 
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.
Prerequisite(s):
One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
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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.
Prerequisite(s):
One or more of Ontario Secondary School MCV4U, Mathematics 1600A/B, or the former Linear Algebra 1600A/B.
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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.
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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.
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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.
Prerequisite(s):
One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
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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.
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Mathematics
2120A/B 
Intermediate Linear Algebra

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.
Prerequisite(s):
Mathematics 1600A/B or the former Linear Algebra 1600A/B with a minimum mark of 60%.
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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.
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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.
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Mathematics 2122A/B with a minimum mark of 60%, or permission of the Department.
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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.
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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.
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Mathematics
2155F/G 
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 errorcorrecting codes.
Antirequisite(s):
Mathematics 2151A/B, the former Software Engineering 2251A/B, the former Mathematics 2155A/B.
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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).
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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 problemsolving rather than theoretical development. Cannot be taken for credit by students in honors Mathematics programs.
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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.
Prerequisite(s):
1.0 course of university level Mathematics.
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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.
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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, pgroups, Sylow theorems; direct and semidirect products, wreath products, finite abelian groups; JordanHölder theorem, commutator subgroup, solvable and nilpotent groups; free groups, generators and relations.
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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; CayleyHamilton theorem; bilinear forms; Sylvester's theorem.
Antirequisite(s):
The former Mathematics 2121A/B.
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Mathematics
3122A/B 
Metric Space Topology

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.
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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.
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Mathematics
3124A/B 
Complex Analysis I

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.
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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.
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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):
The former Mathematics 2291.
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Mathematics
3152A/B 
Combinatorial Mathematics

Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.
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Mathematics
3153A/B 
Discrete Optimization

Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.
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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):
The former Mathematics 2292.
Prerequisite(s):
Mathematics 1600A/B, Mathematics 2120A/B, or the former Linear Algebra 1600A/B; Mathematics 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, or the former Differential Equations 2402A.
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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. SpragueGrundy theory of impartial combinatorial games. Modelling preferences with utility functions. Nash equilibria, analysis of twoplayer games.
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Mathematics
3958A/B 
Special Topics in Mathematics

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Permission of the Department.
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Mathematics
3959A/B 
Special Topics in Mathematics

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Permission of the Department.
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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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Mathematics
4121A/B 
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 3132B.
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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 RieszFischer theorem and completeness of the trigonometric system, pointwise convergence of Fourier series, Fejér's theorem.
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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.
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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 Lseries.
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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, MayerVietoris sequence, Brouwer fixed point theorem, JordanBrouwer separation theorem, invariance of domain, Euler characteristic, cell complexes, projective spaces, Poincaré theorem.
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Mathematics
4153A/B 
Algebraic Geometry

Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.
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Mathematics
4154A/B 
Introduction to Functional Analysis

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.
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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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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.
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Mathematics
4157A/B 
Complex Variables III

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).
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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. Firstorder logic: propositional calculus, quantifiers, truth and satisfaction, models of firstorder theories, consistency, completeness and compactness.
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The permission of the Department.
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Mathematics
4958A/B 
Special Topics in Mathematics

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Permission of the Department.
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Mathematics
4959A/B 
Special Topics in Mathematics

Antirequisite(s):
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Permission of the Department.
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