 
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.
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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.
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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.
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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 
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 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

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.
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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 
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.
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Prerequisite(s):
Mathematics 2122A/B or the former Mathematics 2123A/B, each with a minimum mark of 60%.
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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.
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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.
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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.
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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
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, symmetrickey cryptosystems, publickey cryptosystems, RSA encryption, primality and factoring, discrete logarithms, elliptic curves and information theory.
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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 
Topology

Topological spaces, neighbourhoods, bases, subspaces, product and quotient spaces, connectedness, compactness, separation axioms.
Antirequisite(s):
The former Mathematics 3132B.
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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, RadonNikodym Theorem.
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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.
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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, covering spaces, simplicial and singular homology, homotopy invariance, long exact sequence of a pair, excision, MayerVietoris 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.
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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 
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: HahnBanach theorem, bounded linear operators, adjoints, closed graph and Banach Steinhaus theorems.
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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.
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Mathematics
4156A/B 
Complex Analysis II

Linearfractional transformations, Schwarz's lemma, Reflection Principle, the Argument principle, the Riemann mapping theorem, Runge's theorem, the MittagLefler and Weierstrass theorems.
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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

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Permission of the Department.
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