APM236H1
Applications of Linear Programming 39L
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)
APM346H1
Differential Equations 39L
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
APM351Y1
Partial Differential Equations 78L
Diffusion and wave equations. Separation of variables. Fourier series. Laplace's equation; Green's function. Schr”dinger
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
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.
APM421H1
Mathematical Foundations of Quantum 39L
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. Schr”dinger and Heisenberg pictures, complete sets of observables, representations of the canonical
commutative relations; essential self-adjointedness of Schr”dinger operators, density operators, elements of scattring theory.
Prerequisite: (MAT224H1, MAT337H1)/MAT357H1
APM426H1
General Relativity 39L
Einstein's theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manisfestation 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
APM436H1
Fluid Mechanics 39L
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
APM441H1
Asymptotic and Perturbation Methods 39L
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
APM446H1
Applied Nonlinear Equations 39L
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. Propogation of nonlinear waves; dispersion, modulation,
optical bistability. Global behaviour solutions; asymptotics and blow-up.
Prerequisite: APM346H1/APM351Y1
APM456H1
Control Theory and Optimization 39L
Differential systems with controls and reachable sets. Non-commutativity, Lie bracket and controllability. Optimality and
maximum principle. Hamiltonian formalism and symplectic geometry. Integrability. Applications to engineering, mechanics and
geometry.
Prerequisite: MAT357H1 or MAT244H1/MAT267H1, MAT337H1
APM461H1
Combinatorial Methods 39L
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial
identities.
Prerequisite: MAT224H1
Recommended preparation: MAT344H1
APM462H1
Nonlinear Optimization 39L
(formerly APM362H1)
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
APM466H1
Mathematical Theory of Finance 39L
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
APM496H1/497H1/498Y1/499Y1
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
Mathematics Courses
See page 30 for Key to Course Descriptions.
For Distribution Requirement purposes,
all MAT courses except MAT123H1, MAT124H1 and MAT133Y1 are
classified as SCIENCE
courses (see page 22).
SCI199Y1
First Year Seminar 52S
Undergraduate seminar that focuses on specific ideas, questions, phenomena or controversies, taught by a regular Faculty
member deeply engaged in the discipline. Open only to newly admitted first year students. It may serve as a breadth
requirement course; see page 40.
JUM102H1
Mathematics as an Interdisciplinary Pursuit 26L, 13T
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, are considered. (Offered every three years) \
JUM102H1 is particularly
suited as a Science Distribution Requirement course for Humanities and Social
Science students.
JUM103H1
Mathematics as a Recreation 26L, 13T
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)
JUM103H1 is particularly suited as a Science Distribution
Requirement course for Humanities
and Social Science students.
JUM105H1
Mathematical Personalities 26L, 13T
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, G”del, Erd”s, Coxeter, Grothendieck. (Offered every
three years)
JUM105H1
is particularly suited as a Science Distribution Requirement course for Humanities
and Social Science students.
MAT123H1,124H1
See below MAT133Y1
MAT125H1,126H1
See below MAT135Y1
MAT133Y1
Calculus and Linear Algebra for Commerce 78L,
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, MAT136Y1, MAT137Y1, MAT157Y1
Prerequisite: MCB4U,MGA4U/MDM4U
MAT133Y1
counts as a Social Science course
MAT123H1
Calculus and Linear Algebra for Commerce (A) 39L
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, MAT136Y1, 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.
MAT123H1
is a Social Science course
MAT124H1
Calculus and Linear Algebra for Commerce (B) 39L, 13T
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, MAT136Y1, MAT137Y1, MAT157Y1
Prerequisite: MAT123H1 successfully
completed in the preceding Spring Term
MAT124H1 is a Social Science course
MAT135Y1
Calculus I 78L, 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, MAT136Y1, MAT137Y1, MAT157Y1
Prerequisite: MCB4U
MAT125H1
Calculus I (A) 39L
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, MAT136Y1, 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.
MAT126H1
Calculus I (B) 39L, 13T
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, MAT136Y1, MAT137Y1
Prerequisite: MAT125H1 successfully completed in the preceding Spring Term
MAT136Y1
Calculus and its Foundations 104L, 52T
Limited to out-of-province students interested in the biological, physical, or computer sciences, whose high school mathematics
preparation is strong but does not include calculus. Develops the concepts of calculus at the level of MAT135Y1. May include
background material on functions, analytic geometry, and trigonometry, as well as on calculus.
Exclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1
Prerequisite: Solid background in high school mathematics, up to and including Grade 11
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 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, MAT136Y1, MAT157Y1
Prerequisite: MCB4U,MGA4U
MAT157Y1
Analysis I 78L, 52T
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: MCB4U, MGA4U
JMB170Y1
Biology, Models, and Mathematics 52L, 26T
Applications of mathematics to biological problems in physiology, biomechanics, genetics, evolution, growth, population
dynamics, cell biology, ecology and behaviour.
Prerequisite: OAC/U Biology
Co-requisite: BIO150Y1
MAT223H1
Linear Algebra I 39L, 13T
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:MCB4U, MGA4U
MAT224H1
Linear Algebra II 39L, 13T
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
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: MAT237Y1, MAT257Y1
Prerequisite: MAT135Y1/MAT136Y1/MAT137Y1/MAT157Y1
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: MAT235Y1, MAT257Y1
Prerequisite: MAT135Y1(80%)/MAT136Y1/MAT137Y1/MAT157Y1
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.
Prerequisite: MCB4U, MGA4U
Co-requisite: MAT157Y1
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: MAT267H1
Prerequisite: MAT135Y1/MAT136Y1/MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Co-requisite: MAT235Y1/MAT237Y1
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.
Exclusion: MAT257Y1
Prerequisite: MAT133Y1/MAT135Y1/MAT136Y1/MAT137Y1
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.
Prerequisite: MAT240H1, MAT157Y1
MAT257Y1
Analysis II 78L, 52T
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
Co-requisite: MAT247H1
MAT267H1
Advanced 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: MAT244H1
Prerequisite: MAT157Y1, MAT247H1
Co-requisite: MAT257Y1
MAT299Y1
Research Opportunity Program
Credit course for supervised participation in faculty research project. See page 43 for details.
300-Series Courses
MAT301H1
Groups and Symmetries 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: MAT347Y1
Prerequisite: MAT224H1, MAT235Y1/MAT237Y1,MAT246Y1/ (CSC236H1/CSC240H1, PHL245H1)/MAT257Y1
MAT309H1
Introduction to Mathematical Logic 39L
Predicate calculus. Relationship between truth and provability; G”del's completeness theorem. First order arithmetic as an
example of a first-order system. G”del's incompleteness theorem; outline of its proof. Introduction to recursive functions.
Exclusion: CSC438H1
Prerequisite: MAT223H1/MAT240H1, MAT235Y1/MAT237Y1, MAT246Y1/(CSC236H1/CSC240H1, PHL245H1)/MAT257Y1
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: (MAT235Y1/MAT237Y1, MAT223H1/MAT240H1)/MAT257Y1
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: MAT257Y1/(MAT237Y1 and permission of the instructor)
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 2.5
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: MAT354H1
Prerequisite: MAT235Y1/MAT237Y1/MAT257Y1
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: MAT137Y1/200-level calculus
MAT337H1
Introduction to Real Analysis 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: MAT357H1
Prerequisite: (MAT223H1, MAT235Y1/MAT237Y1, MAT246Y1)/MAT257Y1
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: MAT223H1/MAT240H1
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: MAT247H1, MAT257Y1
MAT354H1
Complex Analysis I 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.
Prerequisite: MAT247H1, MAT257Y1
MAT357H1
Real Analysis I 39L, 13T
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: MAT247H1, MAT257Y1, (MAT327H1 or permission of instructor)
MAT363H1
Introduction to Differential Geometry 39L
Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for
surfaces. Surfaces of constant curvature.
Prerequisite: MAT224H1/MAT247H1, MAT237Y1/MAT257Y1
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: HPS309H1, 310Y1, HPS390H1
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: HPS309H1, 310H1, HPS391H1
Prerequisite: At least one full 200-level MAT course
MAT393Y1/394Y1
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
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
MAT398H0/399Y0
Independent Experiential Study Project
An instructor-supervised group project in an off-campus setting. See page 43 for details.
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.
MAT401H1
Polynomial Equations and Fields 39L
(formerly MAT302H1)
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: MAT224H1, MAT235Y1/MAT237Y1,MAT246Y1/ (CSC236H1/238H1/CSC240H1, PHL245H1)/257Y1
MAT402H1
Classical Geometries 39L
(formerly MAT365H1)
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
MAT409H1
Set Theory 39L
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
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: MAT347Y1 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: MAT334H1/MAT354H1/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: MAT257Y1, MAT327H1
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: MAT327H1, MAT347Y1
MAT443H1
Computer Algebra 39L
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, Gr”bner bases. Symbolic manipulators such as Maple and Mathematica are used.
Prerequisite: MAT347Y1
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: MAT347Y1
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: MAT347Y1
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: MAT347Y1, MAT354H1
MAT454H1
Complex Analysis II 39L
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
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: MAT357H1
MAT464H1
Differential Geometry 39L
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
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: MAT267H1, MAT354H1, MAT357H1
MAT477H1
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.
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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