Academic Calendar - 2025
Western University Academic Calendar. - 2025
Mathematics
Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.
Antirequisite(s): Ontario Secondary School MCV4U, any university level calculus course.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 3 hours.
Course Weight: 0.50
This course is also offered at:
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Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.
Antirequisite(s): Ontario Secondary School MCV4U, any university level calculus course.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 3 hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MHF4U, MCR3U, Mathematics 0105A/B, Mathematics 0109A/B, or equivalent
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MHF4U, MCR3U, Mathematics 0105A/B, Mathematics 0109A/B, or equivalent
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): Ontario Secondary School MCV4U or equivalent.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0109A/B, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0109A/B, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, Mathematics 0109A/B, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, Mathematics 0109A/B, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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An enriched version of Mathematics 1600A/B. Complex numbers and the integers modulo a prime; vectors, lines, planes; solving linear systems; spanning sets and linear independence; matrix algebra; elementary matrices; subspaces, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1600A/B, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One of Mathematics 1120A/B (recommended), Calculus 1000A/B or Calculus 1500A/B with a minimum mark of 70% in the case of Calculus 1000A/B or Calculus 1500A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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.
Prerequisite(s): A minimum mark of 60% in one of Calculus 1501A/B, or Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1414A/B, or the former Applied Mathematics 1413, or a minimum mark of 85% in Calculus 1301A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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): A minimum mark of 60% in one of Calculus 1501A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1414A/B, the former Applied Mathematics 1413, or a minimum mark of 85% in Calculus 1301A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Logic, sets and functions, algorithms, mathematical reasoning, counting, relations, graphs, trees, Boolean Algebra, computation, modeling.
Antirequisite(s): Computer Science 2214A/B, Mathematics 2155F/G.
Prerequisite(s): Computer Science 1026A/B or Engineering Science 1036A/B, in each case with at least 60%, and 1.0 courses with at least 60% in each from: Numerical and Mathematical Methods 1411A/B or the former Applied Mathematics 1411A/B, Numerical and Mathematical Methods 1412A/B or the former Applied Mathematics 1412A/B, Numerical and Mathematical Methods 1414A/B or the former Applied Mathematics 1414A/B, or the former Applied Mathematics 1413.Â
Extra Information: 3 lecture hours.
Note: this course is offered only to software engineering students enrolled in the Faculty of Engineering.
Course Weight: 0.50
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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.
Prerequisite(s): 1.0 course from: Mathematics 1120A/B, Mathematics 1600A/B, Mathematics 1700A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1414A/B, the former Applied Mathematics 1413, in each case with a minimum mark of 60%; or permission of the department. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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This course continues the development of logical reasoning and proofs begun in Mathematics 2155F/G. 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 2155F/G.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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 honours Mathematics programs.
Antirequisite(s): Applied Mathematics 2811A/B, Mathematics 2120A/B.
Prerequisite(s): Mathematics 1600A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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This course will provide a historical overview of the emergence of Mathematics and its impacts. Topics include: the origins of counting, Mathematics in the Ancient World (numbers, geometry, axioms); the emergence of algebra, analytic geometry and calculus (15th – 17th century); explosive modern growth (18th – 21st century).
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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 2251F/G.
Prerequisite(s): 1.0 course of university level Mathematics.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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The Gram-Schmidt process; similarity and orthogonal diagonalization; abstract vector spaces and linear transformations over arbitrary fields; change of basis; inner product spaces; norms and distance; least squares and Fourier approximation; singular value decomposition. Applications to differential equations and other topics will be emphasized throughout the course.
Antirequisite(s): Mathematics 2211A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B, the former Mathematics 3121A/B.
Prerequisite(s): One of Mathematics 1700A/B (recommended), Mathematics 1600A/B, Numerical and Mathematical Methods 1411A/B, or the former Applied Mathematics 1411A/B; plus one of Calculus 1301A/B, Calculus 1501A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1413, or the former Applied Mathematics 1414A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B; plus one of Mathematics 2151A/B, Mathematics 2155F/G (recommended) or Computer Science 2214A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3020A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Differentiation, the Mean Value Theorem, and integration. Metric spaces, including topology, convergence, compactness, completeness, and connectedness. Uniform convergence of functions. Selected additional topics.
Prerequisite(s): Mathematics 2122A/B with a minimum mark of 60%.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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): 1.5 courses: Calculus 2503A/B; Mathematics 3122A/B; plus one of Mathematics 2700A/B, the former Applied Mathematics 2811A/B, or the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Divisibility, primes, congruences, theorems of Fermat and Wilson, Chinese remainder theorem, quadratic reciprocity, some functions of number theory, diophantine equations, simple continued fractions.
Prerequisite(s): 1.0 course in Mathematics, Applied Mathematics, or Calculus at the 2100 level or higher.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3150A/B or Mathematics 2156A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.
Prerequisite(s): 0.5 course from: Mathematics 2700A/B, Mathematics 2155F/G, Mathematics 2211A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B, or permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.
Prerequisite(s): One of: Mathematics 2156A/B, Mathematics 3152A/B, or permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2700A/B or the former Mathematics 2120A/B; one of Mathematics 2122A/B, Mathematics 2124A/B, Mathematics 2155F/G, Mathematics 3700A/B or the former Mathematics 3121A/B; an additional 0.5 course in Mathematics, Applied Mathematics, Calculus at the 2100 level or above.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B, Calculus 1301A/B or Calculus 1501A/B, and one of Mathematics 1120A/B, Mathematics 2122A/B, Mathematics 2124A/B, Mathematics 2155F/G, Mathematics 2700A/B, the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B and one of Mathematics 2124A/B, Mathematics 2151A/B, Mathematics 2155F/G, Mathematics 2700A/B, Mathematics 3150A/B, Computer Science 2214A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Invariant subspaces and the Cayley-Hamilton theorem; normal and self-adjoint operators; unitary and orthogonal operators; the principal axis theorem; orthogonal projections and the spectral theorem; singular value decomposition; bilinear forms; Sylvester's theorem; Jordan canonical form and the minimal polynomial.
Antirequisite(s): the former Mathematics 3121A/B.
Prerequisite(s): Mathematics 2155F/G and one of Mathematics 2700A/B or the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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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 3020A/B and Mathematics 3120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Topological spaces, neighbourhoods, bases, subspaces, product and quotient spaces, connectedness, compactness, separation axioms.
Prerequisite(s): Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Lebesgue measure, measurable sets and functions, Littlewood principles; the Lebesgue integral, basic convergence theorems, approximation theorems; measure spaces, signed measures, Radon-Nikodym Theorem.
Prerequisite(s): Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3020A/B.
Extra Information: 3 lecture hours.
Note: It is recommended that Mathematics 3700A/B or the former Mathematics 3121A/B be taken before or concurrently with Mathematics 4123A/B.
Course Weight: 0.50
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The course will be an introduction to commutative algebra. It provides students with necessary algebraic tools to conduct research in a number of areas of modern mathematics. Topics include: Hom and tensor, localisation, primary decomposition, Jacobson rings and the Nullstellensatz, integral dependence, dimension theory, Artinian rings and Dedekind domains.
Antirequisite(s): Mathematics 4959A/B, if taken in 2020-21 or 2022-23.
Prerequisite(s): Mathematics 4123A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 4120A/B; Mathematics 3151A/B strongly recommended but not required.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3120A/B and Mathematics 4121A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.
Prerequisite(s): Mathematics 4123A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2700A/B or the former Mathematics 2120A/B, Mathematics 3122A/B.
Pre-or Corequisite(s): Mathematics 3124A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Calculus 2503A/B and Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Linear-fractional transformations, Schwarz's lemma, Reflection Principle, the Argument principle, the Riemann mapping theorem, Runge's theorem, the Mittag-Lefler and Weierstrass theorems.
Prerequisite(s): Mathematics 3124A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Mathematics
Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.
Antirequisite(s): Ontario Secondary School MCV4U, any university level calculus course.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 3 hours.
Course Weight: 0.50
This course is also offered at:
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Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.
Antirequisite(s): Ontario Secondary School MCV4U, any university level calculus course.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 3 hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of mathematical operations and linear equations; introduction to functions; introductory finite mathematics, including combinatorics and probability; introductory financial mathematics, including compound interest and annuities.
Antirequisites: Ontario Secondary School MCV4U, Mathematics 0105A/B, or any university-level calculus course.
Prerequisite(s): Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MHF4U, MCR3U, Mathematics 0105A/B, Mathematics 0109A/B, or equivalent
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1225A/B, Mathematics 1230A/B, Calculus 1000A/B, Calculus 1500A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): One or more of Ontario Secondary School MHF4U, MCR3U, Mathematics 0105A/B, Mathematics 0109A/B, or equivalent
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): Ontario Secondary School MCV4U or equivalent.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; 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, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, Mathematics 1230A/B, Numerical and Mathematical Methods 1412A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413.
Prerequisite(s): Ontario Secondary School MCV4U or Mathematics 0110A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0109A/B, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, MHF4U, MDM4U, Mathematics 0109A/B, Mathematics 0110A/B, Mathematics 1225A/B, Mathematics 1229A/B.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, Mathematics 0109A/B, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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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): Mathematics 1600A/B, Mathematics 1700A/B, Mathematics 2155F/G, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One or more of Ontario Secondary School MCF3M, MCR3U, Mathematics 0109A/B, or equivalent.
Extra Information: 2 lecture hours, 1 tutorial hour (Main campus); 3 lecture hours (Huron, King's).
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.
Antirequisite(s): Applied Mathematics 1201A/B, Calculus 1000A/B, Calculus 1301A/B, Calculus 1500A/B, Calculus 1501A/B, Mathematics 1225A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1413, the former Applied Mathematics 1414A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 0110A/B.
Extra Information: 4 lecture hours.
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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Complex numbers; vectors, lines and planes; solving linear systems; spanning sets and linear independence; matrix algebra; the fundamental subspaces of a matrix, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1700A/B, Numerical and Mathematical Methods 1411A/B, Applied Mathematics 2811A/B, the former Applied Mathematics 1411A/B.
Prerequisite(s): One or more of Ontario Secondary School MCV4U, Mathematics 1229A/B, Mathematics 1120A/B, Calculus 1000A/B or Calculus 1500A/B. Calculus 1000A/B or Calculus 1500A/B may be taken as a pre-or corequisite.
Extra Information: 3 lecture hours (Main Campus, Huron); 4 hours (King's).
Course Weight: 0.50
This course is also offered at:
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An enriched version of Mathematics 1600A/B. Complex numbers and the integers modulo a prime; vectors, lines, planes; solving linear systems; spanning sets and linear independence; matrix algebra; elementary matrices; subspaces, bases, dimension and rank; introduction to linear transformations; determinants; eigenvalues and eigenvectors; similarity and diagonalization; Markov chains and other applications.
Antirequisite(s): Mathematics 1600A/B, Mathematics 2211A/B, Numerical and Mathematical Methods 1411A/B, the former Applied Mathematics 1411A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Prerequisite(s): One of Mathematics 1120A/B (recommended), Calculus 1000A/B or Calculus 1500A/B with a minimum mark of 70% in the case of Calculus 1000A/B or Calculus 1500A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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.
Prerequisite(s): A minimum mark of 60% in one of Calculus 1501A/B, or Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1414A/B, or the former Applied Mathematics 1413, or a minimum mark of 85% in Calculus 1301A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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): A minimum mark of 60% in one of Calculus 1501A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1414A/B, the former Applied Mathematics 1413, or a minimum mark of 85% in Calculus 1301A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Logic, sets and functions, algorithms, mathematical reasoning, counting, relations, graphs, trees, Boolean Algebra, computation, modeling.
Antirequisite(s): Computer Science 2214A/B, Mathematics 2155F/G.
Prerequisite(s): Computer Science 1026A/B or Engineering Science 1036A/B, in each case with at least 60%, and 1.0 courses with at least 60% in each from: Numerical and Mathematical Methods 1411A/B or the former Applied Mathematics 1411A/B, Numerical and Mathematical Methods 1412A/B or the former Applied Mathematics 1412A/B, Numerical and Mathematical Methods 1414A/B or the former Applied Mathematics 1414A/B, or the former Applied Mathematics 1413.Â
Extra Information: 3 lecture hours.
Note: this course is offered only to software engineering students enrolled in the Faculty of Engineering.
Course Weight: 0.50
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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.
Prerequisite(s): 1.0 course from: Mathematics 1120A/B, Mathematics 1600A/B, Mathematics 1700A/B, Numerical and Mathematical Methods 1412A/B, Numerical and Mathematical Methods 1414A/B, Calculus 1000A/B, Calculus 1500A/B, Calculus 1301A/B, Calculus 1501A/B, the former Applied Mathematics 1412A/B, the former Applied Mathematics 1414A/B, the former Applied Mathematics 1413, in each case with a minimum mark of 60%; or permission of the department. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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This course continues the development of logical reasoning and proofs begun in Mathematics 2155F/G. 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 2155F/G.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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 honours Mathematics programs.
Antirequisite(s): Applied Mathematics 2811A/B, Mathematics 2120A/B.
Prerequisite(s): Mathematics 1600A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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This course will provide a historical overview of the emergence of Mathematics and its impacts. Topics include: the origins of counting, Mathematics in the Ancient World (numbers, geometry, axioms); the emergence of algebra, analytic geometry and calculus (15th – 17th century); explosive modern growth (18th – 21st century).
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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 2251F/G.
Prerequisite(s): 1.0 course of university level Mathematics.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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The Gram-Schmidt process; similarity and orthogonal diagonalization; abstract vector spaces and linear transformations over arbitrary fields; change of basis; inner product spaces; norms and distance; least squares and Fourier approximation; singular value decomposition. Applications to differential equations and other topics will be emphasized throughout the course.
Antirequisite(s): Mathematics 2211A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B, the former Mathematics 3121A/B.
Prerequisite(s): One of Mathematics 1700A/B (recommended), Mathematics 1600A/B, Numerical and Mathematical Methods 1411A/B, or the former Applied Mathematics 1411A/B; plus one of Calculus 1301A/B, Calculus 1501A/B, Numerical and Mathematical Methods 1414A/B, the former Applied Mathematics 1413, or the former Applied Mathematics 1414A/B. Integrated Science 1001X with a minimum mark of 60% can be used in place of Calculus 1301A/B.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B; plus one of Mathematics 2151A/B, Mathematics 2155F/G (recommended) or Computer Science 2214A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3020A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Differentiation, the Mean Value Theorem, and integration. Metric spaces, including topology, convergence, compactness, completeness, and connectedness. Uniform convergence of functions. Selected additional topics.
Prerequisite(s): Mathematics 2122A/B with a minimum mark of 60%.
Extra Information: 3 lecture hours, 1 tutorial hour.
Course Weight: 0.50
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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): 1.5 courses: Calculus 2503A/B; Mathematics 3122A/B; plus one of Mathematics 2700A/B, the former Applied Mathematics 2811A/B, or the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Divisibility, primes, congruences, theorems of Fermat and Wilson, Chinese remainder theorem, quadratic reciprocity, some functions of number theory, diophantine equations, simple continued fractions.
Prerequisite(s): 1.0 course in Mathematics, Applied Mathematics, or Calculus at the 2100 level or higher.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3150A/B or Mathematics 2156A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.
Prerequisite(s): 0.5 course from: Mathematics 2700A/B, Mathematics 2155F/G, Mathematics 2211A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B, or permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.
Prerequisite(s): One of: Mathematics 2156A/B, Mathematics 3152A/B, or permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2700A/B or the former Mathematics 2120A/B; one of Mathematics 2122A/B, Mathematics 2124A/B, Mathematics 2155F/G, Mathematics 3700A/B or the former Mathematics 3121A/B; an additional 0.5 course in Mathematics, Applied Mathematics, Calculus at the 2100 level or above.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B, Calculus 1301A/B or Calculus 1501A/B, and one of Mathematics 1120A/B, Mathematics 2122A/B, Mathematics 2124A/B, Mathematics 2155F/G, Mathematics 2700A/B, the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 1600A/B or Mathematics 1700A/B and one of Mathematics 2124A/B, Mathematics 2151A/B, Mathematics 2155F/G, Mathematics 2700A/B, Mathematics 3150A/B, Computer Science 2214A/B, the former Applied Mathematics 2811A/B, the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Invariant subspaces and the Cayley-Hamilton theorem; normal and self-adjoint operators; unitary and orthogonal operators; the principal axis theorem; orthogonal projections and the spectral theorem; singular value decomposition; bilinear forms; Sylvester's theorem; Jordan canonical form and the minimal polynomial.
Antirequisite(s): the former Mathematics 3121A/B.
Prerequisite(s): Mathematics 2155F/G and one of Mathematics 2700A/B or the former Mathematics 2120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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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 3020A/B and Mathematics 3120A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Topological spaces, neighbourhoods, bases, subspaces, product and quotient spaces, connectedness, compactness, separation axioms.
Prerequisite(s): Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Lebesgue measure, measurable sets and functions, Littlewood principles; the Lebesgue integral, basic convergence theorems, approximation theorems; measure spaces, signed measures, Radon-Nikodym Theorem.
Prerequisite(s): Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3020A/B.
Extra Information: 3 lecture hours.
Note: It is recommended that Mathematics 3700A/B or the former Mathematics 3121A/B be taken before or concurrently with Mathematics 4123A/B.
Course Weight: 0.50
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The course will be an introduction to commutative algebra. It provides students with necessary algebraic tools to conduct research in a number of areas of modern mathematics. Topics include: Hom and tensor, localisation, primary decomposition, Jacobson rings and the Nullstellensatz, integral dependence, dimension theory, Artinian rings and Dedekind domains.
Antirequisite(s): Mathematics 4959A/B, if taken in 2020-21 or 2022-23.
Prerequisite(s): Mathematics 4123A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 4120A/B; Mathematics 3151A/B strongly recommended but not required.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 3120A/B and Mathematics 4121A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.
Prerequisite(s): Mathematics 4123A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Mathematics 2700A/B or the former Mathematics 2120A/B, Mathematics 3122A/B.
Pre-or Corequisite(s): Mathematics 3124A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Calculus 2503A/B and Mathematics 3122A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Linear-fractional transformations, Schwarz's lemma, Reflection Principle, the Argument principle, the Riemann mapping theorem, Runge's theorem, the Mittag-Lefler and Weierstrass theorems.
Prerequisite(s): Mathematics 3124A/B.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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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.
Prerequisite(s): Permission of the Department.
Extra Information: 3 lecture hours.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.
Course Weight: 0.50
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Extra Information: 3 lecture hours.
Prerequisite(s): Permission of the Department.