First Year Seminars
The 199Y1 and 199H1 seminars are designed to provide the opportunity to
work closely with an instructor in a class of no more than twenty-four students.
These interactive seminars are intended to stimulate the students’ curiosity and provide an opportunity to get to know a member of the professorial staff in a seminar environment during the first year of study. Details here.
JMB170Y1
Biology, Models, and Mathematics [48L, 24T]
Applications of mathematics to biological problems in physiology, biomechanics, genetics, evolution, growth, population dynamics, cell biology, ecology and behaviour.
Co-requisite: BIO150Y1
DR=SCI; BR=4+5
JUM202H1
Mathematics as an Interdisciplinary Pursuit (formerly JUM102H1) [24L, 12T]
A study of the interaction of mathematics with other fields of inquiry: how mathematics influences, and is influenced by, the evolution of science and culture. Art, music, and literature, as well as the more traditionally related areas of the natural and social sciences may be considered. (Offered every three years)
Exclusion: JUM102H1
JUM202H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.
DR=SCI; BR=5
JUM203H1
Mathematics as a Recreation (formerly JUM103H1) [24L, 12T]
A study of games, puzzles and problems focusing on the deeper principles they illustrate. Concentration is on problems arising out of number theory and geometry, with emphasis on the process of mathematical reasoning. Technical requirements are kept to a minimum. A foundation is provided for a continuing lay interest in mathematics. (Offered every three years)
Exclusion: JUM103H1
JUM203H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.
DR=SCI; BR=5
JUM205H1
Mathematical Personalities (formerly JUM105H1) [24L, 12T]
An in-depth study of the life, times and work of several mathematicians who have been particularly influential. Examples may include Newton, Euler, Gauss, Kowalewski, Hilbert, Hardy, Ramanujan, Gödel, Erdös, Coxeter, Grothendieck. (Offered every three years)
Exclusion: JUM105H1
JUM205H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.
DR=SCI; BR=5
Applied Mathematics Courses
Key to Course Descriptions.
APM236H1
Applications of Linear Programming [36L]
Introduction to linear programming including a rapid review of linear algebra (row reduction, linear independence), the simplex method, the duality theorem, complementary slackness, and the dual simplex method. A selection of the following topics are covered: the revised simplex method, sensitivity analysis, integer programming, the transportation algorithm.
Prerequisite: MAT223H1/MAT240H1 (Note: no waivers of Prerequisites will be granted)
DR=SCI; BR=5
APM346H1
Partial Differential Equations [36L]
Sturm-Liouville problems, Green’s functions, special functions (Bessel, Legendre), partial differential equations of second order, separation of variables, integral equations, Fourier transform, stationary phase method.
Prerequisite: MAT235Y1/MAT237Y1/MAT257Y1, MAT244H1
DR=SCI; BR=TBA
APM351Y1
Partial Differential Equations [72L]
Diffusion and wave equations. Separation of variables. Fourier series. Laplace’s equation; Green’s function. Schrödinger equations. Boundary problems in plane and space. General eigenvalue problems; minimum principle for eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction to nonlinear equations (shock waves, solitary waves).
Prerequisite: MAT267H1
Co-requisite: MAT334H1/MAT354H1
DR=SCI; BR=TBA
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 graduate brochure for more details.
APM421H1
Mathematical Foundations of Quantum Mechanics [36L]
The general formulation of non-relativistic quantum mechanics based on the theory of linear operators in a Hilbert space, self-adjoint operators, spectral measures and the statistical interpretation of quantum mechanics; functions of compatible observables. Schrödinger and Heisenberg pictures, complete sets of observables, representations of the canonical commutative relations; essential self-adjointedness of Schrödinger operators, density operators, elements of scattering theory.
Prerequisite: (MAT224H1, MAT337H1)/MAT357H1
DR=SCI; BR=TBA
APM426H1
General Relativity [36L]
Einstein’s theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein’s equations. Cosmological implications: big bang and inflationary universe. Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from black hole dynamics and gravity waves.
Prerequisite: MAT363H1
DR=SCI; BR=TBA
APM436H1
Fluid Mechanics [36L]
Boltzmann, Euler and Navier-Stokes equations. Viscous and non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propagation. Analysis of one dimensional gas flow.
Prerequisite: APM351Y1
DR=SCI; BR=TBA
APM441H1
Asymptotic and Perturbation Methods [36L]
Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained co-ordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering)
Prerequisite: APM346H1/APM351Y1, MAT334H1
DR=SCI; BR=TBA
APM446H1
Applied Nonlinear Equations [36L]
Nonlinear partial differential equations and their physical origin. Fourier transform; Green’s function; variational methods; symmetries and conservation laws. Special solutions (steady states, solitary waves, travelling waves, self-similar solutions). Calculus of maps; bifurcations; stability, dynamics near equilibrium. Propagation of nonlinear waves; dispersion, modulation, optical bistability. Global behaviour solutions; asymptotics and blow-up.
Prerequisite: APM346H1/APM351Y1
DR=SCI; BR=TBA
APM461H1
Combinatorial Methods [36L]
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisite: MAT224H1
Recommended preparation: MAT344H1
DR=SCI; BR=TBA
APM462H1
Nonlinear Optimization [36L]
An introduction to first and second order conditions for finite and infinite dimensional optimization problems with mention of available software. Topics include Lagrange multipliers, Kuhn-Tucker conditions, convexity and calculus variations. Basic numerical search methods and software packages which implement them will be discussed.
Prerequisite: MAT223H1, MAT235Y1
DR=SCI; BR=TBA
APM466H1
Mathematical Theory of Finance [36L]
Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.
Prerequisite: APM346H1, STA347H1
Co-requisite: STA457H1
DR=SCI; BR=TBA
APM496H1
Readings in Applied Mathematics [TBA]
APM497H1
Readings in Applied Mathematics [TBA]
APM498Y1
Readings in Applied Mathematics [TBA]
APM499Y1
Readings in Applied Mathematics [TBA]
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Prerequisite: minimum GPA 3.5 for math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisor
DR=SCI; BR=TBA
Mathematics Courses
NOTE: Transfer students who have received MAT1**H1 – Calculus with course exclusion to MAT133Y1/MAT135Y1 may
take MAT137Y1/MAT157Y1 without forfeiting the half credit in Calculus.
High school Prerequisites for students coming from outside the Ontario
high school system:
-
MAT133Y1: high school level calculus and (algebra-geometry or finite
math or discrete math)
- MAT135Y1:
high school level calculus
- MAT137Y1: high school level
calculus and algebra-geometry
- MAT157Y1:
high school level calculus and algebra-geometry
- MAT223H1: high school
level calculus and algebra-geometry
MAT123H1, MAT124H1
See below MAT133Y1
MAT125H1, MAT126H1
See below MAT135Y1
MAT133Y1
Calculus and Linear Algebra for Commerce [72L, 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 MAT133Y1 may be deregistered from this course.
Exclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT135Y1, MAT137Y1, MAT157Y1
Prerequisite: MCV4U, MHF4U
DR=SOC SCI; BR=N/A
MAT123H1
Calculus and Linear Algebra for Commerce (A) [36L]
First term of MAT133Y1. Students in academic difficulty in MAT133Y1 who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT133Y1 and enrol in MAT123H1 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 MAT133Y1 in the Fall Term are not allowed to enrol in MAT123H1. MAT123H1 together with MAT124H1 is equivalent for program and Prerequisite purposes to MAT133Y1.
Exclusion: MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1
NOTE: students who enrol in MAT133Y1 after completing MAT123H1 but not MAT124H1 do not receive degree credit for MAT133Y1; it is counted ONLY as an “Extra Course.”
Prerequisite: Enrolment in MAT133Y1, and withdrawal from MAT133Y1 after two midterms, with a mark of at least 20% in the second midterm.
DR=SOC SCI; BR=N/A
MAT124H1
Calculus and Linear Algebra for Commerce (B) [36L, 12T]
Second Term content of MAT133Y1; the final examination includes topics covered in MAT123H1. Offered in the Summer Session only; students not enrolled in MAT123H1 in the preceding Spring Term will NOT be allowed to enrol in MAT124H1. MAT123H1 together with MAT124H1 is equivalent for program and Prerequisite purposes to MAT133Y1.
Exclusion: MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1
Prerequisite: MAT123H1 successfully completed in the preceding Spring Term
DR=SOC SCI; BR=N/A
MAT135Y1
Calculus I [72L, 24T]
Review of trigonometric functions; trigonometric identities and trigonometric limits. Review of differential calculus; applications. Integration and fundamental theorem; applications. Series. Introduction to differential equations.
Exclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT133Y1, MAT137Y1, MAT157Y1
Prerequisite: MCV4U, MHF4U
DR= SCI; BR=5
MAT125H1
Calculus I (A) [36L]
First term of MAT135Y1. Students in academic difficulty in MAT135Y1 who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT135Y1 and enrol in MAT125H1 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 MAT135Y1 in the Fall Term will not be allowed to enrol in MAT125H1. MAT125H1 together with MAT126H1 is equivalent for program and Prerequisite purposes to MAT135Y1.
Exclusion: MAT123H1, MAT124H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1
NOTE: students who enrol in MAT135Y1 after completing MAT125H1 but not MAT126H1 do not receive degree credit for MAT135Y1; it is counted ONLY as an “Extra Course.”
Prerequisite: Enrolment in MAT135Y1, and withdrawal from MAT135Y1 after two midterms, with a mark of at least 20% in the second midterm.
DR= SCI; BR=5
MAT126H1
Calculus I (B) [36L, 12T]
Second Term content of MAT135Y1; the final examination includes topics covered in MAT125H1. Offered in the Summer Session only; students not enrolled in MAT125H1 in the preceding Spring Term will NOT be allowed to enrol in MAT126H1. MAT125H1 together with MAT126H1 is equivalent for program and Prerequisite purposes to MAT135Y1.
Exclusion: MAT123H1, MAT124H1, MAT133Y1, MAT135Y1, MAT137Y1
Prerequisite: MAT125H1 successfully completed in the preceding Spring Term
DR=SCI; BR=5
MAT137Y1
Calculus [72L, 24T]
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 first a review of trigonometric functions followed by discussion of trigonometric identities. 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: MAT125H1, MAT126H1, MAT135Y1, MAT157Y1
Prerequisite: MCV4U, MHF4U
DR= SCI; BR=5
MAT157Y1
Analysis I [72L, 48T]
A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. 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: MAT137Y1
Prerequisite: MCV4U, MHF4U
DR= SCI; BR=5
MAT223H1
Linear Algebra I [36L, 12T]
Matrix arithmetic and linear systems. Rn subspaces, linear independence, bases, dimension; column spaces, null spaces, rank and dimension formula. Orthogonality orthonormal sets, Gram-Schmidt orthogonalization process; least square approximation. Linear transformations Rn—>Rm. The determinant, classical adjoint, Cramer’s Rule. Eigenvalues, eigenvectors, eigenspaces, diagonalization. Function spaces and application to a system of linear differential equations.
Exclusion: MAT240H1
Prerequisite: MCV4U, MHF4U
DR= SCI; BR=5
MAT224H1
Linear Algebra II [36L, 12T]
Abstract vector spaces: subspaces, dimension theory. Linear mappings: kernel, image, dimension theorem, isomorphisms, matrix of linear transformation. Changes of basis, invariant spaces, direct sums, cyclic subspaces, Cayley-Hamilton theorem. Inner product spaces, orthogonal transformations, orthogonal diagonalization, quadratic forms, positive definite matrices. Complex operators: Hermitian, unitary and normal. Spectral theorem. Isometries of R2 and R3.
Exclusion: MAT247H1
Prerequisite: MAT223H1/MAT240H1
DR= SCI; BR=5
MAT235Y1
Calculus II [72L]
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: MAT237Y1, MAT257Y1
Prerequisite: MAT135Y1/MAT137Y1/MAT157Y1
DR= SCI; BR=5
MAT237Y1
Multivariable Calculus [72L]
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: MAT235Y1, MAT257Y1
Prerequisite: MAT137Y1/MAT157Y1/MAT135Y1(90%),MAT223H1/MAT240H1
DR= SCI; BR=5
MAT240H1
Algebra I [36L, 24T]
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.
Prerequisite: MCV4U, MHF4U
Co-requisite: MAT157Y1
DR= SCI; BR=5
MAT244H1
Introduction to Ordinary Differential Equations [36L]
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. Applications in life and physical sciences and economics.
Exclusion: MAT267H1
Prerequisite: MAT135Y1/MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Co-requisite: MAT235Y1/MAT237Y1
DR= SCI; BR=5
MAT246H1
Concepts in Abstract Mathematics (formerly MAT246Y1) [36L]
Designed to introduce students to mathematical proofs and abstract mathematical concepts. Topics may include modular arithmetic, sizes of infinite sets, and a proof that some angles cannot be trisected with straightedge and compass.
Exclusion: MAT157Y1, 246Y1
Prerequisite: MAT133Y1/MAT135Y1/MAT137Y1,MAT223H1
DR= SCI; BR=5
MAT247H1
Algebra II [36L, 12T]
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.
Prerequisite: MAT240H1
Co-requisite: MAT157Y1
DR= SCI; BR=5
MAT257Y1
Analysis II [72L, 48T]
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; Stokes’ theorem for differential forms and classical versions.
Prerequisite: MAT157Y1, MAT240H1, MAT247H1
DR= SCI; BR=5
MAT267H1
Advanced Ordinary Differential Equations I [36L, 12T]
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: MAT244H1
Prerequisite: MAT157Y1, MAT247H1
Co-requisite: MAT257Y1
DR= SCI; BR=5
MAT271H1
Insights from Mathematics [36L, 6T]
This breadth course is accessible to students with limited mathematical background. Various mathematical techniques will be illustrated with examples from humanities and social science disciplines. Some of the topics will incorporate user friendly computer explorations to give participants the feel of the subject without requiring skill at calculations.
DR= SCI; BR=5
MAT299Y1
Research Opportunity Program
Credit course for supervised participation in faculty research project. Details here.
DR= SCI; BR=TBA
300-Series Courses
MAT301H1
Groups and Symmetries [36L]
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: MAT347Y1
Prerequisite: MAT224H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1. (These Prerequisites will be waived for students who have MAT257Y1)
DR= SCI; BR=TBA
MAT309H1
Introduction to Mathematical Logic [36L]
Predicate calculus. Relationship between truth and provability; Gödel’s completeness theorem. First order arithmetic as an example of a first-order system. Gödel’s incompleteness theorem; outline of its proof. Introduction to recursive functions.
Exclusion: CSC438H1
Prerequisite: MAT223H1/MAT240H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1 (These Prerequisites will be waived for students who have MAT257Y1)
DR= SCI; BR=TBA
MAT315H1
Introduction to Number Theory [36L]
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: MAT223H1/MAT240H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1 (These Prerequisites will be waived for students who have MAT257Y1)
DR= SCI; BR=TBA
MAT327H1
Introduction to Topology [36L]
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. Students in the math specialist program wishing to take additional topology courses are advised to obtain permission to take MAT1300Y. Students must meet minimum GPA requirements as set by SGS and petition with their college.
Prerequisite: MAT257Y1/(MAT224H1, MAT237Y1, MAT246H1 and permission of the instructor)
DR= SCI; BR=TBA
MAT329Y1
Concepts in Elementary Mathematics [72L]
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 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 2.5
DR= SCI; BR=TBA
MAT332H1
Introduction to Graph Theory [36L]
This course will explore the following topics: Graphs, Subgraphs, Isomorphism, Trees, Connectivity, Euler and Hamiltonian Properties, Matchings, Vertex and Edge Colourings, Planarity, Network Flows and Strongly Regular Graphs. Participants will be encouraged to use these topics and execute applications to such problems as timetabling, tournament scheduling, experimental design and finite geometries. Students are invited to replace MAT344H1 with MAT332H1.
Prerequisite: MAT224H1/MAT247H1
Recommended Corequisite: MAT301H1/MAT347Y1
DR= SCI; BR=TBA
MAT334H1
Complex Variables [36L]
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: MAT354H1
Prerequisite: MAT223H1, MAT235Y1/MAT237Y1
DR= SCI; BR=TBA
MAT335H1
Chaos, Fractals and Dynamics [36L]
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: MAT137Y1/200-level calculus, MAT223H1
DR= SCI; BR=TBA
MAT337H1
Introduction to Real Analysis [36L]
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: MAT357H1
Prerequisite: MAT224H1, MAT235Y1/MAT237Y1,MAT246H1
NOTE: These Prerequisites will be waived for students who have MAT257Y1
DR= SCI; BR=TBA
MAT344H1
Introduction to Combinatorics [36L]
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: MAT223H1/MAT240H1
DR= SCI; BR=TBA
MAT347Y1
Groups, Rings and Fields [72L, 24T]
Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hölder 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: MAT257Y1
DR= SCI; BR=TBA
MAT354H1
Complex Analysis I [36L]
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.
Prerequisite: MAT257Y1
DR= SCI; BR=TBA
MAT357H1
Real Analysis I [36L]
Function spaces; Arzela-Ascoli theorem, 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.
Prerequisite: MAT257Y1/(MAT327H1 and permission of instructor)
DR= SCI; BR=TBA
MAT363H1
Introduction to Differential Geometry [36L]
Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.
Prerequisite: MAT224H1, MAT237Y1/MAT257Y1
DR= SCI; BR=TBA
MAT390H1
History of Mathematics up to 1700 [36L]
A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)
Exclusion: HPS309H1, 310Y1, HPS390H1
Prerequisite: at least one full MAT 200-level course
DR= SCI; BR=TBA
MAT391H1
History of Mathematics afTER 1700 [24L, 12T]
A survey of the development of mathematics fROM 1700 to the present with emphasis on technical development. (Offered in alternate years)
Exclusion: HPS309H1, 310H1, HPS391H1
Prerequisite: At least one fULL 200-level MAT course
DR= SCI; BR=TBA
MAT393Y1
Independent Work in Mathematics [TBA]
MAT394Y1
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
DR= SCI; BR=TBA
MAT395H1
Independent Work in Mathematics [TBA]
MAT396H1
Independent Work in Mathematics [TBA]
MAT397H1
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
DR= SCI; BR=TBA
MAT398H0
Independent Experiential Study Project
MAT399Y0
Independent Experiential Study Project
An instructor-supervised group project in an off-campus setting. Details here.
DR= SCI; BR=TBA
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 graduate brochure for more details.
MAT401H1
Polynomial Equations and Fields [36L]
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. Insolvability of quintics by radicals.
Exclusion: MAT347Y1
Prerequisite: MAT301H1
DR= SCI; BR=TBA
MAT402H1
Classical Geometries [36L]
Euclidean and non-euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.
Prerequisite: MAT301H1/MAT347Y1, MAT235Y1/MAT237Y1/MAT257Y1
DR= SCI; BR=TBA
MAT409H1
Set Theory [36L]
Set theory and its relations with other branches of mathematics. ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible sets and the continuum hypothesis. Introduction to independence proofs. Topics from large cardinals, infinitary combinatorics and descriptive set theory.
Prerequisite: MAT357H1
DR= SCI; BR=TBA
MAT415H1
Topics in Algebraic Number Theory [36L]
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: MAT347Y1 or permission of instructor
DR= SCI; BR=TBA
MAT417H1
Topics in Analytic Number Theory [36L]
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: MAT334H1/MAT354H1/permission of instructor
DR= SCI; BR=TBA
MAT425H1
Differential Topology [36L]
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: MAT257Y1, MAT327H1
DR= SCI; BR=TBA
MAT427H1
Algebraic Topology [36L]
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: MAT327H1, MAT347Y1
DR= SCI; BR=TBA
MAT443H1
Computer Algebra [36L]
Introduction to algebraic algorithms used in computer science and computational mathematics. Topics may include: generating sequences of random numbers, fast arithmetic, Euclidean algorithm, factorization of integers and polynomials, primality tests, computation of Galois groups, Gröbner bases. Symbolic manipulators such as Maple and Mathematica are used.
Prerequisite: MAT347Y1
DR= SCI; BR=TBA
MAT445H1
Representation Theory [36L]
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: MAT347Y1
DR= SCI; BR=TBA
MAT448H1
Introduction to Commutative Algebra and Algebraic Geometry [36L]
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: MAT347Y1
DR= SCI; BR=TBA
MAT449H1
Algebraic Curves [36L]
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: MAT347Y1, MAT354H1
DR= SCI; BR=TBA
MAT454H1
Complex Analysis II [36L]
Harmonic functions, Harnack’s principle, Poisson’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.
Prerequisite: MAT354H1
DR= SCI; BR=TBA
MAT457H1
Real Analysis I (formerly MAT457Y1) [36L]
Lebesque measure and integration; convergence theorems, Fubini’s theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem. Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, Lp spaces, Holder and Minkowski inequalities.
Prerequisite: MAT357H1
Exclusion: MAT457Y1
DR= SCI; BR=TBA
MAT458H1
Real Analysis II (formerly MAT457Y1) [36L]
Fourier series and transform, convergence results, Fourier inversion theorem, L2 theory, estimates, convolutions. Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.
Prerequisite: MAT457H1
Exclusion: MAT457Y1
DR= SCI; BR=TBA
MAT464H1
Differential Geometry [36L]
Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar 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: MAT363H1
DR= SCI; BR=TBA
MAT468H1
Ordinary Differential Equations II [36L]
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: MAT267H1, MAT354H1, MAT357H1
DR= SCI; BR=TBA
MAT475H1
Problem Solving Seminar [TBA]
This course addresses the question: “How do you attack a problem the likes of which you’ve never seen before?” Students will apply Polya’s principles of mathematical problem solving, draw upon their previous mathematical knowledge, and explore the creative side of mathematics in solving a variety of interesting problems and explaining those solutions to others.
Prerequisite: MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, and at least ONE 300-level MAT or APM course
DR= SCI; BR=TBA
MAT477Y1
Seminar in Mathematics [TBA]
Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required)
Prerequisite: MAT347Y1, MAT354H1, MAT357H1; or permission of instructor.
DR= SCI; BR=TBA
MAT495H1
Readings in Mathematics [TBA]
MAT496H1
Readings in Mathematics [TBA]
MAT497H1
Readings in Mathematics [TBA]
MAT498Y1
Readings in Mathematics [TBA]
MAT499Y1
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
DR= SCI; BR=TBA
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