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University of Calgary Calendar 2018-2019 COURSES OF INSTRUCTION Course Descriptions M Mathematics MATH
Mathematics MATH

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 and Statistics.
  • Not all Senior courses are offered every year.
  • Effective Fall 2018, all Applied Mathematics and Pure Mathematics courses have been renamed as Mathematics with a change in course number in some cases. Please refer to the descriptions of the individual Mathematics courses for details. Students enrolled in any program that requires any Applied Mathematics or Pure Mathematics course should use the corresponding Mathematics course as replacement.
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 211 and 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, or Mathematics 2 (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 209       Applied and Computational Linear Algebra for Energy Engineers
System of Linear Equations with Applications, Matrices and Matrix Operations, Determinants, Vectors in Two and Three – Space with Geometrical Applications. Emphasis on applications and computing techniques. Students will complete a software-based project.
Course Hours:
3 units; H(3-2)
Prerequisite(s):
Admission to the Energy Engineering Program.
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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(3-1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 30-1. (Alternatives are presented in C. Mathematics Competency Equivalents in the Academic Regulations section of this Calendar).
Antirequisite(s):
Credit for Mathematics 211 and 213 will not be allowed.
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Mathematics 213       Linear Algebra I
Systems of equations and matrices, vector spaces, subspaces, bases and dimension, linear transformations, determinants, eigenvalues and eigenvectors.
Course Hours:
3 units; H(3-1)
Prerequisite(s):
A grade of 80 per cent or higher in Mathematics 30-1.
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(4-1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 30-1. (Alternatives are presented in C. Mathematics Competency Equivalents in the Academic Regulations section of this Calendar).
Antirequisite(s):
Not open to students with 50 per cent or higher in Mathematics 31 or a grade of "C" or higher in Mathematics 3 offered through University of Calgary Continuing Education, except with special departmental permission. Credit for Mathematics 249 and either 265 or 275 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(3-1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 30-1; and a grade of 50 per cent or higher in Mathematics 31 or a grade of "C" or higher in Mathematics 3 offered through University of Calgary Continuing Education. (Alternatives to Mathematics 30-1 are presented in C. Mathematics Competency Equivalents in the Academic Regulations section of this Calendar).
Antirequisite(s):
Credit for Mathematics 265 and either 249 or 275 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(3-1)
Prerequisite(s):
One of Mathematics 249, 265 or 275.
Antirequisite(s):
Credit for Mathematics 267 and 277 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 or 213.
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Mathematics 273       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. (Alternatives are presented in C. Mathematics Competency Equivalents in the Academic Regulations section of this Calendar).
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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 Mathematics 30-1; and credit in Mathematics 31 or Mathematics 3 offered through University of Calgary Continuing Education. Alternatively, admission to the Faculty of Engineering including credit in Mathematics 30-1; and Mathematics 31 or Mathematics 3 offered through University of Calgary Continuing Education.
Antirequisite(s):
Credit for Mathematics 275 and either 249 or 265 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 211 and 275.
Antirequisite(s):
Credit for Mathematics 277 and 267 will not be allowed.
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Senior Courses
Mathematics 305       Inside Mathematics
An exploration of the usually tacit elements of mathematical concepts and processes, the course focuses on strategies for unpacking concepts and for sustained engagement in inquiry.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 211 or 213; and 271 or 273.
Notes:
This course will be co-taught by scholars from the Faculty of Science and Werklund School of Education.
Also known as:
(Education 305)
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Mathematics 307       Complex Analysis I
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):
Mathematics 267.
Antirequisite(s):
Credit for Mathematics 307 and 423 will not be allowed.
Also known as:
(formerly Mathematics 421)
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Mathematics 311       Linear Methods II
Vector spaces and subspaces. Linear independence, matrix representations of linear transformations, Gram-Schmidt orthogonalization.
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.
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Mathematics 313       Linear Algebra II
Diagonalization, canonical forms, inner products, orthogonalization, spectral theory.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 213.
Antirequisite(s):
Credit for Mathematics 311 and 313 will not be allowed.
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Mathematics 315       Algebra I
Basic ring theory: rings and fields, the integers modulo n, Polynomial rings, polynomials over the integers and rationals, homomorphisms, ideals and quotients, principal ideal domains, adjoining the root of an irreducible polynomial; basic group theory: groups, examples including cyclic, symmetric, alternating and dihedral groups, subgroups, cosets and Lagrange’s theorem, normal subgroups and quotients, group homomorphisms, the isomorphism theorems, further topics as time permits, e.g., group actions, Cayley’s theorem.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 315 and Pure Mathematics 317 will not be allowed.
Notes:
(formerly Pure Mathematics 315)
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Mathematics 318       Introduction to Cryptography
The basics of cryptography, with emphasis on attaining well-defined and practical notions of security. Symmetric and public-key cryptosystems; one-way and trapdoor functions; mechanisms for data integrity; digital signatures; key management; applications to the design of cryptographic systems. Assessment will primarily focus on mathematical theory and proof-oriented homework problems; additional application programming exercises will be available for extra credit.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 318 and any of Pure Mathematics 329, Computer Science 418, 429, or 557 will not be allowed.
Also known as:
(formerly Pure Mathematics 418)
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Mathematics 319       Transformation Geometry
Geometric transformations in the Euclidean plane. Frieze patterns. Wallpaper patterns. Tessellations.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 211 or 213; and one other 200-level course from the Field of Mathematics.
Also known as:
(formerly Pure Mathematics 319)
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Mathematics 322       Differential Geometry
Curvature, connections, parallel transport, Gauss-Bonnet theorem.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 271 or 273; Mathematics 367 or 377; Mathematics 375 or 376.
Also known as:
(formerly Pure Mathematics 423)
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Mathematics 325       Introduction to Optimization
Examples of optimization problems. Quadratic forms, minimum energy and distance. Least squares, generalized inverse. Location and classification of critical points. Variational treatment of eigenvalues. Lagrange multipliers. Linear programming.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 311 or 313; one of Mathematics 367, 377 or 331.
Also known as:
(formerly Applied Mathematics 425)
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Mathematics 327       Number Theory
Divisibility and the Euclidean algorithm, modular arithmetic and congruences, quadratic reciprocity, arithmetic functions, distribution of primes.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 271 or 273.
Also known as:
(formerly Pure Mathematics 427)
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Mathematics 331       Advanced Calculus for the Natural Sciences
Linear ordinary differential equations, and systems of ordinary differential equations. Calculus of functions of several variables, double and triple integrals. Introduction to vector analysis, theorems of Green, Gauss and Stokes. Notions of probability and normal distribution. Introduction to the Fourier Transform.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 267 or 277; Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 331 and either 367 or 377 will not be allowed.
Notes:
This course is not part of the Field of Mathematics.
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Mathematics 335       Analysis I
Functions, countable and uncountable sets. The axioms and basic topology of the real numbers. Convergence of sequences, limits of functions, continuity and uniform continuity. Differentiability and the mean value theorem. The Riemann integral and the fundamental theorem of calculus. Series and convergence tests.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 267 or 277; Mathematics 271 or 273.
Antirequisite(s):
Credit for Mathematics 335 and 355 will not be allowed.
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Mathematics 355       Enriched Analysis I
Functions, countable and uncountable sets. The axioms and basic topology of the real numbers. Convergence of sequences, limits of functions, continuity and uniform continuity. Differentiability and the mean value theorem. The Riemann integral and the fundamental theorem of calculus. Series and convergence tests.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 267 or 277; 271 or 273.
Antirequisite(s):
Credit for Mathematics 355 and 335 will not be allowed.
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Mathematics 361       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 or a grade of "B+" or higher in Mathematics 213; Mathematics 267 or 277.
Antirequisite(s):
Credit for Mathematics 361 and 313 will not be allowed.
Also known as:
(formerly Mathematics 411)
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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):
Mathematics 267 or 277; Mathematics 211 or 213.
Antirequisite(s):
Credit for Mathematics 367 and either 331 or 377 will not be allowed.
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Mathematics 371       Combinatorics and Graph Theory
Counting techniques, generating functions, inclusion-exclusion, introduction to graph theory.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 271 or 273; and one of Mathematics 249, 265 or 275.
Also known as:
(formerly Pure Mathematics 471)
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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 277 or both Mathematics 267 and 177.
Antirequisite(s):
Credit for Mathematics 375 and either 376 or Applied Mathematics 311 will not be allowed.
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Mathematics 376       Differential Equations I
Classification of ordinary differential equations, first and second order equations with applications, series solutions about regular points and singular points, special functions, Laplace transform.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 267 or 277.
Antirequisite(s):
Credit for Mathematics 376 and 375 or Applied Mathematics 307 will not be allowed.
Also known as:
(formerly Applied Mathematics 311)
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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’ 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 Mathematics 377 and either 331 or 367 will not be allowed.
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Mathematics 383       Introduction to Mathematical Finance
An introduction to the fundamental concepts of mathematical finance in an elementary setting. Topics include: risk, return, no arbitrage principle; basic financial derivatives: options, forwards and future contracts; risk free assets, time value of money, zero coupon bonds; risky assets, binomial tree model, fundamental theorem of asset pricing; portfolio management and capital asset pricing model; no arbitrage pricing of financial derivatives; hedging.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Statistics 321.
Also known as:
(formerly Applied Mathematics 481)
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Mathematics 391       Numerical Analysis I
Interpolation and approximation, numerical integration and differentiation, numerical methods for the solution of non-linear equations, systems of linear equations and the eigenvalue problem, introduction to a scientific computing software.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
Mathematics 211 or 213; Mathematics 267 or 277; and one of Computer Science 217, 231, 235 or Data Science 211.
Antirequisite(s):
Credit for Mathematics 391 and Computer Science 491 will not be allowed.
Also known as:
(formerly Applied Mathematics 491)
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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.
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; Mathematics 267 or 277 or both Economics 387 and 389.
Notes:
This course is not part of the Field of Mathematics.
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Mathematics 413       Introduction to Partial Differential Equations
First order partial differential equations, Sturm-Liouville systems, Fourier series, Double Fourier series, Fourier integrals, Applications to boundary value problems in bounded and unbounded domains, Bessel function with applications.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
One of Mathematics 375, 376, 331 or Applied Mathematics 311; and Mathematics 331 or 367 or 377
Also known as:
(formerly Applied Mathematics 413)
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Mathematics 415       Mathematical Methods
Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
One of Mathematics 331, 367, 375, 376 or Applied Mathematics 311.
Notes:
This course is not part of the Field of Mathematics.
Also known as:
(formerly Applied Mathematics 415)
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Mathematics 429       Cryptography: Design and Analysis of Cryptosystems
Review of basic algorithms and complexity. Designing and attacking public key cryptosystems based on number theory. Basic techniques for primality testing, factoring and extracting discrete logarithms. Elliptic curve cryptography. Additional topics may include knapsack systems, zero knowledge, attacks on hash functions, identity-based cryptography, and quantum cryptography.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
One of Mathematics 315, Pure Mathematics 315 or 317; one of Mathematics 318, Pure Mathematics 329, 418 or Computer Science 418.
Also known as:
(formerly Pure Mathematics 429)
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Mathematics 431       Algebra II
Group theory: Sylow theorems, solvable, nilpotent and p-groups, simplicity of alternating groups and PSL(n,q), structure theory of finite abelian groups; field theory: fields, algebraic and transcendental extensions, separability and normality, Galois theory, insolvability of the general quintic equation, computation of Galois groups over the rationals.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 311 or 313; one of Mathematics 315, Pure Mathematics 315 or 317.
Also known as:
(formerly Pure Mathematics 431)
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Mathematics 433       Mathematical Methods in Physics
Fourier analysis. Laplace Transforms. Partial differential equations. Complex analysis. Residue integrals. Extensive physical applications.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
One of Mathematics 375, 376 or Applied Mathematics 311; Mathematics 367 or 377; Mathematics 211 or 213.
Notes:
This course is not part of the Field of Mathematics.
Also known as:
(formerly Applied Mathematics 433)
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Mathematics 445       Analysis II
Basic topology of metric spaces and normed vector spaces. Complete metric spaces and the Baire category theorem. Continuous functions on compact metric spaces and uniform convergence. The contraction mapping principle and applications. The Stone-Weierstrass and Arzelàa-Ascoli theorems. Differentiability on Euclidean spaces. The implicit and inverse function theorems.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 367 or 377; one of Mathematics 311, 313, 351 or 411; Mathematics 335 or 355.
Antirequisite(s):
Credit for Mathematics 445 and 447 will not be allowed.            
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Mathematics 447       Enriched Analysis II
Basic topology of metric spaces and normed vector spaces. Complete metric spaces and the Baire category theorem. Continuous functions on compact metric spaces and uniform convergence. The contraction mapping principle and applications. The Stone-Weierstrass and Arzelà-Ascoli theorems. Differentiability on Euclidean spaces. The implicit and inverse function theorems.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 367 or 377; Mathematics 313 or "B+" or higher in Mathematics 311; Mathematics 355 or "B+" or higher in Mathematics 335.
Antirequisite(s):
Credit for Mathematics 447 and either Mathematics 445 or Pure Mathematics 545 will not be allowed.
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Mathematics 476       Differential Equations II
Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
One of Mathematics 375, 376 or Applied Mathematics 311; one of Mathematics 367, 377 or 331; Mathematics 335 or 355.
Also known as:
(formerly Applied Mathematics 411)
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Mathematics 493       Numerical Analysis II
Numerical solution of ordinary differential equations, single and multi-step methods, numerical solution of boundary value problems, numerical solution of partial differential equations, stability analysis.
Course Hours:
3 units; H(3-1T)
Prerequisite(s):
One of Mathematics 331, 375, 376 or Applied Mathematics 311; and one of Mathematics 413, 375 or Applied Mathematics 413; and one of Mathematics 391, Applied Mathematics 491, Computer Science 491 or Geophysics 419.
Also known as:
(formerly Applied Mathematics 493)
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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 445 or 447; and 3 units of Mathematics in the Field of Mathematics at the 400 level or higher.
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 502       Topics in Mathematics
Topics will be chosen according to the interests of instructors and students.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Any two Mathematics courses in the Field of Mathematics at the 400 level or above.
Also known as:
(formerly Pure Mathematics 503)
MAY BE REPEATED FOR CREDIT
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Mathematics 503       The Mathematics of Wavelets, Signal and Image Processing
Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
One of Mathematics 391, Applied Mathematics 491, Computer Science 491 or Geophysics 419; 6 units of Mathematics in the Field of Mathematics at the 400 level or above.
Also known as:
(formerly Applied Mathematics 503)
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Mathematics 505       Calculus on Manifolds
Integral and differential calculus on manifolds including tensor fields, covariant differentiation, Lie differentiation, differential forms, Frobenius' theorem, Stokes' theorem, flows of vector fields.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 445 or 447; and one of Mathematics 375, 376 or Applied Mathematics 311.
Also known as:
(formerly Applied Mathematics 505)
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Mathematics 511       Algebra III
Linear algebra: Modules, direct sums and free modules, tensor products, linear algebra over modules, finitely generated modules over PIDs, canonical forms, computing invariant factors from presentations; projective, injective and flat modules.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
One of Mathematics 313, 351, 411, 431 or Pure Mathematics 431; 3 units of Mathematics in the Field of Mathematics at the 400 level or higher.
Antirequisite(s):
Credit for Mathematics 511 and either 607 or Pure Mathematics 611 will not be allowed.
Also known as:
(formerly Pure Mathematics 511)
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Mathematics 515       Foundations
Set theory, mathematical logic, and category theory. Topics covered will vary based on interests of students and instructor.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 431 or Pure Mathematics 431; 3 units of Mathematics from the Field of Mathematics at the 400 level or above.
Also known as:
(formerly Pure Mathematics 415)
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Mathematics 516       Senior Project
A capstone course intended for Mathematics students in the final year of study (excluding those in the honours program or the statistics concentration). Students will investigate scientific or social issues by applying mathematical methods acquired in previous mathematics courses. A final project will be submitted at the end of the term and its contents summarized in a presentation.
Course Hours:
3 units; H(1.5-3)
Prerequisite(s):
6 units in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 516 and 518 will not be allowed.
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Mathematics 518       Honours Thesis
A capstone course intended for Honours Mathematics students in their final year of study. Students will produce and present a substantial thesis under the supervision of faculty. The emphasis is on how to address theoretical or real world scientific or social issues by applying the various mathematical methods acquired in the earlier years in a unified and appropriate way.
Course Hours:
3 units; H(1.5-3)
Prerequisite(s):
6 units in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 518 and 516 will not be allowed.
Notes:
This course extends over the Fall and Winter Terms. Students will meet regularly with their thesis supervisors during the terms. Students submit a thesis, and the course culminates in a series of student presentations. A grade of "B" or higher is required for the Honours program. Students are advised to consult with the Undergraduate Director for information and advice before registration into the course. Students earning an Honours degree in Mathematics along with a concentration in Statistics must complete both Mathematics 518 and Statistics 517.
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Mathematics 521       Complex Analysis II
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):
One of Mathematics 307, 421 or 423; Mathematics 445 or 447; 3 units of Mathematics in the Field of Mathematics at the 400 level or above.
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Mathematics 525       Introduction to Algebraic Topology
An introduction to the algebraic invariants that distinguish topological spaces. Specifically, the course focuses on the fundamental group and its applications, and homology. Students will be introduced to the basics of homological algebra.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 431 or Pure Mathematics 445; Mathematics 445 or 447.
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Mathematics 527       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):
Mathematics 431 or Pure Mathematics 431; Mathematics 429 or Pure Mathematics 429 or Mathematics 327 or Pure Mathematics 427.
Antirequisite(s):
Credit for Mathematics 527 and either Pure Mathematics 527 or 643 will not be allowed.
Also known as:
(formerly Pure Mathematics 527)
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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 445 or 447; 3 units of Mathematics in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 545 and 603 will not be allowed.
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Mathematics 581       Stochastic Calculus for Finance
Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanov’s theorem, 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):
Mathematics 383 or Applied Mathematics 481; 6 units of Mathematics in the Field of Mathematics at the 400 level or above.
Antirequisite(s):
Credit for Mathematics 581 and Applied Mathematics 681 will not be allowed.
Also known as:
(formerly Applied Mathematics 581)
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Mathematics 583       Computational Finance
Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Mathematics 381 or Applied Mathematics 481; Mathematics 413 or Applied Mathematics 413; Mathematics 493 or Applied Mathematics 493.
Antirequisite(s):
Credit for Mathematics 583 and Applied Mathematics 683 will not be allowed.
Also known as:
(formerly Applied Mathematics 583)
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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)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 601 and either Mathematics 501 or Pure Mathematics 501 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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 607 and any of Pure Mathematics 511, 607 or 611 will not be allowed.   
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Mathematics 617       Functional Analysis
Introduction to Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum.
Course Hours:
3 units; H(3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 617 and Applied Mathematics 617 will not be allowed.      
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 621 and 521 will not be allowed.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 625 and either Mathematics 525 or Pure Mathematics 607 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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or 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):
Admission to a graduate program in Mathematics and Statistics or 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):
Admission to a graduate program in Mathematics and Statistics or 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 behaviour 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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 643 and any of Mathematics 527, Pure Mathematics 527, or Pure Mathematics 627 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 Automorphic 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 Representations


Course Hours:
3 units; H(3-0)
Prerequisite(s):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or 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):
Admission to a graduate program in Mathematics and Statistics or 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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 681 and any one of Mathematics 581, Applied Mathematics 681, or Applied Mathematics 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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 683 and any one of Mathematics 583, Applied Mathematics 683, or Applied Mathematics 583 will not be allowed.    
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):
Admission to a graduate program in Mathematics and Statistics or consent of the Department.
Antirequisite(s):
Credit for Mathematics 685 and Statistics 761  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 and admission to a graduate program in Mathematics and Statistics or consent of the Department.
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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 and admission to a graduate program in Mathematics and Statistics or consent of the Department.
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