Mathematics (S)

Calculus courses and are offered jointly by the Departments of Applied Mathematics and Mathematics. Please refer to the CALCULUS (S) section for the first and second year course offerings.

 Mathematics 0110A/B - Introductory Calculus
Introduction to differential calculus including limits, continuity, definition of derivative, rules for differentiation, implicit differentiation, velocity, acceleration, related rates, maxima and minima, exponential functions, logarithmic functions, differentiation of exponential and logarithmic functions, curve sketching.
Antirequisite(s):
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 4 lecture hours, 0.5 course.
 Mathematics 1120A/B - Fundamental Concepts in Mathematics
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):
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1600A/B, or the former Linear Algebra 1600A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 4 lecture hours, 0.5 course.
 Mathematics 1225A/B - Methods of Calculus
Elementary techniques of integration; applications of Calculus such as area, volume, probability; functions of several variables, Lagrange multipliers. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules. It may not be used as a prerequisite for any Calculus course numbered 1300 or above.
Antirequisite(s): Applied Mathematics 1201A/B or the former Calculus 1201A/B, Applied Mathematics 1413, Calculus 1301A/B, 1501A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B, Calculus 1000A/B, 1100A/B, 1500A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 1228A/B - Methods of Finite Mathematics
Permutations and combinations; probability theory. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules.
Antirequisite(s): Mathematics 2124A/B, 2155A/B, Statistical Sciences 2035, 2141A/B, 2857A/B, the former Statistical Sciences 2657A.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0110A/B, 1225A/B, 1229A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 1229A/B - Methods of Matrix Algebra
Matrix algebra including vectors and matrices, linear equations, determinants. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules.
Antirequisite(s): Applied Mathematics 1411A/B, 2811B, Mathematics 1600A/B, 2120A/B, 2155A/B, 2211A/B, the former Linear Algebra 1600A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 1600A/B - Linear Algebra I
Properties and applications of vectors; matrix algebra; solving systems of linear equations; determinants; vector spaces; orthogonality; eigenvalues and eigenvectors.
Antirequisite(s): Applied Mathematics 1411A/B, 2811B, the former Linear Algebra 1600A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Calculus 1000A/B or 1500A/B, the former Calculus 1100A/B. Calculus 1000A/B or 1500A/B may be taken as a pre- or corequisite.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 laboratory hour, 0.5 course.
 Mathematics 2120A/B - Intermediate Linear Algebra I
A rigorous development of lines and planes in Rn; linear transformations and abstract vector spaces. Determinants and an introduction to diagonalization and its applications including the characteristic polynomials, eigenvalues and eigenvectors.
Antirequisite(s):
Prerequisite(s): Mathematics 1600A/B or the former Linear Algebra 1600A/B with a minimum mark of 60% or Mathematics 1120A/B with a minimum mark of 70% or permission of the Mathematics Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 2121A/B - Intermediate Linear Algebra II
A continuation of the material of Mathematics 2120A/B including properties of complex numbers and the principal axis theorem; singular value decomposition; linear groups; similarity; Jordan canonical form; Cayley-Hamilton theorem; bilinear forms; Sylvester's theorem.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 2122A/B - Real Analysis I
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.
Antirequisite(s):
Prerequisite(s): Calculus 1501A/B or Applied Mathematics 1413, with a minimum mark of 60%, or Calculus 1301A/B with a minimum mark of 85%.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 4 lecture hours, 0.5 course.
 Mathematics 2123A/B - Real Analysis II
A continuation of the rigorous introduction to analysis on the real line, begun in Mathematics 2122A/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):
Prerequisite(s): Mathematics 2122A/B with a minimum mark of 60%, or permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 2124A/B - Introduction to Mathematical Problems
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.
Antirequisite(s):
Prerequisite(s): Calculus 1501A/B or Applied Mathematics 1413, with a minimum mark of 60%, or Calculus 1301A/B with a minimum mark of 85% or permission of the instructor.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 2151A/B - Discrete Structures for Engineering
Logic, sets and functions, algorithms, mathematical reasoning, counting, relations, graphs, trees, Boolean Algebra, computation, modeling.
Antirequisite(s): Mathematics 2155A/B, the former Software Engineering 2251A/B.
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Note: this course is offered only to software engineering students enrolled in the Faculty of Engineering.
 Mathematics 2155A/B - Discrete Structures I
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 error-correcting codes.
Antirequisite(s): Mathematics 2151A/B or the former Software Engineering 2251A/B.
Prerequisite(s): 1.0 course from: Mathematics 1120A/B, Applied Mathematics 1413, Calculus 1000A/B, 1100A/B or 1500A/B Calculus 1301A/B or 1501A/B, Mathematics 1600A/B or the former Linear Algebra 1600A/B (in each case with a minimum mark of 60%); or permission of the department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 4 lecture hours, 0.5 course.
 Mathematics 2156A/B - Discrete Structures II
This course continues the development of logical reasoning and proofs begun in Mathematics 2155A/B. 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).
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
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Extra Information: 4 lecture hours, 0.5 course.
 Mathematics 2211A/B - Linear Algebra
Linear transformations, matrix representation, rank, change of basis, eigenvalues and eigenvectors, inner product spaces, quadratic forms and conic sections. Emphasis on problem-solving rather than theoretical development. Cannot be taken for credit by students in honors Mathematics programs.
Antirequisite(s):
Prerequisite(s): Mathematics 1600A/B or Mathematics 1120A/B with a minimum mark of 70%, or the former Linear Algebra 1600A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 2251F/G - Conceptual Development of Mathematics
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):
Prerequisite(s): 1.0 course of university level Mathematics.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3020A/B - Introduction to Abstract Algebra
Sets and binary operations, commutativity, associativity, distributivity, groups and subgroups, cyclic groups, permutation groups, cosets, Lagrange’s theorem, normal subgroups, quotient groups, first isomorphism theorem, rings, integral domains, fields, polynomial rings, unique factorization of polynomials over a field, finite fields.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3120A/B - Group Theory
An introduction to the theory of groups: cyclic, dihedral, symmetric, alternating; subgroups, quotient groups, homomorphisms, cosets, Lagrange's theorem, isomorphism theorems; group actions, class equation, p-groups, Sylow theorems; direct and semidirect products, wreath products, finite abelian groups; Jordan-Hölder theorem, commutator subgroup, solvable and nilpotent groups; free groups, generators and relations.
Antirequisite(s):
Prerequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3122A/B - Metric Space Topology
An introduction to the theory of metric spaces with emphasis on the point-set topology of Euclidean n-space, including convergence, compactness, completeness, continuity, uniform continuity, homeomorphism, equivalence of metrics, connectedness, path-connectedness, fixed-point theorem for contractions, separability, complete normality, product spaces, category.
Antirequisite(s):
Prerequisite(s): Either Mathematics 2122A/B or Mathematics 2123A/B, each with a minimum mark of 60%.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3123A/B - Differential Equations
Rigorous introduction to ordinary differential equations. Existence, uniqueness, and continuation of solutions. Linear systems with constant coefficients. Flows and dynamical systems. Series solutions.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3124A/B - Complex Analysis I
The Cauchy-Riemann 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):
Prerequisite(s):
Corequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3132B - General Topology
Topological spaces, operations on subsets (e.g. closure), neighbourhoods, bases, subspaces, quotient spaces, product spaces, connectedness, compactness, countability and separation axioms, function spaces.
Antirequisite(s): The former Mathematics 4121A.
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3150A/B - Elementary Number Theory I
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 2291.
Prerequisite(s): 1.0 course in Mathematics, Applied Mathematics, or Calculus at the 2100 level or higher.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3151A/B - Elementary Number Theory II
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 2291.
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3152A/B - Combinatorial Mathematics
Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.
Antirequisite(s):
Prerequisite(s): 0.5 course from: Mathematics 2120A/B, 2156A/B, 2211A/B, Applied Mathematics 2811B, or permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3153A/B - Discrete Optimization
Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.
Antirequisite(s):
Prerequisite(s): One of: Mathematics 2156A/B, 3152A/B, or permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3154A/B - Introduction to Algebraic Curves
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 2292.
Prerequisite(s): Mathematics 1600A/B, Mathematics 2120A/B, or the former Linear Algebra 1600A/B; Mathematics 2121A/B, 2122A/B, 2124A/B or 2155A/B; an additional 0.5 course in Mathematics, Applied Mathematics, Calculus at the 2100 level or above, or the former Differential Equations 2402A.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3157A/B - Introduction to Game Theory
A first course in the mathematical theory of games. Topics begin with the modelling of games: extensive and strategic forms; perfect information; chance. Sprague-Grundy theory of impartial combinatorial games. Modelling preferences with utility functions. Nash equilibria, analysis of two-player games.
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3958A/B - Special Topics in Mathematics
Antirequisite(s):
Prerequisite(s): Permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3959A/B - Special Topics in Mathematics
Antirequisite(s):
Prerequisite(s): Permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4120A/B - Field Theory
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.
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4122A/B - Lebesgue Integration and Fourier Series
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 Riesz-Fischer theorem and completeness of the trigonometric system, pointwise convergence of Fourier series, Fejér's theorem.
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Prerequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4123A/B - Rings and Modules
Commutative rings, ring homomorphisms 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):
Prerequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4151A/B - Algebraic Number Theory
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 L-series.
Antirequisite(s):
Prerequisite(s): Mathematics 4120A/B; Mathematics 3151A/B strongly recommended but not required.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4152A/B - Algebraic Topology
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, Mayer-Vietoris sequence, Brouwer fixed point theorem, Jordan-Brouwer separation theorem, invariance of domain, Euler characteristic, cell complexes, projective spaces, Poincaré theorem.
Antirequisite(s):
Prerequisite(s): Mathematics 3120A/B and either Mathematics 3132B or the former Mathematics 4121A.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4153A/B - Algebraic Geometry
Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.
Antirequisite(s):
Prerequisite(s): Mathematics 4120A/B; Mathematics 3154A/B is recommended but not required.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4154A/B - Introduction to Functional Analysis
Banach and Hilbert spaces, dual spaces, annihilators, Hahn-Banach theorem, Riesz representation theorems, bounded linear operators, adjoints, closed graph and Banach-Steinhaus theorems, compact operators, the Fredholm alternative, the operational calculus, spectral resolution of compact normal operators, applications to integral equations.
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4155A/B - Multivariable Calculus
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.
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4156A/B - Complex Variables II
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.
Antirequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4157A/B - Complex Variables III
Entire and meromorphic functions, infinite products, canonical products, the Weierstrass factorization and Mittag-Leffler 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).
Antirequisite(s):
Prerequisite(s): Mathematics 4156A/B or Mathematics 3124A/B with the permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4158A/B/Y - Foundations of Mathematics
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. First-order logic: propositional calculus, quantifiers, truth and satisfaction, models of first-order theories, consistency, completeness and compactness.
Antirequisite(s):
Prerequisite(s): The permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4958A/B - Special Topics in Mathematics
Antirequisite(s):
Prerequisite(s): Permission of the Department.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.