# Mathematics Course Descriptions

MATH 051 - Mathematics Co-op Work Term
Four-month co-op work term approved by the department and arranged by the co-op coordinator.

MATH 052 - Mathematics Co-op Work Term
Four-month co-op work term #2 approved by the department and arranged by the co-op coordinator. *** Prerequisite: MATH 051 ***

MATH 053 - Mathematics Co-op Work Term
Four month co-op work term #3 approved by the department and arranged by the co-op coordinator. *** Prerequisite: MATH 052 ***

MATH 054 - Mathematics Co-op Work Term
Four month co-op work term #4 approved by the department and arranged by the coop coordinator. *** Prerequisite: MATH 053 ***

MATH 101 - Introductory Finite Mathematics I
This is an introductory course intended to familiarize the students with the basic concepts of arithmetic, number theory, set theory, symbolic logic, and finite mathematics. Topics include logic, sets, numeration systems, arithmetic in non-decimal systems, system of integers, elementary number theory and modular arithmetic. There will be a strong emphasis on critical thinking, problem solving, understanding concepts and their applications. ***Prerequisite: University Admission.*** *Note: Students who have received credit for any mathematics or statistics course (other than MATH 108) cannot take this class for credit, unless it is explicitly required in their program or they have received consent from the Head of the Department of Mathematics and Statistics.*

MATH 102 - Mathematical Modelling and Precalculus
A course in problem solving through the use of mathematical models involving algebraic, exponential, logarithmic, and trigonometric functions and their graphs. The purpose of this course is to enhance students’ abilities to express, visualize, and model real-world problems through an understanding of common functions and their graphs. ***Prerequisite: Any one of the following: Foundations of Mathematics 30, Precalculus 20, Math B30, Math C30, or AMTH 092*** *Note: Students who have received credit in Math 103 or 110 cannot take Math 102 for credit*

MATH 103 - Applied Calculus I
Differentiation of algebraic, exponential, and logarithmic functions. Optimization, curve sketching, and integration by substitution. ***Prerequisite: Precalculus 30 or Mathematics B30 with a grade of at least 65%, or PMTH 092 with a grade of at least 70%, or Math102*** *Note: Although Math 103 leads to Math 112, students who require three or more calculus-based courses should take Math 110 instead of Math 103. Students will only receive credit for one of MATH 103 or 110*

MATH 108 - Mathematical Problems, Ideas and Personalities
This course explores some of the most significant and enduring ideas in mathematics: the great theorems, discoveries of beauty and insight that stand today as monuments to the human intellect. *** Prerequisite: Precalculus 20 or Foundations of Math 20 or Math A30 or AMTH 092*** * Note: This course is designed mainly for students in arts or education who wish some exposure to mathematical ideas. It satisfies the critical thinking requirement in the Faculty of Arts.

MATH 110 - Calculus I
An introductory class in the theory and techniques of differentiation and integration of algebraic and trigonometric functions. Topics include limits, optimization, curve sketching, and areas. ***Prerequisite: Precalculus 30 with at least 75%, or Calculus 30 or Mathematics B30 and C30 with a grade of at least 65% in each or Math 102*** *Note: Students can receive credit for only one of MATH 103 or 110*

MATH 111 - Calculus II
Differentiation and integration of exponential and logarithmic functions; methods of integration and applications; indeterminate forms, L'Hospital's rule and improper integrals; sequences, series, power series and Taylor series. ***Prerequisite: MATH 110, or MATH 103 with a grade of at least 80%***

MATH 112 - Applied Calculus II
An introduction to calculus in two and three variables, first-order differential equations, infinite series, and calculus of trigonometric functions. ***Prerequisites: MATH 103 and Precalculus 30, or MATH 110*** *Note: MATH 112 is a terminal course and is not intended for students who require further calculus courses. Students will receive credit for only one of MATH 111 and 112*

MATH 116 - Mathematics of Finance I
his course provides a basis of financial mathematics. Topics include measurement of interest, basic and general annuities, yield rates, amortization schedules, and sinking funds. ***Prerequisite: MATH 103 or 110.*** *Note: Students can receive credit for only one of MATH 116 and ACSC 116*

MATH 122 - Linear Algebra I
A course intended to introduce students to elementary linear algebra, particularly at a computational and applied level. Topics include matrices and systems of equations, inversion, determinants, vectors, inner products, eigenvectors and eigenvalues. *** Prerequisite: Precalculus 30, Mathematics B30 and C30, or Math 102.***

MATH 124 - The Art and Science of Secret Writing
The course examines methods of message encryption and cryptoanalysis. Attention will be given to the history of cryptology and the public-policy questions raised by its use in conjunction with the Internet. However, the focus will be on the mathematical tools needed to develop and analyze encryption algorithms. *** Prerequisites: Foundations of Math 30 or Precalculus 20 or Math B30 or AMTH 092 ***

MATH 127 - Introductory Finite Mathematics II
Elementary linear programming, counting methods involving permutations and combinations, probability, statistics, regression, and consumer mathematics including interest calculations, annuities and amortizations. ***Prerequisite: Foundations of Math 20 or Precalculus 20 or Math B30 or AMTH 092 or MATH 101*** *Note: Mathematics 127 is not algebra, nor pre-calculus, nor calculus. It satisfies the critical thinking requirement in the Faculty of Arts.* *Note: Students outside of the Faculty of Education cannot take this course for credit if they have received credit for a MATH, STAT, or ACSC course above 200.*

MATH 184 - Problem Solving Techniques
A course on hands-on training in mathematical problem solving, with a view towards the Putnam Competition. The course covers strategies to tackle problems, as well as selected topics from algebra, combinatorics, number theory, geometry, and analysis. ***Prerequisite: Precalculus 30 with at least a 75%; or Calculus 30; or Mathematics B30 and C30 with a grade of at least 65% in each; or MATH 102.*** *Note: This course carries only one credit hour.*

MATH 213 - Vector Calculus
A study of vector functions and functions of several variables and their derivatives; Applied maximum and minimum problems, Lagrange multipliers, multiple integration, integration in polar, cylindrical and spherical coordinates; Green's, Stokes' and the Divergence Theorem. ***Prerequisite: MATH 111 and 122***

MATH 216 - Mathematics of Finance II
This course is a continuation of Mathematics of Finance I and covers more advanced topics ranging from the theory of interest to principles of corporate finance. Specific topics include bonds, securities, analysis of risk and basic principles of pricing theory. ***Prerequisite: MATH 116 or ACSC 116*** *Note: Students can receive credit for only one of Math 216 and ACSC 216*

MATH 217 - Differential Equations I
Ordinary differential equations, modelling with differential equations, Laplace transforms. ***Prerequisite: MATH 111 and MATH 122***

MATH 221 - Introduction to Proofs and Problem Solving
An introductory course intended to familiarize students with mathematical reasoning and proof techniques, including direct reasoning, indirect reasoning, and mathematical induction. Topics include elementary number theory, logic, sets, functions, and relations. ***Prerequisite: Two university math courses beyond MATH 102.***

MATH 222 - Linear Algebra II
A second course in linear algebra. There will be some emphasis on proofs. Topics include matrices, abstract vector spaces, subspaces, bases, inner product spaces, linear transformations, matrix factorizations, symmetric matrices, quadratic forms, and applications of linear algebra. ***Prerequisite: MATH 122, and one of MATH 103 or 110.***

MATH 223 - Introduction to Abstract Algebra
An introductory course in abstract algebra. Topics include number systems and an introduction to groups, and some other mathematical structures. ***Prerequisite: MATH 221.***

MATH 231 - Euclidean Geometry
This course is intended to familiarize the student with Euclidean geometry. Topics include the postulates and theorems of both classical and modern Euclidean geometry. ***Prerequisite: MATH 221*** *Note: This course is designed for students without a strong background in Euclidean geometry. Students with a mark of at least 70% in either Mathematics C30 or Precalculus 30 should register in MATH 331*

MATH 251 - Introduction to Probability
Basic notions of probability; discrete and continuous random variables; expectation; moment generating functions; joint discrete random variables. ***Prerequisites: MATH 111 or MATH 112 and one of STAT 160 or STAT 200*** *Note: Students can receive credit for only one of Math 251 and Stat 251*

MATH 261 - Methods of Numerical Analysis
Topics will include number systems and errors, solutions of polynomial and other nonlinear equations, interpolation, numerical differentiation and integration, and the cubic spline. ***Prerequisite: MATH 111, MATH 122, and CS 110.*** *Note: Students will receive credit for only one of CS 261, CS 345 or MATH 261.*

MATH 284 - Problem Solving Techniques
A course on hands-on training in mathematical problem solving, with a view towards the Putnam Competition. The course covers strategies to tackle problems, as well as selected topics from algebra, combinatorics, number theory, geometry, and analysis. ***Prerequisite: MATH 184 or permission of the instructor.*** *Note: This course carries only one credit hour.*

MATH 300 - Introduction to Set Theory
Sets, relations, and operations on them. Natural numbers. Finite and infinite sets, ordinals and cardinals. Recursion theorems. Arithmetic of cardinals and ordinals. A brief introduction to set-theoretic topology. Construction of the real numbers and basic properties. ***Prerequisite: MATH 221.***

MATH 301 - Introduction to Mathematical Logic
Propositional and first-order predicate logic from a mathematical viewpoint. Axiomatically built theories and their models. Detailed study of one or more simple mathematical theories. Recursive functions. Basic ideas of automated theorem proving. ***Prerequisite: MATH 221.***

MATH 305 - Introductory Mathematical Analysis
Cardinality, real numbers and their topology, sequences, limits, continuity, and differentiation for functions of one real variable. ***Prerequisite: MATH 221 and 111.*** *Note: This course is designed for students interested in majoring in Mathematics. Students considering a degree in Mathematics with Honours are encouraged to complete this course by the end of their second year.*

MATH 308 - Topics in the History of Mathematics
A survey of the history of mathematics, focusing on mathematics developed before 1850. Topics may include mathematics of ancient cultures, cultural aspects of mathematics, how mathematics developed around the world, famous mathematicians and classical mathematics texts. This course is designed for majors in mathematics or mathematics education with a solid background in mathematics. It will be offered in the winter semester, alternating with MATH 309. ***Prerequisite: MATH 111, 122, and 221.***

MATH 309 - Topics in Modern Mathematics
A survey of modern mathematics, examining the objectives of mathematical advancement, important modern results in mathematics, mathematicians of the modern era, and the influences of modern mathematics on contemporary science. The focus of this course will be on mathematics after Gauss (post 1850). The emphasis will be on general modern approaches to mathematical problems and the philosophy of mathematics, rather than specific results. Topics will include (but are not limited to): the nature of mathematical knowledge, origins of modern mathematics, biographies of mathematicians and the influence of mathematics on contemporary science. ***Prerequisite: MATH 111, 122 and 221.*** *Note: This course is designed for majors in mathematics or mathematics education with a solid background in mathematics. It will be offered in the winter semester, alternating with MATH 308.*

MATH 312 - Complex Analysis I
Complex numbers, analytic functions, contour integration, Cauchy's theorem, infinite series, calculus of residues, basic theory of conformal mappings. ***Prerequisite: MATH 213.***

MATH 313 - Mathematical Analysis II
The Riemann integral for functions of one variable, sequences and series of functions, differentiation and integration for functions of several variables. ***Prerequisites: MATH 213 and MATH 305.***

MATH 316 - Mathematics of Finance III
This course covers the theory and pricing of financial derivatives such as Puts and Calls, with particular emphasis on the Black-Scholes model. ***Prerequisite: ACSC 216 or MATH 216, and STAT 251*** *Note: Students can receive credit for only one of MATH 316, STAT 316, and ACSC 316.*

MATH 317 - Real Analysis
Construction of the real numbers, structure of metric spaces, continuous functions on metric spaces, convergence of series, differential equations. ***Prerequisite: MATH 217 and 313.***

MATH 321 - Number Theory I
This is an introductory course in number theory. Topics include divisibility, primes, congruences, number theoretic functions, and diophantine equations. ***Prerequisite: MATH 221***

MATH 322 - Matrix Theory
Topics include: positive definiteness, Jordan canonical form, nonnegative matrices, and applications in matrix analysis. ***Prerequisite: MATH 222.***

MATH 323 - Modern Algebra I
A course in abstract algebra dealing with groups, rings, unique factorization domains and fields. ***Prerequisite: MATH 223.***

MATH 327 - Introductory Combinatorics
A first course in Combinatorics. Topics include counting, permutations and combinations, inclusion and exclusion, binomial theorem and identities with binomial coefficients, generating functions and recurrence relations, and a brief introduction to design theory. ***Prerequisite: MATH 221 and 111.***

MATH 328 - Introduction to Graph Theory
A first course in Graph Theory. Topics include isomorphism, Graph Algorithms, Trees, Digraphs and Networks, Planar graphs, Connectivity, Independence number, cliques and graph colouring. ***Prerequisite: MATH 221 and 111.***

MATH 329 - Linear and Discrete Optimization
A course in the theory and techniques of linear programming; convexity and extreme points of polyhedral sets, the simplex method, duality and selected applications will be covered. ***Prerequisite: CS 110, MATH 122 and MATH 221 or permission of Department Head.***

MATH 331 - Non-Euclidean Geometry
This course gives an explaination of the nature and foundations of geometry and uses for this purpose the systems of non-Euclidean geometry. It outlines the concept of mathematical models and the historical and logical significance of the parallel postulate. ***Prerequisite: MATH 110, and one of MATH 122 or MATH 231.*** *Note: Students may receive credit for only one of Math 232 and Math 331.*

MATH 335 - Introduction to Differential Geometry
Differential invariants of curves and surfaces in Euclidean three-space. ***Prerequisite: MATH 217.***

MATH 361 - Numerical Analysis I
Least squares and other approximations. Difference equations. Solutions of algebraic systems. Symbol manipulators. ***Prerequisite: MATH 213 and either MATH 261 or CS 261.***

MATH 381 - Differential Equations II
Series solutions of linear equations, systems of linear first-order equations, Fourier series, boundary-value problems, integral transforms, and numerical methods. ***Prerequisite: MATH 217.***

MATH 382 - Ordinary Differential Equations
Existence and uniqueness of solutions, linear systems, non-linear equations, stability, Liapunov's method, applications. ***Prerequisite: MATH 217.***

MATH 384 - Problem Solving Techniques
A course on hands-on training in mathematical problem solving, with a view towards the Putnam Competition. The course covers strategies to tackle problems, as well as selected topics from algebra, combinatorics, number theory, geometry, and analysis. ***Prerequisite: MATH 284 or permission of the instructor.*** *Note: This course carries only one credit hour.*

MATH 395AB - Directed Readings in Probability Theory
Selected advanced topics concerning multivariate random variables and distributions, and stochastic processes. ***Permission of the Department Head is required to register***

MATH 401 - Matrix Groups
An introduction to Lie group theory through study of the classical groups. *** Prerequisite: MATH 305 and 323 ***

MATH 411 - Measure and Integration
Measurable functions, Lebesgue integrals, Lp spaces, modes of convergence, signed measures, Radon-Nikodym Theorem. ***Prerequisite: MATH 313.***

MATH 412 - Complex Analysis II
This is a continuation of MATH 312. Topics include conformal mappings, argument principle, Rouche's theorem, harmonic functions, Riemann Mapping Theorem, infinite products, asymptotic expansions. ***Prerequisite: MATH 312.***

MATH 416 - Introduction to Quantum Information Theory
A first course in the mathematics of quantum information theory. Topics include information measures, quantum states and observables, qubits, entanglement, quantum channels, entropy, and measurements. ***Prerequisite: MATH 305 or MATH 312 or MATH 322, or permission of the Department Head***

MATH 418 - Introduction to Lie Algebras and Representation Theory
This course is an introduction to the structure of finite dimensional complex semisimple Lie algebras, via root systems, as well as their finite dimensional irreducible representations, through highest weight modules. ***Prerequisites: MATH 222 and 323*** *Note: Students can only receive credit for one of MATH 418, 495AD, and MATH 818.*

MATH 420 - Introduction To Commutative Algebra
A first course in commutative algebra. Topics include prime and maximal ideal, radicals, Nakayama's Lemma, exact sequences, tensor products, localization, Noetherian and Artinian rings. Additional topics may vary. This class is designed for advanced students in disciplines such as Mathematics and Computer Science who want to learn some commutative algebra. ***Prerequisite: MATH 222 and MATH 323***

MATH 421 - Number Theory II
This course is a second course in number theory. Topics include quadratic reciprocity, arithmetic functions, distribution of primes, and the prime number theorem. ***Prerequisite: MATH 321, 305, and 312.***

MATH 422 - Abstract Linear Algebra
A course which presents linear algebra in a theoretical setting. Topics include vector spaces, dual spaces, linear transformations, Jordan canonical form, the spectral theorem, and selected topics. ***Prerequisite: MATH 222 and MATH 323. ***

MATH 423 - Modern Algebra II
Continuation of Modern Algebra I with further study of rings, groups and fields. ***Prerequisite: MATH 323.***

MATH 424 - Applied Algebra
This is a course in applications of algebra to a selection of topics concerning enumeration, coding, finite state machines and cryptography. ***Prerequisite: MATH 223.***

MATH 425 - Matrix Analysis
A survey of some of the important topics from Matrix Theory with emphasis on matrix canonical forms, norms, spectral theory, perturbation theory of matrices, and special classes of matrices ***Prerequisite: MATH 305, MATH 322, and MATH 323.***

MATH 426 - Combinatorial Matrix Theory
A survey of some of the topics from combinatorial matrix theory including: spectral graph theory and algebraic graph theory. ***Prerequisite MATH 222 and 328.***

MATH 427 - Graph Theory
This course is a survey of topics in graph theory. Topics may include the following: matchings and factorizations, connectivity, colouring, isomorphisms, homomorphisms, automorphism groups and transitive graphs, extremal problems, adjacency matrices, spectral graph theory, strongly regular graphs, Cayley graphs, Ramsey theory and random graphs ***Prerequisite: MATH 223 and 328.***

MATH 431 - Differential Geometry I
Differentiable manifolds, the tangent bundle, differential forms, and the general Stokes' theorem. ***Prerequisite: MATH 313 or MATH 335.***

MATH 438 - Associative Algebras, Groups, and Representation Theory
An introductory course on the fundamental results concerning associative algebras, groups, and the representation theory of groups and algebras. ***Prerequisite: MATH 222 and MATH 323***

MATH 441 - General Topology
An introduction to point set topology including separation axioms, compactness, connectedness, continuous functions and metric spaces. ***Prerequisite: MATH 305.***

MATH 442 - Algebraic Topology
A first course in algebraic topology. Topics include, homotopy type, more detailed information on the fundamental group, and the homology and cohomology of topological spaces. ***Prerequisite MATH 441, or approval of the department chair.***

MATH 443 - Homological Algebra
A first course in homological algebra. Topics include modules over rings, chain complexes, homology, projective and injective resolutions, derived functors, abelian categories, derived categories, and selected additional topics ***Prerequisite: MATH 222 and MATH 323.*** *Note: Students may receive credit for one of MATH 443 or MATH 843.*

MATH 461 - Numerical Analysis II
Numerical solutions of ordinary differential equations; numerical solutions of partial differential equations; linear and non-linear problems. ***Prerequisite: MATH 361 or CS 361.***

MATH 481 - Partial Differential Equations
Classification and basic properties of equations, separation of variables, Fourier series, Sturm-Liouville theory, Fourier and Laplace transforms. ***Prerequisite: MATH 381.***

MATH 482 - Laplace Transforms
Properties of the Laplace Transform. Convolutions. The inversion integral. Applications to solutions of differential equations. *** Prerequisite: MATH 217 and 312.***

MATH 485 - Introduction to Functional Analysis
Metric, normed linear and inner-product spaces, linear operators and fixed point theorems. Spectral decompositions the Stone-Weierstrass theorem, applications. ***Prerequisite: MATH 312 and 313.***

MATH 495AB - Topology II
Topology II consists of Tietze extension theorem, Urysohn metrization theorem, tychonoff theorem, compact metric spaces.

MATH 495AC - Introduction to Continuum Mechanics
his course is an introduction to the physical concepts and mathematical methods of continuum mechanics with the aim of preparing the student for further studies and research in fluid dynamics. ***Prerequisite: MATH 381***

MATH 495AE - Introduction to von Neumann algebras
In this course we introduce von Neumann algebras and we emphasize connections with ergodic&group theory. We present constructions of von Neumann algebras from groups&actions; explain how group theoretical aspects (e.g. amenability) and orbit equivalence are connected to von Neumann algebras. We assume no background except basic knowledge of real analysis.

MATH 497 - Honours Seminar I
This is the first of two honours seminars. This course must be taken by all honours students in their fourth year. Students are required to attend the seminars and to work in consultation with an assigned supervisor on an independent research project. To receive credit for MATH 497, students must present a seminar on their preliminary work. *Note: This seminar is restricted to honours standing students in mathematics.*

MATH 498 - Honours Seminar II
This is the second of two honours seminars. This course must be taken by all honours students in their fourth year. Students are required to attend the seminars and to work in consultation with an assigned supervisor on an independent research project. To receive credit for MATH 498, students must present their project in both written form and as a seminar. *Note: This seminar is restricted to honours standing students in mathematics.*