
Mathematics
Calculus 050a/b, 051a/b, 081a/b, 250a/b, 251a/b, 280a/b, and 281a/b are offered jointly by the Departments of Applied Mathematics and Mathematics.
Calculus 091a/b is offered by the Department of Applied Mathematics.
Differential Equations 215a is offered jointly by the Departments of Applied Mathematics and Mathematics.
Linear Algebra 040a/b is offered jointly by the Departments of Applied Mathematics and Mathematics.
Please refer to the CALCULUS (S), DIFFERENTIAL EQUATIONS (S), and LINEAR ALGEBRA (S) Subjects for those first and second year course offerings.
Note: Combinations of the courses Mathematics 012a/b, 017a/b, 028a/b, 030, 031, Calculus 081a/b, Calculus 091a/b will not satisfy the mathematics prerequisites of certain Honors Programs offered by the Faculty of Science. 
Mathematics 012a/b, Introductory Calculus  
 Description: Introduction to differential and integral calculus including maxima and minima, calculus of the trigonometric functions, and practical problems involving extrema and related rates.  Antirequisite(s): OAC Calculus, Grade 12U Advanced Functions and Introductory Calculus (MCB4U).  Prerequisite(s): Ontario Grade 12 Advanced Level Mathematics or Grade 11 Functions (MCF3M), or Functions and Relations (MCR3U) or equivalent.  4 lecture hours, 0.5 course.  back to top 
Mathematics 017a/b, Algebra and Geometry  
 Description: Transformations and matrices, conic sections, vectors, linear combinations, equations of lines and planes, complex numbers, mathematical induction.  Antirequisite(s): OAC Algebra and Geometry; Grade 12U Geometry and Discrete Mathematics (MGA4U); Mathematics 030, Mathematics 031; Applied Mathematics 025a/b; all courses in Mathematics numbered 100 or above.  Prerequisite(s): Ontario Grade 12 Advanced Level Mathematics, or Grade 11 Functions (MCF3M), or Functions and Relations (MCR3U), or equivalent.  3 lecture hours, 1 tutorial hour, 0.5 course.  back to top 
Mathematics 028a/b, Finite Mathematics  
 Description: Permutations and combinations; probability theory.  Antirequisite(s): Mathematics 031, Mathematics 208a/b, Mathematics 222a, Statistical Sciences 135, Statistical Sciences 241a/b, the former Mathematics 206a, the former Statistical Sciences 255.  Prerequisite(s): Any OAC Mathematics, or Grade 12U Geometry and Discrete Mathematics (MGA4U), or Grade 12U Advanced Functions and Introductory Calculus (MCB4U), or Grade 12U Mathematics of Data Management (MDM4U), or Mathematics 012a/b or Mathematics 017a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 030, Calculus and Linear Algebra  
 Description: Calculus including elementary techniques of integration; applications such as area, volume, probability; functions of several variables, Lagrange multipliers. Linear algebra including vectors and matrices, linear equations, determinants.  Antirequisite(s): Calculus 050a/b, Calculus 051a/b, Calculus 081a/b, Linear Algebra 040a/b, Applied Mathematics 025a/b, Applied Mathematics 026, Applied Mathematics 213b, Mathematics 031, Mathematics 203b, Mathematics 283b, the former Applied Mathematics 212a, 214a/b, the former Mathematics 202a.  Prerequisite(s): OAC Calculus or Grade 12U Advanced Functions and Introductory Calculus (MCB4U), or Mathematics 012a/b.  3 lecture hours, 1.0 course.  back to top 
Mathematics 031, Finite Mathematics  
 Description: Set theory and logic, permutations and combinations, probability, discrete and continuous random variables, vectors and matrices, linear equations, determinants.  Antirequisite(s): Applied Mathematics 025a/b, Applied Mathematics 213b, the former Applied Mathematics 212a, 214a/b; Linear Algebra 040a/b; Mathematics 028a/b, Mathematics 030, Mathematics 208a/b, Mathematics 222a, Mathematics 283b; the former Mathematics 202a, 203b; Statistical Sciences 241a/b, the former Statistical Sciences 255.  Prerequisite(s): Any OAC Mathematics, or Grade 12U Geometry and Discrete Mathematics (MGA4U), or Grade 12U Advanced Functions and Introductory Calculus (MCB4U), or Grade 12U Mathematics of Data Management (MDM4U), or Mathematics 012a/b or Mathematics 017a/b.  3 lecture hours, 1.0 course.  back to top 
Mathematics 060a/b, Fundamental Concepts in Mathematics  
 Description: Primarily for students interested in pursuing a degree in one of the mathematical sciences. Logic, set theory, relations, functions and operations, careful study of the integers, discussion of the real and complex numbers, polynomials, and infinite sets.  Antirequisite(s): Mathematics 222a.  Prerequisite(s): OAC Algebra and Geometry or Mathematics 017a/b or Grade 12 Geometry and Discrete Mathematics (MGA4U).  4 lecture hours, 0.5 course.  back to top 
Mathematics 200a/b, Intermediate Linear Algebra I  
 Description: A rigorous development of lines and planes in R^{n}; linear transformations and abstract vector spaces. Determinants and an introduction to diagonalization and its applications including the characteristic polynomials, eigenvalues and eigenvectors.  Antirequisite(s): Mathematics 283b and the former Mathematics 203b.  Prerequisite(s): Linear Algebra 040a/b with a minimum mark of 60% or Mathematics 060a/b with a minimum mark of 70% or permission of the Mathematics Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 201a/b, Intermediate Linear Algebra II  
 Description: A continuation of the material of Mathematics 200a/b including properties of complex numbers and the principal axis theorem; singular value decomposition; linear groups; similarity; Jordan canonical form; CayleyHamilton theorem; bilinear forms; Sylvester's theorem.  Prerequisite(s): Mathematics 200a/b  3 lecture hours, 0.5 course.  back to top 
Mathematics 207a/b, Real Analysis I  
 Description: A rigorous introduction to analysis on the real line, primarily for honors students. Sets, functions, natural numbers. Axioms for the real numbers. Completeness and its consequences. Sequences and limits. Continuous and differentiable functions. The Mean Value Theorem.  Prerequisite(s): Calculus 050a/b plus one of Calculus 051a/b or Calculus 081a/b, or Applied Mathematics 026, in each case with a minimum mark of 60%.  4 lecture hours, 0.5 course.  back to top 
Mathematics 208a/b, Introduction to Mathematical Problems  
 Description: Primarily for Mathematics students, but will interest other students with ability in and curiosity about mathematics in the modern world as well as in the past. Stresses development of students' abilities to solve problems and construct proofs. Topics will be selected from: counting, recurrence, induction; number theory; graph theory; parity, symmetry; geometry.  Prerequisite(s): Calculus 050a/b and Calculus 051a/b or Calculus 081a/b, or Applied Mathematics 026, (in each case with a minimum mark of 60%); or permission of the instructor.  3 lecture hours, 0.5 course.  back to top 
Mathematics 217a/b, Real Analysis II  
 Description: A continuation of the rigorous introduction to analysis on the real line, begun in Mathematics 207a/b, primarily for honors students. Uniform continuity. The Riemann integral. Series of numbers, convergence theory. Power series and Taylor series. Sequences and series of functions. Uniform convergence.  Antirequisite(s): Mathematics 306a/b.  Prerequisite(s): Mathematics 207a/b with a minimum mark of 60%, or permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 221F/G, Conceptual Development of Mathematics  
 Description: A survey of some important basic concepts of mathematics in a historical setting, and in relation to the broader history of ideas. Topics may include: the evolution of the number concept, the development of geometry, Zeno's paradoxes.  Antirequisite(s): Philosophy 221F/G, the former Philosophy 121  Prerequisite(s): 1.0 course of university level mathematics  3 lecture hours, 0.5 course.  back to top 
Mathematics 222a, Discrete Structures I  
 Description: This course provides an introduction to logical reasoning and proofs. Topics include sets, counting (permutations and combinations), mathematical induction, relations and functions, partial order relations, equivalence relations, groups and applications to errorcorrecting codes.  Antirequisite(s): Software Engineering 251a/b  Prerequisite(s): One full course equivalent chosen from the following: Mathematics 030, Applied Mathematics 026, Calculus 050a/b, Calculus 051a/b, Calculus 081a/b, Linear Algebra 040a/b, (in each case with a minimum mark of 60%)  4 lecture hours, 0.5 course.  back to top 
Mathematics 223b, Discrete Structures II  
 Description: This course continues the development of logical reasoning and proofs begun in Mathematics 222a. Topics include elementary number theory (gcd, lcm, Euclidean algorithm, congruences, Chinese remainder theorem) and graph theory (connectedness, complete, regular and bipartite graphs; trees and spanning trees, Eulerian and Hamiltonian graphs, planar graphs; vertex, face and edge colouring; chromatic polynomials).  Prerequisite(s): Mathematics 222a.  4 lecture hours, 0.5 course.  back to top 
Mathematics 227b, Complex Variables  
 Description: Complex numbers, CauchyRiemann equations, elementary functions, integrals, Cauchy's theorem and integral formula and applications, Taylor and Laurent expansions, isolated singularities, residue theorem and applications. Cannot be taken for credit by students in honors Mathematics programs.  Antirequisite(s): Mathematics 306a/b, Mathematics 307b, Applied Mathematics 301a.  Prerequisite(s): Calculus 250a/b or Calculus 280a/b, or the former Applied Mathematics 200 or the former Mathematics 204a.  3 lecture hours, 0.5 course.  back to top 
 Description: An introduction to abstract algebra, with principal emphasis on the structure of groups, rings, integral domains and fields. Cannot be taken for credit by students in honors Mathematics programs.  Antirequisite(s): Mathematics 302a, the former Mathematics 303b.  Prerequisite(s): Linear Algebra 040a/b, or the former Applied Mathematics 212a, 214a/b, or the former Mathematics 202a.  3 lecture hours, 1.0 course.  back to top 
Mathematics 231, Synthetic Geometry  
 Description: Groups of transformations of the Euclidean plane, inversion, the projective plane.  Antirequisite(s): The former Mathematics 319a/b, Mathematics 419a/b.  Prerequisite(s): One full course or equivalent, chosen from the following: Calculus 050a/b, Calculus 051a/b or Calculus 081a/b, Applied Mathematics 026, Mathematics 030, Mathematics 031, Mathematics 028a/b; Linear Algebra 040a/b.  3 lecture hours, 1.0 course.  back to top 
Mathematics 236, Elementary Operations Research with Applications  
 Description: Linear programming, basic probability and statistical distributions, networks, decision analysis, utility, game theory, inventory analysis, queuing theory, simulation, Markovian decision model, forecasting. Cannot be taken for credit by students in honors Mathematics programs.  Antirequisite(s): The former Actuarial Science 325a, or 326b, Statistical Sciences 437a/b. This course is equivalent to the former Statistical Sciences 236 and has the same antirequisites as the former Statistical Sciences 236.  Prerequisite(s): a) One full course or equivalent, chosen from the following: Calculus 050a/b, Calculus 051a/b or Calculus 081a/b, Applied Mathematics 026, Mathematics 030, Mathematics 031, Mathematics 028a/b; Linear Algebra 040a/b; and b) if Mathematics 028a/b or Mathematics 031 was not taken in a), one of the following must have been taken previously or concurrently during the fall term: Economics 122a/b, Economics 222a/b, Statistical Sciences 135, Statistical Sciences 241a/b, the former Statistical Sciences 232a, 255.  3 lecture hours, 1.0 course.  back to top 
Mathematics 240, Elementary Theory of Numbers  
 Description: Euclidean algorithm, congruences, indices, continued fractions, Gaussian integers, partitions and Diophantine equations.  Antirequisite(s): Mathematics 310a, the former Mathematics 311b.  Prerequisite(s): One full course or equivalent, chosen from the following: Calculus 050a/b, either Calculus 051a/b or Calculus 081a/b, Applied Mathematics 026, Mathematics 030, Mathematics 031, Mathematics 028a/b; Linear Algebra 040a/b.  3 lecture hours, 1.0 course.  back to top 
Mathematics 283b, Linear Algebra  
 Description: Linear transformations, matrix representation, rank, change of basis, eigenvalues and eigenvectors, inner product spaces, quadratic forms and conic sections. Emphasis on problemsolving rather than theoretical development. Cannot be taken for credit by students in honors Mathematics programs.  Antirequisite(s): Applied Mathematics 213b, the former Applied Mathematics 214a/b, the former Mathematics 203b.  Prerequisite(s): Linear Algebra 040a/b or the former Applied Mathematics 212a, or the former Mathematics 202a.  3 lecture hours, 0.5 course.  back to top 
Mathematics 302a, Group Theory  
 Description: An introduction to the theory of groups: cyclic, dihedral, symmetric, alternating, matrix, and quaternion groups; subgroups, quotient groups and homomorhpisms, cosets and Lagrange's theorem, isomorphism theorems, solvable groups; group actions and representations, the class equation, Sylow theorems; direct products, the fundamental theorem of finitelygenerated abelian groups.  Prerequisite(s): A minimum mark of 60% in Applied Mathematics 213b or the former Mathematics 203b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 304a/b, Metric Space Topology  
 Description: An introduction to the theory of metric spaces with emphasis on the pointset topology of Euclidean nspace, including convergence, compactness, completeness, continuity, uniform continuity, homeomorphism, equivalence of metrics, connectedness, pathconnectedness, fixedpoint theorem for contractions, separability, complete normality, product spaces, category.  Prerequisite(s): Either Mathematics 207a/b or Mathematics 217a/b, each with a minimum mark of 60%.  3 lecture hours, 0.5 course.  back to top 
Mathematics 305a/b, Differential Equations  
 Description: Rigorous introduction to ordinary differential equations. Existence, uniqueness, and continuation of solutions. Linear systems with constant coefficients. Flows and dynamical systems. Series solutions.  Prerequisite(s): Calculus 251a/b or the former Mathematics 205b; Mathematics 217a/b or Mathematics 306a/b; Mathematics 304a/b; Applied Mathematics 213b or the former Mathematics 203b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 307a/b, Complex Analysis I  
 Description: The CauchyRiemann equations, elementary functions, branches of the logarithm and argument, Cauchy's integral theorem and formula, winding number, Liouville's theorem and the fundamental theorem of algebra, the identity theorem, the maximum modulus theorem, Taylor and Laurent expansions, isolated singularities, the residue theorem and applications, the argument principle and applications.  Antirequisite(s): Applied Mathematics 301a.  Prerequisite(s): Mathematics 217a/b or Mathematics 306a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 308a/b, Rings and Modules  
 Description: Commutative rings, ring homorphisms and quotient rings, ideals, rings of fractions, the Chinese remainder theorem; Euclidean domains, principal ideal domains, unique factorization domains; polynomial rings over fields; modules, direct sums of modules, free modules; modules over a principal ideal domain, the rational canonical form, the Jordan canonical form.  Antirequisite(s): The former Mathematics 303b.  Prerequisite(s): Mathematics 302a.  3 lecture hours, 0.5 course.  back to top 
Mathematics 310a, Elementary Number Theory I  
 Description: Divisibility, primes, congruences, theorems of Fermat and Wilson, Chinese remainder theorem, quadratic reciprocity, some functions of number theory, diophantine equations, simple continued fractions.  Antirequisite(s): Mathematics 240.  Prerequisite(s): Any course in Mathematics or Applied Mathematics at the 200 level or higher.  3 lecture hours, 0.5 course.  back to top 
Mathematics 313a/b, Elementary Number Theory II  
 Description: Arithmetic functions, perfect numbers, the Möbius inversion formula, introduction to Dirichlet series and the Riemann zeta function, some methods of combinatorial number theory, primitive roots and their relationship with quadratic reciprocity, the Gaussian integers, sums of squares and Minkowski's theorem, square and triangular numbers, Pell's equation, introduction to elliptic curves.  Antirequisite(s): Mathematics 240  Prerequisite(s): Mathematics 310a or Mathematics 223b  3 lecture hours, 0.5 course.  back to top 
Mathematics 329a/b, Introduction to Algebraic Curves  
 Description: Geometry of algebraic curves over the rational, real and complex fields. Classification of affine conics, singularities, intersection numbers, tangents, projective algebraic curves, multiplicity of points, flexes. Some discussion of cubic curves.  Antirequisite(s): Mathematics 231 and the former Mathematics 319a/b.  Prerequisite(s): Linear Algebra 040a/b and 1.0 course in Mathematics or Applied Mathematics at the 200 level or higher.  3 lecture hours, 0.5 course.  back to top 
Mathematics 343a/b, Combinatorial Mathematics  
 Description: Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.  Prerequisite(s): One of Mathematics 223b, Mathematics 283b, Applied Mathematics 213b, the former Mathematics 203b or permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 344a/b, Discrete Optimization  
 Description: Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.  Prerequisite(s): One of Mathematics 223b, Mathematics 236, Mathematics 343a/b, the former Statistical Sciences 236, the former Actuarial Science 325a/b, or permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 398a/b, Special Topics in Mathematics  
 Description:  Prerequisite(s): Permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 399a/b, Special Topics in Mathematics  
 Description:  Prerequisite(s): Permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 402a/b, Linear Algebra III  
 Description: Advanced topics in linear algebra: linear functionals and dual spaces; determinants; eigenvalues and eigenvectors, diagonalizable and triangulable operators, similarity, rational and Jordan canonical forms, invariant factors; bilinear, quadratic and Hermitian forms, congruence; positive forms and operators; inner product spaces, orthogonal and unitary operators, selfadjoint and normal operators, the spectral theorem.  Prerequisite(s): The former Mathematics 303b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 403a/b, Field Theory  
 Description: Automorphisms of fields, separable and normal extensions, splitting fields, fundamental theorem of Galois theory, primitive elements, Lagrange's theorem. Finite fields and their Galois groups, cyclotomic extensions and polynomials, applications of Galois theory to geometric constructions and solution of algebraic equations.  Prerequisite(s): Mathematics 308a/b or the former Mathematics 303b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 404a, General Topology  
 Description: Topological spaces, operations on subsets (e.g. closure), neighbourhoods, bases, subspaces, quotient spaces, product spaces, connectedness, compactness, countability and separation axioms, function spaces.  Prerequisite(s): Mathematics 304a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 406a/b, Lebesgue Integration and Fourier Series  
 Description: Lebesgue measure, measurable sets and functions, approximation theorems, the Lebesgue integral, comparison with the Riemann integral, the basic convergence theorems and continuity properties, the space L2, the RieszFischer theorem and completeness of the trigonometric system, pointwise convergence of Fourier series, Fejér's theorem.  Prerequisite(s): Mathematics 304a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 411a/b, Algebraic Number Theory  
 Description: Algebraic numbers, cyclotomic fields, low dimensional Galois cohomology, Brauer groups, quadratic forms, local and global class fields, class field theory, Galois group representations, modular forms and elliptic curves, zeta function and Lseries.  Antirequisite(s): The former Mathematics 311b.  Prerequisite(s): Mathematics 403a/b; Mathematics 313a/b strongly recommended but not required.  3 lecture hours, 0.5 course.  back to top 
Mathematics 414b, Algebraic Topology  
 Description: Homotopy, fundamental group, Van Kampen's theorem, fundamental theorem of algebra, Jordan curve theorem, singular homology, homotopy invariance, long exact sequence of a pair, excision, MayerVietoris sequence, Brouwer fixed point theorem, JordanBrouwer separation theorem, invariance of domain, Euler characteristic, cell complexes, projective spaces, Poincaré theorem.  Prerequisite(s): Mathematics 302a, Mathematics 404a.  3 lecture hours, 0.5 course.  back to top 
Mathematics 415a/b, Ordinary Differential Equations  
 Description: Laplace transforms and their application to solving differential equations. SturmLiouville systems; eigenvalue problems, expansions, Fourier series, autonomous systems; linear and nonlinear problems, types of critical points, stability.  Prerequisite(s): Mathematics 305a/b or Differential Equations 215a.  3 lecture hours, 0.5 course.  back to top 
Mathematics 416a, Complex Variables II  
 Description: Moebius transformations; local behavior of analytic functions, open and inverse mapping theorems; Schwarz's lemma; harmonic functions, solution of the Dirichlet problem on the disk, the Jensen and Poisson Jensen formulas, the Schwarz reflection principle; analytic continuation; normal families, the Riemann mapping theorem, the homotopic version of Cauchy's theorem; conformal mapping.  Prerequisite(s): Mathematics 307a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 417b, Complex Variables III  
 Description: Entire and meromorphic functions, infinite products, canonical products, the Weierstrass factorization and MittagLeffler theorems, the Hadamard factorization theorem; simply periodic and doubly periodic functions, elliptic functions; the Picard theorems (with Schottky's, Montel's, and Landau's theorems); the prime number theorem (with the Gamma and Riemann Zeta functions).  Prerequisite(s): Mathematics 416a or Mathematics 307a/b with the permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 418a/b, Introduction to Functional Analysis  
 Description: Banach and Hilbert spaces, dual spaces, annihilators, HahnBanach theorem, Riesz representation theorems, bounded linear operators, adjoints, closed graph and BanachSteinhaus theorems, compact operators, the Fredholm alternative, the operational calculus, spectral resolution of compact normal operators, applications to integral equations.  Prerequisite(s): Mathematics 203b, Mathematics 304a/b, Mathematics 307a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 419a/b, Algebraic Geometry  
 Description: Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.  Prerequisite(s): Mathematics 403a/b; Mathematics 329a/b is recommended but not required.  3 lecture hours, 0.5 course.  back to top 
Mathematics 420a/b, Foundations of Mathematics  
 Description: Set theory: axioms, ordinal numbers, transfinite induction, cardinality, the axiom of choice. Foundations of mathematics: construction of the real numbers from the natural numbers by one of the standard methods. Firstorder logic: propositional calculus, quantifiers, truth and satisfaction, models of firstorder theories, consistency, completeness and compactness.  Prerequisite(s): The permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 424a/b, Multivariable Calculus  
 Description: Review of differentiability in Euclidean space, inverse and implicit function theorems, integration in Euclidean space, Fubini's theorem, partitions of unity, change of variable, multilinear functions, tensor and wedge product, vector fields, differential forms, Poincaré's lemma, Stokes' theorem, manifolds, fields and forms on manifolds, Stokes' theorem on manifolds.  Prerequisite(s): Calculus 251a/b or the former Mathematics 205b, and Mathematics 304a/b.  3 lecture hours, 0.5 course.  back to top 
Mathematics 425a/b, Linear Ordinary Differential Equations  
 Description: First order vector systems and nth order single equations; adjoint systems and boundary value problems; Green's functions and self adjoint eigenvalue problems; expansion theory and spectral decomposition.  Prerequisite(s): Mathematics 305a/b or Differential Equations 215a.  3 lecture hours, 0.5 course.  back to top 
Mathematics 498a/b, Special Topics in Mathematics  
 Description:  Prerequisite(s): Permission of the Department.  3 lecture hours, 0.5 course.  back to top 
Mathematics 499a/b, Special Topics in Mathematics  
 Description:  Prerequisite(s): Permission of the Department.  3 lecture hours, 0.5 course.  back to top 

