
Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.
Note: For listings of related courses, see Actuarial Science, Applied Mathematics, Pure Mathematics, and Statistics.

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(31)
Prerequisite(s):
Mathematics 301, Mathematics 302, 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
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(32)
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(31T1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 301 or Pure Mathematics 30. (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

Honours 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(31T1)
Prerequisite(s):
A grade of 80 per cent or higher in Mathematics 301 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(41T1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 301 or Pure Mathematics 30. (Alternatives are presented in C. Mathematics Competency Equivalents 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 University of Calgary 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(31T1)
Prerequisite(s):
A grade of 70 per cent or higher in Mathematics 301 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 University of Calgary Continuing Education. (Alternatives to Pure Mathematics 30 are presented in C. Mathematics Competency Equivalents 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(31T1)
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(31T1)
Prerequisite(s):
Mathematics 211 or 213.

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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(31T1)
Prerequisite(s):
A grade of 80 per cent or higher in Mathematics 301 or Pure Mathematics 30. (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(31T1.5)
Prerequisite(s):
A grade of 70 per cent or higher in Pure Mathematics 30 or Mathematics 301; 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 either Pure Mathematics 30 or Mathematics 301; and Mathematics 31 or Mathematics 3 offered through University of Calgary 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; Vectorvalued functions; derivatives and integrals; parametric curves.
Course Hours:
3 units; H(31T1.5)
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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Mathematics
305

Inside Mathematics


An exploration of the usuallytacit 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(31T)
Prerequisite(s):
Mathematics 211 or 213; and 271 or 273.
Notes:
This course will be cotaught by scholars from the Faculty of Science and Werklund School of Education.
Also known as:
(Education 305)

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Mathematics
311

Linear Methods II


Vector spaces and subspaces. Linear independence. Matrix representations of linear transformations. GramSchmidt orthogonalization. Students will complete a project using a computer algebra system.
Course Hours:
3 units; H(31T)
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(31T)
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

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(31T)
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


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(31T)
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


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(31T)
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(31T)
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(31.5T)
Prerequisite(s):
Mathematics 211; and one of Applied Mathematics 219 or Mathematics 277, or both Mathematics 267 and 177, or both Mathematics 253 and 114.
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’ theorem; applications; curvilinear coordinates; Laplace, diffusion and wave equations in three dimensional space.
Course Hours:
3 units; H(31.5T)
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(30)
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 hemicontinuous correspondences. Fixed point theorems, Optimal control.
Course Hours:
3 units; H(30)
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(31T)
Prerequisite(s):
Mathematics 311; and one of Mathematics 331, 353, 367, 377, 381 or Applied Mathematics 309.
Antirequisite(s):
Credit for Mathematics 411 and 313 will not be allowed.

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Mathematics
421

Complex Analysis I


Basic complex analysis – complex numbers and functions, differentiation, CauchyRiemann 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(31T)
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, CauchyRiemann 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(31T)
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 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 StoneWeierstrass and ArzelàaAscoli theorems. Differentiability on Euclidean spaces. The implicit and inverse function theorems.
Course Hours:
3 units; H(30)
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 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 StoneWeierstrass and ArzelàArzelaAscoli theorems. Differentiability on Euclidean spaces. The implicit and inverse function theorems.
Course Hours:
3 units; H(30)
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, RadonNikodym theorem, Lp Spaces, Riesz representation theorems.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 445 or 447.
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


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 selfmaps of the unit disk.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 335 or 355 or Pure Mathematics 435 or 455; Mathematics 421 or 423; or consent of the Department.
Antirequisite(s):
Credit for Mathematics 521 and Pure Mathematics 521 will not be allowed.

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Mathematics
525

Introduction to Algebraic Topology


An introduction to the algebraic invariants that distinguish topological spaces and to the basis of homological algebra. Also, a focus on the fundamental group and its applications, and homotopy.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 445 and Pure Mathematics 431.
Antirequisite(s):
Credit for Mathematics 525 and either 625 and Pure Mathematics 607 is not 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(30)
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 Department is a prerequisite for all 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(3S0)
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, RadonNikodym theorem, Lp spaces, Riesz representation theorem.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 445 or 447.
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(30)
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(30)
Prerequisite(s):
Applied Mathematics 411 and Mathematics 445 or 447.
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; homtensor 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(30)
Prerequisite(s):
Pure Mathematics 431 or Mathematics 411.
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 some basic aspects of Functional Analysis, Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 545 or 603.
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 selfmaps of the unit disk.
Course Hours:
3 units; H(30)
Prerequisite(s):
Mathematics 335 or 355 or Pure Mathematics 435 or 455; and Mathematics 421 or 423; 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(30)
Prerequisite(s):
Pure Mathematics 431 and Mathematics 445 or 447.
Antirequisite(s):
Credit for Mathematics 625 and 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(30)
Prerequisite(s):
Mathematics 607 or Pure Mathematics 511 or 611.

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Mathematics
631

Discrete Mathematics


Discrete Geometry: Euclidean, spherical and hyperbolic nspaces, 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(30)
Prerequisite(s):
Consent of the Department.

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Mathematics
635

Geometry of Numbers


The interplay of the grouptheoretic 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(30)
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(30)
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 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(30)
Prerequisite(s):
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(30)
Prerequisite(s):
Pure Mathematics 427 or 429.
Antirequisite(s):
Credit for Mathematics 643 and either Pure Mathematics 527 or 627 will not be allowed.

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Mathematics
647

Modular Forms


Modular forms and automorphic representations and their Lfunctions. 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 Lfunctions attached to modular forms and the modularity theorem. An Introduction to Automorphic Representations: introduction to the Langlands Programme. A study of partial Lfunctions 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(30)
Prerequisite(s):
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(30)
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(30)
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(30)
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 LaxMilgram’s theorem, variational formulation of boundary value problems, finite element methods, Sobolev spaces, distributions, and pseudodifferential operators.
663.01. Interior Point Methods
663.03. Theoretical Numerical Analysis
663.05. Differential Equations
Course Hours:
3 units; H(30)
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, ChoiJamiolkowski isomorphism, the theory of majorization and entanglement, the PeresHorodecki 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 HolevoSchumacherWestmoreland theorem.
Course Hours:
3 units; H(30)
Prerequisite(s):
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 twodimensional 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(30)
Prerequisite(s):
Mathematics 617 is required for Mathematics 669.01.

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Mathematics
681

Stochastic Calculus for Finance


Martingales in discrete and continuous time, riskneutral valuations, discrete and continuoustime (B,S)security markets, the CoxRossRubinstein formula, Wiener and Poisson processes, Itô’s formula, stochastic differential equations, Girsanov’s theorem, the BlackScholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, valueatrisk and risk management.
Course Hours:
3 units; H(30)
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), treebased methods, finitedifference methods, Fourier methods, Monte Carlo simulation and quasiMonte Carlo methods.
Course Hours:
3 units; H(30)
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 FeynmanKac formula, Kolmogorov backward/forward equations, Dynkin’s formula.
Course Hours:
3 units; H(30)
Prerequisite(s):
Statistics 321; one of Mathematics 331, 335, 355 or 367; and one of Mathematics 311 or 313.
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 LévyKhintchine formula, the LévyItô decomposition, subordinators, LP as timechanged Brownian motions, and also dealing with semigroups and generators of LP, the Itô formula for LP, the Girsanov theorem, stochastic differential equations driven by LP, the FeynmanKac 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(30)
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, quasiMonte Carlo methods, computing ‘greeks', valuation of pathdependent and earlyexercise 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(30)
Prerequisite(s):
Mathematics 681 and 683.

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