Program Requirements
The Minor Concentration Mathematics is offered in two versions: an expandable version, for students who wish to leave open the option of expanding the program into a Major Concentration Mathematics, and a nonexpandable version for students who know on entry into the Minor that they do not wish to expand it into a major concentration.
The Minor Concentration Mathematics may be taken in conjunction with a major concentration in some other discipline under option A of the Multitrack System. Students planning on taking the Major Concentration Mathematics and the Minor Concentration Mathematics as part of Multitrack option C should select the Supplementary Minor Concentration in Mathematics in place of this Minor concentration.
Under option C, it is not possible to combine the Minor Concentration Mathematics and the Minor Concentration Statistics. Students wishing to do this should instead take the Major Concentration Mathematics under option B (two major concentrations) and select a large number of statistics complementaries.
For more information about the Multitrack System options please refer to the Faculty of Arts regulations under "Faculty Degree Requirements", "About Program Requirements", and "Departmental Programs".
No overlap is permitted with other programs.
Program Prerequisites
Students who have not completed the program prerequisite courses listed below or their equivalents will be required to make up any deficiencies in these courses over and above the 18 credits required for the program.

MATH 133 Linear Algebra and Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Systems of linear equations, matrices, inverses, determinants; geometric vectors in three dimensions, dot product, cross product, lines and planes; introduction to vector spaces, linear dependence and independence, bases; quadratic loci in two and three dimensions.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Djivede Kelome, William J Anderson, Eric Cormier, Michael Bailey (Fall) William J Anderson, Alexey Eremin (Winter)
 3 hours lecture, 1 hour tutorial
 Prerequisite: a course in functions
 Restriction A: Not open to students who have taken MATH 221 or CEGEP objective 00UQ or equivalent.
 Restriction B: Not open to students who have taken or are taking MATH 123, MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.
 Restriction C: Not open to students who are taking or have taken MATH 134.

MATH 140 Calculus 1 (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of functions and graphs. Limits, continuity, derivative. Differentiation of elementary functions. Antidifferentiation. Applications.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Sidney Trudeau, Stephen W Drury, William G Brown, Thomas F Fox (Fall)
 3 hours lecture, 1 hour tutorial
 Prerequisite: High School Calculus
 Restriction: Not open to students who have taken MATH 120, MATH 139 or CEGEP objective 00UN or equivalent
 Restriction: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics
 Each Tutorial section is enrolment limited

MATH 141 Calculus 2 (4 credits)
Overview
Mathematics & Statistics (Sci) : The definite integral. Techniques of integration. Applications. Introduction to sequences and series.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Antonio Lei (Fall) Sidney Trudeau, Stephen W Drury, Thomas F Fox (Winter)
 Prerequisites: MATH 139 or MATH 140 or MATH 150.
 Restriction: Not open to students who have taken MATH 121 or CEGEP objective 00UP or equivalent
 Restriction Note B: Not open to students who have taken or are taking MATH 122 or MATH 130 or MATH 131, except by permission of the Department of Mathematics and Statistics.
 Each Tutorial section is enrolment limited
Expandable Version: Required Courses (12 credits)
* Note: Credit cannot be received for both MATH 236 and MATH 223 (listed as a required course in the nonexpandable version of this Minor concentration).

MATH 222 Calculus 3 (3 credits)
Overview
Mathematics & Statistics (Sci) : Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Wilbur Jonsson, Mathew Donald Rogers (Fall) Wilbur Jonsson (Winter) Eric Cormier (Summer)
 Prerequisite: MATH 141. Familiarity with vector geometry or Corequisite: MATH 133
 Restriction: Not open to students who have taken CEGEP course 201303 or MATH 150, MATH 151 or MATH 227

MATH 235 Algebra 1 (3 credits)
Overview
Mathematics & Statistics (Sci) : Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; group actions on sets.
Terms: Fall 2012
Instructors: Henri Darmon (Fall)
 Fall
 3 hours lecture; 1 hour tutorial
 Prerequisite: MATH 133 or equivalent

MATH 236 Algebra 2 (3 credits) *
Overview
Mathematics & Statistics (Sci) : Linear equations over a field. Introduction to vector spaces. Linear mappings. Matrix representation of linear mappings. Determinants. Eigenvectors and
eigenvalues. Diagonalizable operators. CayleyHamilton theorem. Bilinear and quadratic forms. Inner product spaces, orthogonal diagonalization of symmetric
matrices. Canonical forms.
Terms: Winter 2013
Instructors: There are no professors associated with this course for the 20122013 academic year.
 Winter
 Prerequisite: MATH 235

MATH 315 Ordinary Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: JianJun Xu (Fall) Alexey Eremin (Winter)
 Prerequisite: MATH 222.
 Corequisite: MATH 133.
 Restriction: Not open to students who have taken or are taking MATH 325.
Expandable Version: Complementary Courses (6 credits)
Students selecting the expandable version of this program complete six complementary courses from the Complementary Course List.
It is strongly recommended that students take MATH 323 as a complementary course.
NonExpandable Version: Required Courses (9 credits)
* Note: Credit cannot be received for both MATH 223 and MATH 236 (listed as a required course in the expandable version of this Minor concentration).

MATH 222 Calculus 3 (3 credits)
Overview
Mathematics & Statistics (Sci) : Taylor series, Taylor's theorem in one and several variables. Review of vector geometry. Partial differentiation, directional derivative. Extreme of functions of 2 or 3 variables. Parametric curves and arc length. Polar and spherical coordinates. Multiple integrals.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Wilbur Jonsson, Mathew Donald Rogers (Fall) Wilbur Jonsson (Winter) Eric Cormier (Summer)
 Prerequisite: MATH 141. Familiarity with vector geometry or Corequisite: MATH 133
 Restriction: Not open to students who have taken CEGEP course 201303 or MATH 150, MATH 151 or MATH 227

MATH 223 Linear Algebra (3 credits) *
Overview
Mathematics & Statistics (Sci) : Review of matrix algebra, determinants and systems of linear equations. Vector spaces, linear operators and their matrix representations, orthogonality. Eigenvalues and eigenvectors, diagonalization of Hermitian matrices. Applications.
Terms: Fall 2012, Winter 2013
Instructors: Layan El Hajj, Wilbur Jonsson (Fall) Wilbur Jonsson (Winter)
 Fall and Winter
 Prerequisite: MATH 133 or equivalent
 Restriction: Not open to students in Mathematics programs nor to students who have taken or are taking MATH 236, MATH 247 or MATH 251. It is open to students in Faculty Programs

MATH 315 Ordinary Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order ordinary differential equations including elementary numerical methods. Linear differential equations. Laplace transforms. Series solutions.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: JianJun Xu (Fall) Alexey Eremin (Winter)
 Prerequisite: MATH 222.
 Corequisite: MATH 133.
 Restriction: Not open to students who have taken or are taking MATH 325.
NonExpandable Version: Complementary Courses (9 credits)
Students selecting the nonexpandable version of this program complete nine complementary courses from the Complementary Course List.
It is strongly recommended that students take MATH 323 as a complementary course.
Complementary Course List
* Note: Either MATH 249 or MATH 316 may be taken but not both.

MATH 249 Honours Complex Variables (3 credits) *
Overview
Mathematics & Statistics (Sci) : Functions of a complex variable; CauchyRiemann equations; Cauchy's theorem and consequences. Taylor and Laurent expansions. Residue calculus; evaluation of real integrals; integral representation of special functions; the complex inversion integral. Conformal mapping; SchwarzChristoffel transformation; Poisson's integral formulas; applications.
Terms: Winter 2013
Instructors: Johannes Walcher (Winter)
 Winter
 Prerequisite: MATH 248.
 Restriction: Intended for Honours Physics and Engineering students
 Restriction: Not open to students who have taken or are taking MATH 316

MATH 314 Advanced Calculus (3 credits)
Overview
Mathematics & Statistics (Sci) : Derivative as a matrix. Chain rule. Implicit functions. Constrained maxima and minima. Jacobians. Multiple integration. Line and surface integrals. Theorems of Green, Stokes and Gauss. Fourier series with applications.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: JianJun Xu (Fall) Wilbur Jonsson (Winter) Charles Roth (Summer)
 Prerequisites: MATH 133, MATH 222
 Restriction: Not open to students who have taken or are taking MATH 248

MATH 316 Complex Variables (3 credits) *
Overview
Mathematics & Statistics (Sci) : Algebra of complex numbers, CauchyRiemann equations, complex integral, Cauchy's theorems. Taylor and Laurent series, residue theory and applications.
Terms: Fall 2012
Instructors: John A Toth (Fall)
 Fall
 Prerequisites: MATH 314 and MATH 243
 Restriction: Not open to students who have taken or are taking MATH 249, MATH 366, MATH 381 or MATH 466.

MATH 317 Numerical Analysis (3 credits)
Overview
Mathematics & Statistics (Sci) : Error analysis. Numerical solutions of equations by iteration. Interpolation. Numerical differentiation and integration. Introduction to numerical solutions of differential equations.
Terms: Fall 2012
Instructors: Peter Bartello (Fall)
 Fall
 Prerequisites: MATH 315 or MATH 325 or MATH 263, and COMP 202 or permission of instructor.

MATH 318 Mathematical Logic (3 credits)
Overview
Mathematics & Statistics (Sci) : Propositional calculus, truthtables, switching circuits, natural deduction, first order predicate calculus, axiomatic theories, set theory.
Terms: Fall 2012
Instructors: Diana Dubrovsky (Fall)
 Fall
 Restriction: Not open to students who are taking or have taken PHIL 210

MATH 319 Introduction to Partial Differential Equations (3 credits)
Overview
Mathematics & Statistics (Sci) : First order equations, geometric theory; second order equations, classification; Laplace, wave and heat equations, SturmLiouville theory, Fourier series, boundary and initial value problems.
Terms: Winter 2013
Instructors: There are no professors associated with this course for the 20122013 academic year.
 Winter
 Prerequisites: MATH 223 or MATH 236, MATH 314, MATH 315

MATH 320 Differential Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of Euclidean geometry. Local theory of plane and space curves: the Frenet formulas. Local theory of surfaces: the first and second fundamental forms, the shape operator, the mean and Gaussian curvatures, surfaces of revolution with prescribed curvature, ruled and developable surfaces. Geodesic curves on surfaces of revolution. The GaussCodazzi equations, rigidity.
Terms: This course is not scheduled for the 20122013 academic year.
Instructors: There are no professors associated with this course for the 20122013 academic year.
 Fall
 Prerequisites: MATH 236 or MATH 223 or MATH 247, and MATH 314 or MATH 248

MATH 323 Probability (3 credits)
Overview
Mathematics & Statistics (Sci) : Sample space, events, conditional probability, independence of events, Bayes' Theorem. Basic combinatorial probability, random variables, discrete and continuous univariate and multivariate distributions. Independence of random variables. Inequalities, weak law of large numbers, central limit theorem.
Terms: Fall 2012, Winter 2013, Summer 2013
Instructors: Masoud AsgharianDastenaei (Fall) David B Wolfson (Winter) Djivede Kelome (Summer)
 Prerequisites: MATH 141 or equivalent.
 Restriction: Intended for students in Science, Engineering and related disciplines, who have had differential and integral calculus
 Restriction: Not open to students who have taken or are taking MATH 356

MATH 324 Statistics (3 credits)
Overview
Mathematics & Statistics (Sci) : Sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, contingency tables, nonparametric inference, regression, Bayesian inference.
Terms: Fall 2012, Winter 2013
Instructors: William J Anderson (Fall) David B Wolfson (Winter)
 Fall and Winter
 Prerequisite: MATH 323 or equivalent
 Restriction: Not open to students who have taken or are taking MATH 357
 You may not be able to receive credit for this course and other statistic courses. Be sure to check the Course Overlap section under Faculty Degree Requirements in the Arts or Science section of the Calendar.

MATH 326 Nonlinear Dynamics and Chaos (3 credits)
Overview
Mathematics & Statistics (Sci) : Linear systems of differential equations, linear stability theory. Nonlinear systems: existence and uniqueness, numerical methods, one and two dimensional flows, phase space, limit cycles, PoincareBendixson theorem, bifurcations, Hopf bifurcation, the Lorenz equations and chaos.
Terms: Fall 2012
Instructors: JeanChristophe Nave (Fall)
 Fall
 Prerequisites: MATH 222, MATH 223
 Restriction: Not open to students who have taken or are taking MATH 376

MATH 327 Matrix Numerical Analysis (3 credits)
Overview
Mathematics & Statistics (Sci) : An overview of numerical methods for linear algebra applications and their analysis. Problem classes include linear systems, least squares problems and eigenvalue problems.
Terms: Winter 2013
Instructors: Antony Raymond Humphries (Winter)
 Winter
 Prerequisites: MATH 223 or MATH 236 or MATH 247 or MATH 251, COMP 202 or consent of instructor.

MATH 340 Discrete Structures 2 (3 credits)
Overview
Mathematics & Statistics (Sci) : Review of mathematical writing, proof techniques, graph theory and counting. Mathematical logic. Graph connectivity, planar graphs and colouring. Probability and graphs. Introductory group theory, isomorphisms and automorphisms of graphs. Enumeration and listing.
Terms: Winter 2013
Instructors: Frederick Shepherd (Winter)
 Winter
 Prerequisites: MATH 235 or MATH 240 or MATH 242.
 Corequisites: MATH 223 or MATH 236.
 Restriction: Not open to students who have taken or are taking MATH 343 or MATH 350.

MATH 346 Number Theory (3 credits)
Overview
Mathematics & Statistics (Sci) : Divisibility. Congruences. Quadratic reciprocity. Diophantine equations. Arithmetical functions.
Terms: Winter 2013
Instructors: Henri Darmon (Winter)
 Winter
 Prerequisite: MATH 235 or consent of instructor
 Restriction: Not open to students who have taken or are taking MATH 377.

MATH 348 Topics in Geometry (3 credits)
Overview
Mathematics & Statistics (Sci) : Selected topics  the particular selection may vary from year to year. Topics include: isometries in the plane, symmetry groups of frieze and ornamental patterns, equidecomposibility, nonEuclidean geometry and problems in discrete geometry.
Terms: Summer 2013
Instructors: There are no professors associated with this course for the 20122013 academic year.
 Prerequisite: MATH 133 or equivalent or permission of instructor.

MATH 407 Dynamic Programming (3 credits)
Overview
Mathematics & Statistics (Sci) : Sequential decision problems, resource allocation, transportation problems, equipment replacement, integer programming, network analysis, inventory systems, project scheduling, queuing theory calculus of variations, markovian decision processes, stochastic path problems, reliability, discrete and continuous control processes.
Terms: This course is not scheduled for the 20122013 academic year.
Instructors: There are no professors associated with this course for the 20122013 academic year.
 Winter
 Prerequisites: COMP 202; MATH 223 or MATH 236, MATH 314, MATH 315 and MATH 323

MATH 417 Mathematical Programming (3 credits)
Overview
Mathematics & Statistics (Sci) : An introductory course in optimization by linear algebra, and calculus methods. Linear programming (convex polyhedra, simplex method, duality, multicriteria problems), integer programming, and some topics in nonlinear programming (convex functions, optimality conditions, numerical methods). Representative applications to various disciplines.
Terms: Fall 2012
Instructors: Frederick Shepherd (Fall)
 Prerequisites: COMP 202, and MATH 223 or MATH 236, and MATH 314 or equivalent
 Restriction: Not open to students who have taken or are taking MATH 487.