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): Mathematics 2155F/G, the former Mathematics 2155A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 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, Applied Mathematics 1413, Calculus 1301A/B, 1501A/B. If Calculus 1000A/B or 1500A/B was completed after September 1, 2016 it is an antirequisite, but not if it was completed before that time.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/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, 2155F/G, the former Mathematics 2155A/B, Statistical Sciences 2035, 2141A/B, 2857A/B.
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, 2155F/G, 2211A/B, the former Mathematics 2155A/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):
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
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):
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. Sets and functions, logic and mathematical proof, the natural and real numbers, completeness and its consequences, limits of sequences, limits of real functions, continuity and uniform continuity.
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: 3 lecture hours, 1 tutorial hour, 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 2155F/G, the former 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 2155F/G - Mathematical Structures
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, binary operations, elementary group theory, and applications to error-correcting codes.
Antirequisite(s): Mathematics 2151A/B, the former Software Engineering 2251A/B, the former Mathematics 2155A/B.
Prerequisite(s): 1.0 course from: Mathematics 1120A/B, 1600A/B, Applied Mathematics 1413, Calculus 1000A/B, 1500A/B, 1301A/B, 1501A/B, or the former Calculus 1100A/B, in each case 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 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): Mathematics 2155F/G or the former Mathematics 2155A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 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):
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
Properties of integers, rational, real and complex numbers: commutativity, associativity, distributivity. Polynomials, prime and irreducible elements. Rings, ideals, integral and Euclidean domains, fields, and unique factorization. First isomorphism theorem, quotient rings and finite fields. Introduction to groups.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
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):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3121A/B - Advanced Linear Algebra
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): The former Mathematics 2121A/B.
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3122A/B - Real Analysis II
Differentiation, the Mean Value Theorem, and integration. Metric spaces, including topology, convergence, compactness, completeness, and connectedness. Uniform convergence of functions. Selected additional topics.
Antirequisite(s):
Prerequisite(s): Mathematics 2122A/B or the former Mathematics 2123A/B, each with a minimum mark of 60%.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 1 tutorial hour, 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):
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):
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):
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, 2155F/G, 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):
Prerequisite(s): Mathematics 1600A/B, 2120A/B, 2122A/B, 2124A/B, 2155F/G, 3121A/B, the former Mathematics 2121A/B, or the former Mathematics 2155A/B; an additional 0.5 course in Mathematics, Applied Mathematics, Calculus at the 2100 level or above.
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.
Antirequisite(s):
Prerequisite(s): Mathematics 1600A/B, Calculus 1301A/B or 1501A/B, and one of Mathematics 1120A/B, 2120A/B, 2122A/B, 2124A/B or 2155F/G, the former Mathematics 2155A/B.
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 3159A/B - Introduction to Cryptography
Modern cryptological algorithms will be discussed with an emphasis placed on their mathematical foundation. Main topics will include: basic number theory, complexity of algorithms, symmetric-key cryptosystems, public-key cryptosystems, RSA encryption, primality and factoring, discrete logarithms, elliptic curves and information theory.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
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.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4121A/B - Topology
Topological spaces, neighbourhoods, bases, subspaces, product and quotient spaces, connectedness, compactness, separation axioms.
Antirequisite(s): The former Mathematics 3132B.
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4122A/B - Introduction to Measure Theory
Lebesgue measure, measurable sets and functions, Littlewood principles; the Lebesgue integral, basic convergence theorems, approximation theorems; measure spaces, signed measures, Radon-Nikodym Theorem.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4123A/B - Rings and Modules
Rings: fractions and localization, Chinese Remainder Theorem, factorization in commutative rings, Euclidean algorithm, PIDs, algebraic integers, polynomials and formal power series, factorization in polynomial rings; Modules: generation, direct products and sums, freeness, presentations, tensor algebras, exact sequences, projectivity, injectivity, Hom and duality, Zorn's Lemma, chain conditions, modules over PIDs.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
Pre-or Corequisite(s):
Extra Information: 3 lecture hours, 0.5 course.
Note: It is recommended that Mathematics 3121A/B (or the former Math 2121A/B) be taken before or concurrently with Mathematics 4123A/B.
 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, covering spaces, simplicial and singular homology, homotopy invariance, long exact sequence of a pair, excision, Mayer-Vietoris sequence, degree, Euler characteristic, cell complexes, projective spaces. Applications include the fundamental theorem of algebra, the Brouwer fixed point theorem, division algebras, and invariance of domain.
Antirequisite(s):
Prerequisite(s): Mathematics 3120A/B and either Mathematics 4121A/B or the former Mathematics 3132B.
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):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4154A/B - Functional Analysis
Hilbert spaces: L^2 spaces, orthogonal complements, dual spaces, Riesz representation theorem, the Fredholm alternative, spectral resolution of compact normal operators. Banach spaces: Hahn-Banach theorem, bounded linear operators, adjoints, closed graph and Banach Steinhaus theorems.
Antirequisite(s):
Prerequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4155A/B - Calculus on Manifolds
Manifolds (definition, examples, constructions), orientation, functions, partitions of unity, tangent bundle, cotangent bundle, vector fields, integral curves, differential forms, integration, manifolds with boundary, Stokes' theorem, submersions, immersions, embeddings, submanifolds, Sard's theorem, Whitney embedding theorem.
Antirequisite(s):
Prerequisite(s):
Corequisite(s):
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Extra Information: 3 lecture hours, 0.5 course.
 Mathematics 4156A/B - Complex Analysis II
Linear-fractional transformations, Schwarz's lemma, Reflection Principle, the Argument principle, the Riemann mapping theorem, Runge's theorem, the Mittag-Lefler and Weierstrass theorems.
Antirequisite(s):
Prerequisite(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.