT125H. Offered in the Summer Session only; students not enrolled in MAT125H in the preceding Spring Term will NOT be allowed to enrol in MAT126H. MAT125H together with MAT126H is equivalent for program and prerequisite purposes to MAT135Y.
Exclusion: MAT123H, 124H, 133Y, 135Y, 137Y
Prerequisite: MAT125H successfully completed in the preceding Spring Term


MAT133Y1
Calculus and Linear Algebra for Commerce 52L, 24T

Mathematics of finance. Matrices and linear equations. Review of differential calculus; applications. Integration and fundamental theorem; applications. Introduction to partial differentiation; applications.
NOTE: please note prerequisites listed below. Students without the proper prerequisites for MAT133Y may be deregistered from this course.
Exclusion: MAT123H, 124H, 125H, 126H, 135Y, 137Y, 157Y
Prerequisite: Calc + A&G/FM


MAT135Y1
Calculus I 52L, 24T

Review of differential calculus; applications. Integration and fundamental theorem; applications. Series. Introduction to differential equations.
Exclusion: MAT123H, 124H, 125H, 126H, 133Y, 137Y, 157Y
Prerequisite: Calc


MAT137Y1
Calculus! 78L, 26T

A conceptual approach for students with a serious interest in mathematics. Geometric and physical intuition are emphasized but some attention is also given to the theoretical foundations of calculus. Material covers the basic concepts of calculus: limits and continuity, the mean value and inverse function theorems, the integral, the fundamental theorem, elementary transcendental functions, Taylor's theorem, sequence and series, uniform convergence and power series.
Exclusions: MAT123H, 124H, 125H, 126H, 133Y, 135Y, 157Y
Prerequisite: Calc + A&G/FM


MAT157Y1
Analysis I 78L, 52T

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor's theorem; sequences and series; uniform convergence and power series.
Exclusion: MAT137Y
Prerequisite: Calc + A&G


MAT223H1
Linear Algebra I 39L

Matrices, linear systems, elementary matrices and the inverse of a matrix. Vector spaces over R, subspaces, basis and dimension. Real inner product spaces, geometry in Rn, lines and hyperplanes. Linear transformation, kernel, range, matrix representation, isomorphisms. The determinant, Cramer's rule, the adjoint matrix. Eigenvalues, eigenvectors, similarity, diagonalization. Projections, Gram-Schmidt process, orthogonal transformations and orthogonal diagonalization, isometries, quadratic forms, conics, quadric surfaces.
Exclusion: MAT240H
Prerequisite: (Calc + A&G)/MAT133Y/134Y/135Y/137Y


MAT224H1
Linear Algebra II 39L

Fields. Vector spaces over a field. Linear transformations, dual spaces. Diagonalizability, direct sums. Invariant subspaces, Cayley-Hamilton theorem. Complex inner product, orthogonality, the adjoint of a linear operator, the projection matrix and the method of least squares. Normal, self-adjoint and unitary operators. Spectral theorem. Conditioning and Rayleigh quotient. Jordan canonical form.
Prerequisite: MAT223H/240H


MAT235Y1
Calculus II 78L

Differential and integral calculus of functions of several variables. Line and surface integrals, the divergence theorem, Stokes' theorem. Sequences and series, including an introduction to Fourier series. Some partial differential equations of Physics.
Exclusion: MAT237Y, 239Y, 257Y
Prerequisite: MAT135Y/137Y/157Y


MAT237Y1
Multivariable Calculus 78L

Sequences and series. Uniform convergence. Convergence of integrals. Elements of topology in R2 and R3. Differential and integral calculus of vector valued functions of a vector variable, with emphasis on vectors in two and three dimensional euclidean space. Extremal problems, Lagrange multipliers, line and surface integrals, vector analysis, Stokes' theorem, Fourier series, calculus of variations.
Exclusion: MAT235Y, 239Y, 257Y
Prerequisite: MAT135Y(80%)/137Y/157Y


MAT240H1
Algebra I 39L, 26T

A theoretical approach to: vector spaces over arbitrary fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.
Exclusion: MAT223H
Prerequisite: (Calc + A&G)/MAT135Y/137Y/157Y


MAT244H1
Introduction to Ordinary Differential Equations 39L

Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; non-linear equations; difference equations.
Exclusion: MAT267H
Prerequisite: MAT135Y/137Y/157Y, 223H/240H
Co-requisite: MAT235Y/237Y/239Y


MAT246Y1
Concepts in Abstract Mathematics 78L

Designed to introduce students to mathematical proofs and abstract mathematical concepts. Topics may include modular arithmetic, prime numbers, sizes of infinite sets, a proof that some angles cannot be trisected with straightedge and compass, an introduction to group theory, or an introduction to topology.
Prerequisite: MAT133Y/135Y/137Y


MAT247H1
Algebra II 39L, 13T

A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra. Examples of symmetry groups and linear groups, stochastic matrices, matrix functions.
Exclusion: MAT224H
Prerequisite: MAT240H


MAT257Y1
Analysis II 78L, 26T

Topology of Rn; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integrals; Fubini's theorem, partitions of unity, change of variables. Differential forms. Manifolds in Rn; integration on manifolds; Stoke's theorem for differential forms and classical versions.
Exclusion: MAT237Y
Prerequisite: MAT157Y, 240H
Co-requisite: MAT247H


MAT267H1
Ordinary Differential Equations I 39L, 13T

First-order equations. Linear equations and first-order systems. Non-linear first-order systems. Existence and uniqueness theorems for the Cauchy problem. Method of power series. Elementary qualitative theory; stability, phase plane, stationary points. Examples of applications in mechanics, physics, chemistry, biology and economics.
Exclusion: MAT244H
Prerequisite: MAT157Y, 247H
Co-requisite: MAT257Y


MAT299Y1
Research Opportunity Program

Credit course for supervised participation in faculty research project. See page 42 for details.


MAT301H1
Groups and Symmetry 39L

Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange's theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.
Exclusion: MAT347Y
Prerequisite: MAT224H, 246Y/(CSC238H, PHL245H)/MAT257Y


MAT302H1
Polynomial Equations and Fields 39L

Commutative rings; quotient rings. Construction of the rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of roots of a polynomial. Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Solvable groups, simple groups. Insolvability of quintics by radicals.
Exclusion: MAT347Y
Prerequisite: MAT301H


MAT309H
Introduction to Mathematical Logic 39L

Proof theory: formal logic and G”del's incompleteness theorems. Introduction to the theory of recursive functions.
Exclusion: CSC438H
Prerequisite: MAT223H/240H, 235Y/237Y/239Y, 246Y/(CSC238H, PHL245H)/MAT257Y


MAT315H1
Introduction to Number Theory 39L

Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.
Prerequisite: MAT(235Y/237Y, 223H/240H)/257Y


MAT327H1
Introduction to Topology 39L

Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Topology of function spaces. Fundamental group and covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point theorem.
Prerequisite: MAT257Y


MAT329Y1
Concepts in Elementary Mathematics 78L

The formation of mathematical concepts and techniques, and their application to the everyday world. Nature of mathematics and mathematical understanding. Role of observation, conjecture, analysis, structure, critical thinking and logical argument. Numeration, arithmetic, geometry, counting techniques, recursion, algorithms.
This course is specifically addressed to students intending to become elementary school teachers and is strongly recommended by the Faculty of Education. Previous experience working with children is useful. The course is taught jointly by the Department of Mathematics and the Faculty of Education. The course content is considered in the context of elementary school teaching. In particular, the course may include a practicum in school classrooms. The course has an enrolment limit of 40, and students are required to ballot.
Prerequisite: Any 7 full courses with a CGPA of at least 3.0


MAT334H1
Complex Variables 39L

Theory of functions of one complex variable, analytic and meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.
Exclusion: MAT357Y
Prerequisite: MAT235Y/237Y/239Y/257Y


MAT335H1
Chaos, Fractals and Dynamics 39L

An elementary introduction to a modern and fast-developing area of mathematics. One-dimensional dynamics: iterations of quadratic polynomials. Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia sets. History and applications.
Prerequisite: MAT137Y/200-level calculus


MAT337H1
Introduction to Real Analysis (replaces MAT338H) 39L

Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension.
Exclusion: MAT338H, 357H, 357Y
Prerequisite: MAT223H/240H, (237Y, 246Y)/257Y


MAT344H1
Introduction to Combinatorics 39L

Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows). Combinatorial structures including block designs and finite geometries.
Prerequisite: MAT223H/240H


MAT347Y1
Groups, Rings and Fields 78L, 26T

Groups, subgroups, quotient groups, Sylow theorems, Jordan-H”lder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic.
Prerequisite: MAT247H, 257Y


MAT354H1
Complex Analysis I (formerly part of MAT357Y) 39L, 13T

Complex numbers, the complex plane and Riemann sphere, Mobius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy's theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz's lemma, residue theorem and residue calculus.
Exclusion: MAT357Y
Prerequisite: MAT247H, 257Y


MAT357H1
Real Analysis I (formerly part of MAT357Y) 39L, 13T

Function spaces; Arzela-Ascoli thoerem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, Lp spaces. applications to probability.
Exclusion: MAT337H, 338H, 357Y
Prerequisite: MAT247H, 257Y, 327H


MAT363H1
Introduction to Differential Geometry 39L

Geometry of curves and surfaces in 3-spaces. Curvature and geodisics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.
Prerequisite: MAT224H/247H, 237Y/257Y


MAT365H1
Classical Geometries (replaces MAT325H) 39L

Euclidean and non-Euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.
Exclusion: MAT325H
Co-requisite: MAT301H/347Y


MAT390H1
History of Mathematics up to 1700 39L

A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)
Exclusion: HPS309H, 310Y, 390H
Prerequisite: at least one full MAT 200-level course


MAT391H1
History of Mathematics after 1700 26L, 13T

A survey of the development of mathematics from 1700 to the present with emphasis on technical development. (Offered in alternate years)
Exclusion: HPS309H, 310H, 391H
Prerequisite: At least one full 200-level MAT course


MAT395H1/396H1/397H1
Independent Work in Mathematics TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Prerequisite: Minimum GPA of 3.5 in math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisor


MAT398Y1/399Y1
Independent Work in Mathematics TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Prerequisite: Minimum GPA of 3.5 in math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisor
400-SERIES COURSES
NOTE Some courses at the 400-level are cross-listed as graduate courses and may not be offered every year. Please see the Department's undergraduate brochure for more details.


MAT415H1
Topics in Algebraic Number Theory 39L

A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; diophantine equations.
Prerequisite: MAT347Y or permission of instructor


MAT417H1
Topics in Analytic Number Theory 39L

A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; diophantine approximation, modular forms.
Prerequisite: MAT334H/354H or permission of instructor


MAT425H1
Differential Topology 39L

Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.
Prerequisite: MAT257Y, 327H


MAT427H1
Algebraic Topology 39L

Introduction to homology theory: singular and simplicial homology; homotopy invariance, long exact sequence, excision, Mayer-Vietoris sequence; applications. Homology of CW complexes; Euler characteristic; examples. Singular cohomology; products; cohomology ring. Topological manifolds; orientation; Poincare duality.
Prerequisite: MAT327H, 347Y


MAT445H1
Representation Theory 39L

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.
Prerequisite: MAT347Y


MAT448H1
Introduction to Commutative Algebra and Algebraic Geometry 39L

Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers.
Prerequisite: MAT347Y


MAT449H1
Algebraic Curves 39L

Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem.
Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities.
Prerequisite: MAT347Y, 354H


MAT454H1
Complex Analysis II (formerly MAT437H) 39L

Harmonic functions, Harnack's principle, Poission's integral formula and Dirichlet's problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.
Exclusion: MAT437H
Prerequisite: MAT354H


MAT457Y1
Real Analysis II 78L

Measure theory and Lebesgue integration; convergence theorems. Riesz representation theorem, Fubini's theorem, complex measures. Banach spaces; Lp spaces, density of continuous functions. Hilbert spaces; weak and strong topologies; self-adjoint, compact and projection operators. Hahn-Banach theorem, open mapping and closed graph theorems. Inequalities. Schwartz space; introduction to distributions; Fourier transforms on Rn (Schwartz space and L2). Spectral theorem for bounded normal operators.
Prerequisite: MAT357H


MAT464H1
Differential Geometry (formerly MAT364H) 39L

Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scarlar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.
Prerequisite: MAT363H


MAT468H1
Ordinary Differential Equations II 39L

Sturm-Liouville problem and oscillation theorems for second-order linear equations. Qualitative theory; integral invariants, limit cycles. Dynamical systems; invariant measures; bifurcations, chaos. Elements of the calculus of variations. Hamiltonian systems. Analytic theory; singular points and series solution. Laplace transform.
Prerequisite: MAT267H, 354H, 357H


MAT477H1
Seminar in Mathematics (formerly MAT467H) TBA

Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required)
Prerequisite: MAT347Y, 354H, 357H; or permission of instructor.


MAT495H1/496H1/497H1/498Y1/499Y1
Readings in Mathematics TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Prerequisite: Minimum GPA of 3.5 in math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisor


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