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 Áù¾ÅÉ«Ìà Calendar 2016-2017 COURSES OF INSTRUCTION Course Descriptions M Mathematics MATH
Mathematics MATH
MATH 113 177 205 211 213 249 265 267 271 273 275 277 311 313 331 335 355 367 375 377 401 403 411 421 423 445 447 501 521 545 600 601 603 605 607 617 621 625 627 631 635 637 641 643 647 651 653 661 663 667 669 681 683 685 691 693
 

Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.

Notes:

  • For listings of related courses, see Actuarial Science, Applied Mathematics, Pure Mathematics, and Statistics.
  • Effective Fall 2014, Mathematics 265, 267, 367, Mathematics 275, 277, 375 and 377 replaced respectively Mathematics 251, 253, 353, Applied Mathematics 217, 219, 307 and 309 and serves as prerequisites for appropriate courses. In some special cases, Mathematics 267 replaces Mathematics 349 or 353. For these and other deviations from the general rule, see individual course entries for details. Mathematics 267 supplemented by Mathematics 177 will be accepted as equivalent to Mathematics 277.
 
Mathematics 113       Eigenvalues and Eigenvectors
A review of these particular topics for students who have completed Mathematics 211 or equivalent.
Course Hours: 0.75 units; E(8 hours)
Notes: Open to students with credit in Mathematics 211 or equivalent.
NOT INCLUDED IN GPA
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Mathematics 177       Further Topics from Mathematics 277
Vector functions and differentiation, curves and parametrization, functions of several variables, partial differentiation, differentiability, implicit functions, extreme values.
Course Hours: 0.75 units; E(16 hours)
Prerequisite(s): Mathematics 267.
Notes: Designed to rectify a deficiency for those students whose Calculus I and II courses covered all the topics from Mathematics 265 and 267 but did not cover some of the topics on the calculus of functions of several variables from Mathematics 277.
NOT INCLUDED IN GPA
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Junior Courses

Note: Students who have not studied mathematics for some time are strongly advised to review high school material thoroughly prior to registering in any junior level mathematics course.
 
Mathematics 205       Mathematical Explorations
A mathematics appreciation course. Topics selected by the instructor to provide a contemporary mathematical perspective and experiences in mathematical thinking. May include historical material on the development of classical mathematical ideas as well as the evolution of recent mathematics.
Course Hours: 3 units; H(3-1)
Prerequisite(s): Mathematics 30-1, Mathematics 30-2, Pure Mathematics 30, Applied Mathematics 30, or Mathematics II (offered by Continuing Education).
Notes: For students whose major interests lie outside the sciences. Highly recommended for students pursuing an Elementary School Education degree.  It is not a prerequisite for any other course offered by the Department of Mathematics and Statistics, and cannot be used for credit towards any Major or Minor program in the Faculty of Science except for a major in General Mathematics.
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Mathematics 211       Linear Methods I
Systems of equations and matrices, vectors, matrix representations and determinants. Complex numbers, polar form, eigenvalues, eigenvectors. Applications.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar).
Antirequisite(s): Credit for Mathematics 211 and either 213 or 221 will not be allowed.
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Mathematics 213       Honours Linear Algebra I
Systems of equations and matrices, vectors, linear transformations, determinants, eigenvalues and eigenvectors.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30.
Antirequisite(s): Credit for Mathematics 213 and 211 will not be allowed.
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Mathematics 249       Introductory Calculus
Algebraic operations. Functions and graphs. Limits, derivatives, and integrals of exponential, logarithmic and trigonometric functions. Fundamental theorem of calculus. Improper integrals. Applications.
Course Hours: 3 units; H(4-1T-1)
Prerequisite(s): A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar).
Antirequisite(s): Not open to students with 60 per cent or higher in Mathematics 31 or a grade of "C" or higher in Mathematics 3 offered through Áù¾ÅÉ«Ìà Continuing Education, except with special departmental permission. Credit for Mathematics 249 and any one of Mathematics 251, 265, 275, 281, or Applied Mathematics 217 will not be allowed.
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Mathematics 265       University Calculus I
Limits, derivatives, and integrals; the calculus of exponential, logarithmic, trigonometric and inverse trigonometric functions. Applications including curve sketching, optimization, exponential growth and decay, Taylor polynomials. Fundamental theorem of calculus. Improper integrals. Introduction to partial differentiation.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): A grade of 70 per cent or higher in Mathematics 30-1 or Pure Mathematics 30; and a grade of 50 per cent or higher in Mathematics 31 or a grade of "C" or higher in Mathematics 3 offered through Áù¾ÅÉ«Ìà Continuing Education. (Alternatives to Pure Mathematics 30 are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar).
Antirequisite(s): Credit for Mathematics 265 and any one of Mathematics 249, 251, 275, 281, or Applied Mathematics 217 will not be allowed.
Notes: This course provides the basic techniques of differential calculus as motivated by various applications. Students performing sufficiently well in a placement test may be advised to transfer directly to Mathematics 267.
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Mathematics 267       University Calculus II
Sequences and series, techniques of integration, multiple integration, applications; parametric equations.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): Mathematics 249 or 251 or 265 or 275 or 281 or Applied Mathematics 217.
Antirequisite(s): Credit for Mathematics 267 and any one of Mathematics 277, 349, or Applied Mathematics 219 will not be allowed.
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Mathematics 271       Discrete Mathematics
Proof techniques. Sets and relations. Induction. Counting and probability. Graphs and trees.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): Mathematics 211.
Antirequisite(s): Credit for Mathematics 271 and 273 will not be allowed.
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Mathematics 273       Honours Mathematics: Numbers and Proofs
Introduction to proofs. Functions, sets and relations. The integers: Euclidean division algorithm and prime factorization; induction and recursion; integers mod n. Real numbers: sequences of real numbers; completeness of the real numbers; open and closed sets. Complex numbers.
Course Hours: 3 units; H(3-1T-1)
Prerequisite(s): A grade of 80 per cent or higher in Mathematics 30-1 or Pure Mathematics 30. (Alternatives are presented in C.1 Mathematics Diagnostic Test in the Academic Regulations section of this Calendar).
Antirequisite(s): Credit for Mathematics 273 and 271 will not be allowed.
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Mathematics 275       Calculus for Engineers and Scientists
Calculus of functions of one real variable; derivative and Riemann integral; Mean Value Theorem; the Fundamental Theorem of Calculus; techniques of integration; Applications; Improper integrals; Power series, Taylor series.
Course Hours: 3 units; H(3-1T-1.5)
Prerequisite(s): A grade of 70 per cent or higher in Pure Mathematics 30 or Mathematics 30-1; and credit in Mathematics 31 or Mathematics 3 offered through Áù¾ÅÉ«Ìà Continuing Education. Alternatively, admission to the Faculty of Engineering including credit in either Pure Mathematics 30 or Mathematics 30-1; and Mathematics 31 or Mathematics 3 offered through Áù¾ÅÉ«Ìà Continuing Education.
Antirequisite(s): Credit for Mathematics 275 and any one of Mathematics 249 or 251 or 265 or 281 or Applied Mathematics 217 will not be allowed. 
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Mathematics 277       Multivariable Calculus for Engineers and Scientists
Calculus of functions of several real variables; differentiation, implicit functions, double and triple integrals; applications; Vector-valued functions; derivatives and integrals; parametric curves.
Course Hours: 3 units; H(3-1T-1.5)
Prerequisite(s): Mathematics 275 or Applied Mathematics 217.
Antirequisite(s): Credit for Mathematics 277 and any one of Mathematics 253 or 267 or 283 or Applied Mathematics 219 will not be allowed.
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Senior Courses
 
Mathematics 311       Linear Methods II
Vector spaces and subspaces. Linear independence. Matrix representations of linear transformations. Gram-Schmidt orthogonalization. Students will complete a project using a computer algebra system.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Mathematics 211 or 213.
Antirequisite(s): Credit for Mathematics 311 and 313 will not be allowed.
Notes: Mathematics 271 is highly recommended as a prerequisite.
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Mathematics 313       Honours Linear Algebra II
Diagonalization. Canonical forms. Inner products, orthogonalization. Spectral theory. Students will be required to complete a project using a computer algebra system.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Mathematics 213 or a grade of "B+" or better in Mathematics 211.
Antirequisite(s): Credit for Mathematics 311 and 313 will not be allowed.
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Mathematics 331       Multivariate Calculus
Linear ordinary differential equations, and systems of ordinary differential equations. Calculus of functions of several variables. Introduction to vector analysis, theorems of Green, Gauss and Stokes.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): One of Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219; and Mathematics 211 or 213.
Antirequisite(s): Credit for Mathematics 331 and any of Mathematics 353 or 367 or 377 or 381 or Applied Mathematics 309 will not be allowed.
Notes: This course is not a member of the list of courses constituting the fields of Actuarial Science, Applied Mathematics, Pure Mathematics, or Statistics and cannot normally be substituted for Mathematics 353 or 367 or 377 or 381 in degree programs in any of those fields.
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Mathematics 335       Analysis I
The real numbers, sequences, series, functions, continuity and uniform continuity, differentiation, intermediate and mean value theorems, the Riemann integral, integrability of continuous functions on closed intervals.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219.
Antirequisite(s): Credit for Mathematics 335 and any one of Mathematics 355, Pure Mathematics 435 or 455 will not be allowed.
Notes: Students with a grade of "B+" or higher in Mathematics 267 or 277 are encouraged to consider taking Mathematics 355.
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Mathematics 355       Honours Analysis I
The real numbers, sequences, series, functions, continuity and uniform continuity, differentiation, intermediate and mean value theorems, the Riemann integral, integrability of continuous functions on closed intervals.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Mathematics 283 or 267 or 277; or a grade of "B+" or better in Mathematics 253 or Applied Mathematics 219.
Antirequisite(s): Credit for Mathematics 355 and any one of Mathematics 335, Pure Mathematics 435 or 455 will not be allowed.
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Mathematics 367       University Calculus III
Functions of several variables; limits, continuity, differentiability, partial differentiation, applications including optimization and Lagrange multipliers. Vector functions, line integrals and surface integrals, Green’s theorem, Stokes’ theorem. Divergence theorem. Students will complete a project using a computer algebra system.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): One of Mathematics 267 or 283 or 349 or Applied Mathematics 219; and Mathematics 211 or 213.
Antirequisite(s): Credit for Mathematics 367 and any one of Mathematics 353, 331, 377, 381 or Applied Mathematics 309 will not be allowed.
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Mathematics 375       Differential Equations for Engineers and Scientists
Definition, existence and uniqueness of solutions; first order and higher order equations and applications; Homogeneous systems; Laplace transform; partial differential equations of mathematical physics.
Course Hours: 3 units; H(3-1.5T)
Prerequisite(s): Mathematics 211 or 213; and one of the following: both Mathematics 267 and 177; or both Mathematics 253 and 114; or Applied Mathematics 219 or Mathematics 277.
Antirequisite(s): Credit for  Mathematics 375 and either Applied Mathematics 307 or 311 will not be allowed.
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Mathematics 377       Vector Calculus for Engineers and Scientists
Review of calculus of functions of several variables. Vector fields, line integrals, independence of path, Green’s theorem; Surface integrals, divergence theorem, Stokes’s theorem; applications; curvilinear co-ordinates; Laplace, diffusion and wave equations in three dimensional space.
Course Hours: 3 units; H(3-1.5T)
Prerequisite(s): Mathematics 375.
Antirequisite(s): Credit for more than one of Mathematics 377, 331, 353, 367, 381 or Applied Mathematics 309 will not be allowed.
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Mathematics 401       Special Topics
Higher level topics which can be repeated for credit.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Notes: This course is designed to add flexibility to completion of an undergraduate pure mathematics or general mathematics program.
MAY BE REPEATED FOR CREDIT
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Mathematics 403       Topics in Mathematics for Economics
Techniques of integration. Multiple integrals. Analysis of functions. Continuity. Compact sets. Convex sets. Separating hyperplanes. Lower and upper hemi-continuous correspondences. Fixed point theorems, Optimal control.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 211 or 213; and Mathematics 253 or 267 or 277 or 283 or Applied Mathematics 219. Alternatively, both Economics 387 and 389.
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Mathematics 411       Linear Spaces with Applications
Canonical forms. Inner product spaces, invariant subspaces and spectral theory. Quadratic forms.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Mathematics 311; and one of Mathematics 331, 353, 367, 377, 381 or Applied Mathematics 309.
Antirequisite(s): Credit for more than one of Mathematics 411, 313 or Applied Mathematics 441 will not be allowed.
Notes: May not be offered every year. Consult the Department for listings.
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Mathematics 421       Complex Analysis I
Basic complex analysis – complex numbers and functions, differentiation, Cauchy-Riemann equations, line integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor’s theorem, the residue theorem, applications to computation of definite integrals.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Both Mathematics 349 and 353; or both Mathematics 283 and 381; or Mathematics 267.
Antirequisite(s): Credit for Mathematics 421 and any one of Mathematics 423, Pure Mathematics 421 or 521 will not be allowed.     
Notes: For students with credit in Mathematics 267, it is strongly recommended that they take Mathematics 367 before or while taking Mathematics 421.
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Mathematics 423       Honours Complex Analysis
Basic complex analysis – complex numbers and functions, differentiation, Cauchy-Riemann equations, line integration, Cauchy’s theorem and Cauchy’s integral formula, Taylor’s theorem, the residue theorem, applications to computation of definite integrals.
Course Hours: 3 units; H(3-1T)
Prerequisite(s): Both Mathematics 349 and 353; or both Mathematics 283 and 381; or Mathematics 267.
Antirequisite(s): Credit for Mathematics 423 and any one of Mathematics 421, Pure Mathematics 421 or 521 will not be allowed.
Notes: Open only to Honours Applied Mathematics and Honours Pure Mathematics students. For students with credit in Mathematics 267 it is strongly recommended that they take Mathematics 367 before or while taking Mathematics 423.
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Mathematics 445       Analysis II
Basic topology of Euclidean space, Fubini’s theorem, the total derivative, change of variable in multiple integrals, inverse and implicit function theorems, submanifolds of Euclidean spaces, differential forms, Stokes’ theorem in arbitrary dimension.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 353 or 367 or 377 or 381 or Applied Mathematics 309; and Mathematics 311 or 313; and Mathematics 335 or 355 or Pure Mathematics 435 or 455.
Antirequisite(s): Credit for Mathematics 445 and either Mathematics 447 or Pure Mathematics 545 will not be allowed.         
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Mathematics 447       Honours Analysis II
Basic topology of Euclidean space, Fubini’s theorem, the total derivative, change of variable in multiple integrals, inverse and implicit function theorems, submanifolds of Euclidean spaces, differential forms, Stokes’ theorem in arbitrary dimension.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 367 or 377 or 381 or Applied Mathematics 309 or "B+" or higher in Mathematics 353; and Mathematics 313 or "B+" or higher in Mathematics 311; and Mathematics 355 or Pure Mathematics 455 or "B+" or higher in Mathematics 335 or Pure Mathematics 435. Alternatively, consent of the Department.
Antirequisite(s): Credit for Mathematics 447 and either Mathematics 445 or Pure Mathematics 545 will not be allowed.
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Mathematics 501       Measure and Integration
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp Spaces, Riesz representation theorems.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 545 or Pure Mathematics 545.
Antirequisite(s): Credit for Mathematics 501 and any one of Mathematics 601, Pure Mathematics 501 or 601 will not be allowed.
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Mathematics 521       Complex Analysis II
Analytic functions as mappings, local properties of analytic functions, Schwarz lemma, Casorati-Weierstrass and Picard theorems, analytic continuation, harmonic and subharmonic functions, approximation theorems, conformal mappings, Riemann surfaces.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 335 or 355 or Pure Mathematics 435 or 455; and Mathematics 421 or 423 or Pure Mathematics 421.
Antirequisite(s): Credit for Mathematics 521 and Pure Mathematics 521 will not be allowed.
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Mathematics 545       Analysis III
Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 447 or a grade of "B+" or better in Pure Mathematics 445 or Mathematics 445.
Antirequisite(s): Credit for Mathematics 545 and Pure Mathematics 545 will not be allowed.
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Graduate Courses Note: In addition to the prerequisites listed below, consent of the Applied Mathematics Department or the Pure Mathematics Department is a prerequisite for these graduate courses.
 
Mathematics 600       Research Seminar
A professional skills course, focusing on the development of technical proficiencies that are essential to succeed as practicing mathematicians in academia, government, or industry. The emphasis is on delivering professional presentations and using modern mathematical research tools. A high level of active student participation is required.
Course Hours: 1.5 units; Q(3S-0)
MAY BE REPEATED FOR CREDIT NOT INCLUDED IN GPA
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Mathematics 601       Measure and Integration
Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp spaces, Riesz representation theorem.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 545 or Pure Mathematics 545.
Antirequisite(s): Credit for more than one of Mathematics 501, 601, Pure Mathematics 501 and 601 will not be allowed.
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Mathematics 603       Analysis III
Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 447 or a grade of "B+" or better in Pure Mathematics 445 or Mathematics 445.
Antirequisite(s): Credit for Mathematics 603 and either Mathematics 545 or Pure Mathematics 545 will not be allowed.
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Mathematics 605       Differential Equations III
Systems of ordinary differential equations.  Existence and uniqueness. Introduction to partial differential equations.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Applied Mathematics 411 and Pure Mathematics 445 or 545.
Antirequisite(s): Credit for Mathematics 605 and Applied Mathematics 605 will not be allowed.
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Mathematics 607       Algebra III
A sophisticated introduction to modules over rings, especially commutative rings with identity. Major topics include: snake lemma; free modules; tensor product; hom-tensor duality; finitely presented modules; invariant factors; free resolutions; and the classification of finitely generated modules over principal ideal domains. Adjoint functors play a large role. The course includes applications to linear algebra, including rational canonical form and Jordan canonical form.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Pure Mathematics 431 or Mathematics 411. Pure Mathematics 431 is recommended.
Antirequisite(s): Credit for more than one of Pure Mathematics 511, 611 and Mathematics 607 will not be allowed.
Also known as: (formerly Pure Mathematics 611)
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Mathematics 617       Functional Analysis
Introduction to some basic aspects of Functional Analysis, Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 545 or Mathematics 603.
Notes: Credit for more than one of Applied Mathematics 617 and Pure Mathematics 617 and Mathematics 617 will not be allowed.
Also known as: (formerly Applied Mathematics 617)
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Mathematics 621       Complex Analysis
A rigorous study of function of a single complex variable. Holomorphic function, Cauchy integral formula and its applications. Conformal mappings. Fractional linear transformations. Argument principle. Schwarz lemma. Conformal self-maps of the unit disk.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 335 or 355 or Pure Mathematics 435 or 455.
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Mathematics 625       Introduction to Algebraic Topology
Introduction to the algebraic invariants that distinguish topological spaces. Focuses on the fundamental group and its applications, and homology. Introduction to the basics of homological algebra.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Pure Mathematics 505 and Pure Mathematics 431.
Antirequisite(s): Credit for Pure Mathematics 607 and Mathematics 625 will not be allowed.
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Mathematics 627       Algebraic Geometry
The objective of this course is to provide an introduction to modern algebraic geometry sufficient to allow students to read research papers in their fields which use the language of schemes. Topics will include Spectra of rings; the Zariski topology; affine schemes; sheaves; ringed spaces; schemes; morphisms of finite type; arithmetic schemes; varieties; projective varieties; finite morphisms, unramified morphisms; etale morphisms.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 607.
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Mathematics 631       Discrete Mathematics

Discrete Geometry: Euclidean, spherical and hyperbolic n-spaces, trigonometry, isometries, convex sets, convex polytopes, (mixed) volume(s), classical discrete groups, tilings, isoperimetric inequalities, packings, coverings. Graph Theory: connectivity; trees; Euler trails and tours; Hamilton cycles and paths; matchings; edge colourings; vertex colourings; homomorphisms; plane and planar graphs; extremal graph theory and Ramsey theory.

631.01. Discrete Geometry

631.03. Graph Theory
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
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Mathematics 635       Geometry of Numbers
The interplay of the group-theoretic notion of lattice and the geometric concept of convex set, the lattices representing periodicity, the convex sets geometry. Topics include convex bodies and lattice points, the critical determinant, the covering constant and the inhomogeneous determinant of a set, Star bodies, methods related to the above, and homogeneous and inhomogeneous forms.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
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Mathematics 637       Infinite Combinatorics
An excursion into the infinite world, from Ramsey Theory on the natural numbers, to applications in Number Theory and Banach Spaces, introduction to tools in Model Theory and Logic, fascinating homogeneous structures such as the rationals and the Rado graph, and possibly further explorations into the larger infinite world.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
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Mathematics 641       Number Theory

Algebraic Number Theory: an introduction to number fields, rings of integers, ideals, unique factorization, the different and the discriminant. The main objective to the course will be to prove the finiteness of the class number and Dirichlet's Unit Theorem.
Analytic Number Theory: students will learn tools to aid in the study of the average behavior of arithmetic functions, including the use of zeta functions, to prove results about the distribution of prime numbers.

641.01. Algebraic Number Theory

641.03. Analytic Number Theory
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Notes: Mathematics 607 is recommended as preparation for Mathematics 641.01, but not required.  Mathematics 421 or equivalent is recommended as preparation for 641.03.
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Mathematics 643       Computational Number Theory
An investigation of major problems in computational number theory, with emphasis on practical techniques and their computational complexity. Topics include basic integer arithmetic algorithms, finite fields, primality proving, factoring methods, algorithms in algebraic number fields.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Pure Mathematics 427 or 429.
Antirequisite(s): Credit for more than one of Pure Mathematics 527, 627 and Mathematics 643 will not be allowed.
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Mathematics 647       Modular Forms

Modular forms and automorphic representations and their L-functions. Modularity Theorem from two perspectives.
Classical Perspective on Modular Forms: introduction to modular curves as moduli spaces for elliptic curves and as differential forms on modular curves. A study of L-functions attached to modular forms and the modularity theorem.
An Introduction to Autmorphic Representations: introduction to the Langlands Programme. A study of partial L-functions attached to automorphic representations and known instances of the Langlands Correspondence.

647.01. Classical Perspective on Modular Forms

647.03. An Introduction to Automorphic Represenations
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 607.
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Mathematics 651       Topics in Applied Mathematics
Topics will be chosen according to the interest of the instructors and students.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Also known as: (formerly Applied Mathematics 603)
MAY BE REPEATED FOR CREDIT
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Mathematics 653       Topics in Pure Mathematics
Topics will be chosen according to the interest of the instructors and students.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Also known as: (formerly Pure Mathematics 603)
MAY BE REPEATED FOR CREDIT
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Mathematics 661       Scientific Modelling and Computation I

The Convex Optimization: an introduction to modern convex optimization, including basics of convex analysis and duality, linear conic programming, robust optimization, and applications.
Scientific Computation: an introduction to both the methodological and the implementation components underlying the modern scientific computations with the natural emphasis on linear algebra, including modern computing architecture and its implications for the numerical algorithms.
Numerical Differential Equations: fundamentals of solving DEs numerically addressing the existence, stability and efficiency of such methods.

661.01. Convex Optimization

661.03. Scientific Computation

661.05. Numerical Differential Equations
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Notes: Mathematics 603 is recommended as preparation for Mathematics 661.01.
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Mathematics 663       Applied Analysis

Interior Point Methods: exposes students to the modern IPM theory with some applications, to the extent that at the end of the course a student should be able to implement a basic IPM algorithm.
Theoretical Numerical Analysis: provides the theoretical underpinnings for the analysis of modern numerical methods, covering topics such as linear operators on normed spaces, approximation theory, nonlinear equations in Banach spaces, Fourier analysis, Sobolev spaces and weak formulations of elliptic boundary value problems, with applications to finite difference, finite element and wavelet methods.
Differential Equations: essential ideas relating to the analysis of differential equations from a functional analysis point of view. General topics include Hilbert spaces and the Lax-Milgram’s theorem, variational formulation of boundary value problems, finite element methods, Sobolev spaces, distributions, and pseudo-differential operators.

663.01. Interior Point Methods

663.03. Theoretical Numerical Analysis

663.05. Differential Equations


Course Hours: 3 units; H(3-0)
Prerequisite(s): Two of Mathematics 601, 603 and 605.
Notes: Mathematics 601, 603 and 605 are recommended as preparation for this course.  Additionally, Mathematics 661.01 and Mathematics 617 are recommended for Mathematics 663.01.
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Mathematics 667       Introduction to Quantum Information

Focus on the mathematical treatment of a broad range of topics in quantum Shannon theory. Topics include quantum states, quantum channels, quantum measurements, completely positive maps, Neumarkís theorem, Stinespring dilation theorem, Choi-Jamiolkowski isomorphism, the theory of majorization and entanglement, the Peres-Horodecki criterion for separability, Shannon’s noiseless and noisy channel coding theorems, Lieb’s theorem and the strong subadditivity of the von Neumann entropy, Schumacher’s quantum noiseless channel coding theorem, and the Holevo-Schumacher-Westmoreland theorem.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Applied Mathematics 411 or Physics 443.
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Mathematics 669       Scientific Modelling and Computation II

Wavelet Analysis: covers the design and implementation of wavelet methods for modern signal processing, particularly for one- and two-dimensional signals (audio and images).
Mathematical Biology: introduction to discrete models of mathematical biology, including difference equations, models of population dynamics and the like. Topics include stability of models describe by difference equations, continuous spatially homogeneous processes and spatially distributed models.

669.01. Wavelet Analysis

669.03. Mathematical Biology
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 617 is required for Mathematics 669.01.  Consent of the Department for Mathematics 669.03.
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Mathematics 681       Stochastic Calculus for Finance
Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, the Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Itô’s formula, stochastic differential equations, Girsanov’s theorem, the Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Applied Mathematics 481.
Antirequisite(s): Credit for more than one of Mathematics 681, Applied Mathematics 681 and 581 will not be allowed.     
Also known as: (formerly Applied Mathematics 681)
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Mathematics 683       Computational Finance
Basic computational techniques required for expertise quantitative finance. Topics include basic econometric techniques (model calibration), tree-based methods, finite-difference methods, Fourier methods, Monte Carlo simulation and quasi-Monte Carlo methods.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Applied Mathematics 481 and 491.
Antirequisite(s): Credit for more than one of Applied Mathematics 683, 583 and Mathematics 683 will not be allowed.
Notes: Although a brief review of asset price and option valuation models is included, it is recommended that students take Mathematics 681 prior to taking this course.
Also known as: (formerly Applied Mathematics 683)
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Mathematics 685       Stochastic Processes
Stochastic processes are fundamental to the study of mathematical finance, but are also of vital importance in many other areas, from neuroscience to electrical engineering. Topics to be covered: Elements of stochastic processes, Markov chains and processes, Renewal processes, Martingales (discrete and continuous times), Brownian motion, Branching processes, Stationary processes, Diffusion processes, The Feynman-Kac formula, Kolmogorov backward/forward equations, Dynkin’s formula.
Course Hours: 3 units; H(3-0)
Prerequisite(s): Consent of the Department.
Antirequisite(s): Credit for Statistics 761 and Mathematics 685 will not be allowed.
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Mathematics 691       Advanced Mathematical Finance I

Topics include specific areas of mathematical finance and build on Mathematics 681. Lévy Processes (LP): fundamental concepts associated with LP such as infinite divisibility, the Lévy-Khintchine formula, the Lévy-Itô decomposition, subordinators, LP as time-changed Brownian motions, and also dealing with semi-groups and generators of LP, the Itô formula for LP, the Girsanov theorem, stochastic differential equations driven by LP, the Feynman-Kac formula, applications of LP and numerical simulation of LP. Credit Risk: corporate bond markets, modelling the bankruptcy risk of a firm, and understanding how corporate bonds are priced.

691.01. Lévy Processes

691.03. Credit Risk


Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 681.
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Mathematics 693       Advanced Mathematical Finance II

Topics include specific areas of mathematical finance and build on Mathematics 681 and 683. Monte Carlo Methods for Quantitative Finance: random number generation, simulation of stochastic differential equations, option valuation, variance reduction techniques, quasi-Monte Carlo methods, computing ‘greeks', valuation of path-dependent and early-exercise options; applications to risk management; Markov Chain Monte Carlo methods. Energy, Commodity and Environmental Finance: energy and commodity markets; spot, futures, forwards and swap contracts; the theory of storage; stochastic models for energy prices; model calibration; emissions market modelling; weather derivatives; energy risk management; energy option valuation.

693.01 Monte Carlo Methods for Quantitative Finance

693.03 Energy, Commodity and Environmental Finance
Course Hours: 3 units; H(3-0)
Prerequisite(s): Mathematics 681 and 683.
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