## MATH - Mathematics

Offered By:

Faculty of Science

Below are the courses available from the MATH code. Select a course to view the available classes, additional class notes, and class times.

#### MATH 100 - Calculus I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-1)**

Review of numbers, inequalities, functions, analytic geometry; limits, continuity; derivatives and applications, Taylor polynomials; log, exp, and inverse trig functions. Integration, fundamental theorem of calculus substitution, trapezoidal and Simpson's rules. Prerequisites: Mathematics 30-1 and Mathematics 31. Notes: (1) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154, or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

#### MATH 101 - Calculus II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-1)**

Area between curves, techniques of integration. Applications of integration to planar areas and lengths, volumes and masses. First order ordinary differential equations: separable, linear, direction fields, Euler's method, applications. Infinite series, power series, Taylor expansions with remainder terms. Polar coordinates. Rectangular, spherical and cylindrical coordinates in 3-dimensional space. Parametric curves in the plane and space: graphing, arc length, curvature; normal binormal, tangent plane in 3- dimensional space. Volumes and surface areas of rotation. Prerequisite: MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

#### MATH 102 - Applied Linear Algebra View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-1)**

Vectors and matrices, solution of linear equations, equations of lines and planes, determinants, matrix algebra, orthogonality and applications (Gram-Schmidt), eigenvalues and eigenvectors and applications, complex numbers. Prerequisite or corequisite: MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 102, 125, or 127. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

#### MATH 111 - Introduction to Discrete Mathematics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

A problem-solving approach to discrete mathematics, covering secret codes, public-key codes, error-correcting codes, enumeration, recurrence relations, induction, graph theory, graph algorithms and parallel algorithms. Prerequisite: MATH 30-1, 30-2, or equivalent. Note : Credit can only be obtained in at most one of MATH 111 or MATH 222.

#### MATH 113 - Elementary Calculus I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

#### MATH 114 - Elementary Calculus I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

#### MATH 115 - Elementary Calculus II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Inverse trigonometric functions. Techniques of integration. Improper integrals. Applications of the definite integral. Introduction to differential equations. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

#### MATH 117 - Honors Calculus I View Available Classes

**★ 3 (fi 6)(FIRST, 4-0-0)**

Functions, continuity, and the derivative. Applications of the derivative. Extended limits and L'Hospital's rule. Prerequisites: Mathematics 30-1 and Mathematics 31, or consent of the Department. Notes: (1) This course is designed for students with at least 80 percent in Pure Mathematics 30 or Mathematics 30-1 and Mathematics 31. (2) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100. (3) Engineering students will receive a weight of 4.0 units for this course.

#### MATH 118 - Honors Calculus II View Available Classes

**★ 3 (fi 6)(SECOND, 4-0-0)**

Integration and the Fundamental Theorem. Techniques and applications of integration. Derivatives and integrals of the exponential, and trigonometric functions. Introduction to infinite series. Introduction to partial derivatives. Prerequisite: MATH 117. (Students with a 100-level calculus course different from MATH 117 may be admitted with consent of the Department.) Notes: (1) Credit can be obtained in at most one MATH 101, 115, 118, 136, 146, 156 or SCI 100. (2) Engineering students will receive a weight of 4.0 units for this course.

#### MATH 125 - Linear Algebra I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Systems of linear equations. Vectors in n-space, vector equations of lines and planes. Matrix algebra, inverses and invertibility. Introduction to linear transformations. Subspaces of n-space. Determinants. Introduction to eigenvalues and eigenvectors. Complex numbers. Dot product, cross product and orthogonality. Applications in a variety of fields. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 102, 125 or 127.

#### MATH 127 - Honors Linear Algebra I View Available Classes

**★ 3 (fi 6)(FIRST, 4-0-0)**

Systems of linear equations; vectors in Euclidean n-space; span and linear independence in Euclidean n-space; dot and cross product; orthogonality; lines and planes; matrix arithmetic; determinants; introduction to eigenvectors and eigenvalues; introduction to linear transformations; complex numbers; vector space axioms; subspaces and quotients. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of Math 102, 125 or 127.

#### MATH 134 - Calculus for the Life Sciences I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Applications in the context of the life sciences. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

#### MATH 136 - Calculus for the Life Sciences II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Techniques and applications of integration. Improper integrals. Differential equations and mathematical modelling. Partial differentiation. Applications in the context of the life sciences. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

#### MATH 144 - Calculus for the Physical Sciences I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Taylor polynomials. Applications in the context of the physical sciences. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

#### MATH 146 - Calculus for the Physical Sciences II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Techniques and applications of integration. Improper integrals. Introduction to differential equations. Partial differentiation. Applications in the context of the physical sciences. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

#### MATH 153 - Elementary Theory of Interest View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Simple and compound interest, simple and general annuities certain; variable annuities and perpetuities, amortization schedules, sinking funds, applications. Prerequisite: Mathematics 30-1. Notes: (1) This course may not be taken for credit if credit has already been obtained in MATH 253. (2) May not be used for credit in any Honors or Specialization degrees offered by the Department of Mathematical and Statistical Sciences.

#### MATH 154 - Calculus for Business and Economics I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Optimization. Applications in the context of business and economics. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100.

#### MATH 156 - Calculus for Business and Economics II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Techniques and applications of integration. Improper integrals. Partial differentiation. Multivariate optimization. Probability and calculus. Applications in the context of business and economics. Prerequisite: One of MATH 100, 113, 114, 117, 134, 144 or 154. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 136, 146, 156 or SCI 100.

#### MATH 160 - Higher Arithmetic View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Elementary Number Theory, Numeration Systems, Number Systems and Elementary Probability Theory. Math Fair. Prerequisite: Mathematics 30-1 or 30-2, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

#### MATH 201 - Differential Equations View Available Classes

**★ 3 (fi 6)(EITH/SP/SU, 3-0-1)**

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 201, 334 or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

#### MATH 201A - Differential Equations View Available Classes

**★ 1.5 (fi 6)(EITH/SP/SU, 3-0-1)**

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) This course may not be taken for credit if credit has already been obtained in any of MATH 205, 334, or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

#### MATH 201B - Differential Equations View Available Classes

**★ 1.5 (fi 6)(EITH/SP/SU, 3-0-1)**

First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) This course may not be taken for credit if credit has already been obtained in any of MATH 205, 334, or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

#### MATH 209 - Calculus III View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-1)**

Partial differentiation, derivatives of integrals. Multiple integration using rectangular, cylindrical, and spherical coordinates. Vector Field Theory. Prerequisite: MATH 101. Prerequisite or corequisite: MATH 102. Notes: (1) This course may not be taken for credit if credit has already been obtained in MATH 215 or 317. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

#### MATH 214 - Intermediate Calculus I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Infinite Series. Plane curves and polar coordinates. Three dimensional analytic geometry. Partial derivatives. Prerequisite: One of MATH 101, 115, 136, 146 or 156, and one of MATH 102, 125 or 127. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 217.

**Starting: 2022-09-01**

#### MATH 214 - Intermediate Calculus III

**★ 3 (fi 6)(EITHER, 3-0-0)**

Sequences and series, convergence tests, and Taylor series. Curves, tangent vectors, and arc length. Applications of partial differentiation. Polar, cylindrical, and spherical coordinates. Multiple integration. Prerequisite: One of MATH 101, 115, 136, 146 or 156. One of MATH 102, 125 or 127 recommended. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 217.

#### MATH 215 - Intermediate Calculus II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

First order and second order linear differential equations with constant coefficients. Curves, tangent vectors, arc length, integration in two and three dimensions, polar cylindrical and spherical coordinates, line and surface integrals. Green's divergence and Stokes' theorems. Prerequisite: MATH 214 or 217. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 317.

**Starting: 2022-09-01**

#### MATH 215 - Intermediate Calculus IV

**★ 3 (fi 6)(EITHER, 3-0-0)**

Vector calculus. Line and surface integrals. The divergence, Green's, and Stokes' theorems. Differential forms. Prerequisite: One of MATH 102, 125 or 127, and either MATH 214 or MATH 217. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 317.

#### MATH 216 - Introduction to Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Sets and functions. Induction. Axiomatic introduction of the real numbers. Sequences and series. Continuity and properties of continuous functions. Differentiation. Riemann integral. Corequisite: One of MATH 101, 115, 136, 146, 156 or SCI 100. Note: This course may not be taken for credit if credit has already been obtained in MATH 118. Credit can only be obtained in one of MATH 216 and MATH 314.

#### MATH 217 - Honors Advanced Calculus I View Available Classes

**★ 3 (fi 6)(FIRST, 4-0-0)**

Axiomatic development of the real number system. Topology of Rn. Sequences, limits and continuity. Multi-variable calculus: differentiation and integration, including integration in spherical and polar coordinates. The differential and the chain rule. Taylor's Formula, maxima and minima. Introduction to vector field theory. Prerequisites: One of MATH 102, 125 or 127, and either MATH 118 or MATH 216. Notes: (1) MATH 216 may be accepted as corequisite with consent of the Department. (2) Engineering students will receive a weight of 4.0 units for this course.

#### MATH 225 - Linear Algebra II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Vector spaces. Inner product spaces. Examples of n-space and the space of continuous functions. Gram-Schmidt process, QR factorization of a matrix and least squares. Linear transformations, change of basis, similarity and diagonalization. Orthogonal diagonalization, quadratic forms. Applications in a variety of fields. Prerequisites: One of MATH 100, 113, 114, 117, 134, 144, 154 or SCI 100, and one of MATH 102, 125 or 127. Note: Credit can be obtained in at most one of MATH 225 or 227.

#### MATH 227 - Honors Linear Algebra II View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Review of vector space axioms, subspaces and quotients; span; linear independence; Gram-Schmidt process; projections; methods of least squares; linear transformations and their matrix representations with respect to arbitrary bases; change of basis; eigenvectors and eigenvalues; triangularization and diagonalization; canonical forms (Schur, Jordan, spectral theorem). Prerequisite: MATH 127. (Students with MATH 102 or 125 may be admitted with consent of the Department.) Note: Credit can be obtained in at most one of MATH 225 or 227.

#### MATH 228 - Algebra: Introduction to Ring Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Integers. Mathematical induction. Equivalence relations. Commutative rings, including the integers mod n, complex numbers and polynomials. The Chinese remainder theorem. Fields and integral domains. Euclidean domains, principal ideal domains and unique factorization. Quotient rings and homomorphisms. Construction of finite fields. Applications such as public domain encryption, Latin squares and designs, polynomial error detecting codes, and/or addition and multiplication of large integers. Prerequisite: MATH 102, 125 or 127. Note: This course may not be taken for credit if credit has already been obtained in MATH 326.

#### MATH 241 - Geometry View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Basic Euclidean geometry, congruence, parallelism, area, and similarity. Sound axiomatic development with emphasis on problem solving. Constructions and loci, inequalities, maxima and minima, circles, isometries, and additional topics. Prerequisite: Any 100-level MATH course or SCI 100.

#### MATH 243 - Transformation Geometry View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Transformation geometry, isometry and homothety, applications in Euclidean geometry; the algebra of transformations, the Classification Theorem, frieze patterns and wall-paper groups. Prerequisite: MATH 241

#### MATH 253 - Theory of Interest View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Accumulation and amount functions, effective, nominal, simple, and compound rates, force of interest and discount, simple and general annuities certain, variable annuities and perpetuities, amortization schedules and sinking funds, bonds and other securities, applications, installment loans, depreciation, depletion, capitalized cost. Prerequisite: One of MATH 101, 115, 118, 136, 146, 156 or SCI 100. Corequisite: MATH 209 or 214.

#### MATH 260 - Mathematical Reasoning for Teachers View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Reasoning and problem solving in the context of logic, algebra, geometry, and combinatorics. Prerequisite: MATH 160, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

#### MATH 298 - Problem Solving Seminar View Available Classes

**★ 1 (fi 2)(EITHER, 0-1S-0)**

Problem solving techniques (pigeonhole principle, invariants, extremal principle, etc.) and survey of problems from various branches of mathematics: calculus, number theory, algebra, combinatorics, probability, geometry, etc. This credit/no-credit course is intended for students interested in mathematics contests and participation in the Putnam Mathematical Competition will be required. Note: This course may be taken for credit up to four times. Prerequisite: consent of the instructor.

#### MATH 300 - Advanced Boundary Value Problems I View Available Classes

**★ 3 (fi 6)(EITH/SP/SU, 3-0-0)**

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

#### MATH 300A - Advanced Boundary Value Problems I View Available Classes

**★ 1.5 (fi 6)(EITH/SP/SU, 3-0-0)**

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

#### MATH 300B - Advanced Boundary Value Problems I View Available Classes

**★ 1.5 (fi 6)(EITH/SP/SU, 3-0-0)**

Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) Credit can be obtained in at most one of MATH 300 and 337. (3) Course cannot be taken for credit if credit has been obtained in ECE 341.

#### MATH 309 - Mathematical Methods for Electrical Engineers View Available Classes

**★ 3 (fi 6)(1 TRM S/S, 3-0-0)**

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

#### MATH 309A - Mathematical Methods for Electrical Engineers View Available Classes

**★ 1.5 (fi 6)(1 TRM S/S, 3-0-0)**

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

#### MATH 309B - Mathematical Methods for Electrical Engineers View Available Classes

**★ 1.5 (fi 6)(1 TRM S/S, 3-0-0)**

Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

#### MATH 311 - Theory of Functions of a Complex Variable View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Complex numbers. Complex series. Functions of a complex variable. Cauchy's theorem and contour integration. Residue Theorem and its applications. Prerequisite or corequisite: A Calculus IV course.

#### MATH 314 - Analysis I View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Construction of real numbers, Heine-Borel and related theorems, differentiation and Riemann integral of functions, topological concepts in metric spaces, sequences, continuous maps, contraction maps, and applications. Prerequisite: MATH 209 or 215 or equivalent. Note: This course may not be taken for credit if credit has already been obtained in MATH 217.

#### MATH 317 - Honors Advanced Calculus II View Available Classes

**★ 3 (fi 6)(SECOND, 4-0-0)**

Implicit function theorem. Transformations of multiple integrals. Line integrals, theorems of Green, Gauss and Stokes. Sequences and series of functions. Uniform convergence. Prerequisite: MATH 217.

#### MATH 322 - Graph Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Graphs, paths and cycles, trees, planarity and duality, coloring problems, digraphs, matching problems, matroid theory. Prerequisite: One of MATH 102, 125 or 127, and any 200-level MATH course. (MATH 216 or MATH 228 recommended.)

#### MATH 324 - Elementary Number Theory View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Divisibility, prime numbers, congruences, quadratic residues, quadratic reciprocity, arithmetic functions and diophantine equations; sums of squares. Prerequisites: MATH 227 or 228.

#### MATH 325 - Linear Algebra III View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Hermitian and unitary matrices, spectral theorem. Jordan canonical form. Cayley-Hamilton Theorem. Bilinear forms, positive-definiteness, Sylvester's Law of inertia, geometric lattices. Numerical methods. Application to discrete system evolution, matrix exponentials and differential equations. Prerequisite: MATH 225. Note: This course may not be taken for credit if credit has already been obtained in MATH 227.

#### MATH 326 - Rings and Modules View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Rings, fields, polynomials, algebras. Homomorphisms, ideals, quotients. Ring extensions, field extensions, construction of finite fields. Integral domains: Euclidean, principal ideal, unique factorization. Chain conditions. Introduction to modules. Modules over a principal ideal domain, finitely generated abelian groups, matrix canonical forms. Prerequisite MATH 227, or both MATH 225 and 228.

#### MATH 328 - Algebra: Introduction to Group Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Groups, subgroups, homomorphisms. Symmetry groups. Matrix groups. Permutations, symmetric group, Cayley's Theorem. Group actions. Cosets and Lagrange's Theorem. Normal subgroups, quotient groups, isomorphism theorems. Direct and semidirect products. Finite Abelian groups. Prerequisite: MATH 227 or 228. This course may not be taken for credit if credit has already been obtained in MATH 229.

#### MATH 334 - Introduction to Differential Equations View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

First order equations, linear equations of higher order. Power series solution. Laplace transform methods. Introduction to special functions. Introduction to linear systems. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Note: Credit can be obtained in at most one of MATH 201, 334 or 336.

#### MATH 336 - Honors Ordinary Differential Equations View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

First order differential equations. Linear systems of differential equations and linear differential equations of higher order. Stability and qualitative theory of 2-dimensional linear and non-linear systems. Laplace transform methods. Existences and uniqueness theorems. Prerequisites: MATH 225 or 227, and either MATH 209, 217, 314 or both 214 and 216. Note: Credit can be obtained in at most one of MATH 201, 334 and 336.

#### MATH 337 - Introduction to Partial Differential Equations View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Boundary value problems of classical Math Physics, orthogonal expansions, classical special functions. Advanced transform techniques. Prerequisites: One of MATH 209, 215, or 217, and one of MATH 201, 334 or 336. Notes: (1) Credit can be obtained in at most one of MATH 300 or 337. (2) Course cannot be taken for credit if credit has been obtained in ECE 341.

#### MATH 341 - Geometry of Convex Sets View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Combinatorial geometry and topology, convex sets, sets with constant width, Helly-type problems, extremal problems. Prerequisite: One of MATH 102, 125, 127, 222 or 241.

#### MATH 343 - Projective and Inversive Geometries View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Projective geometry, Poncelet-Steiner constructions, inverse geometry, Mohr-Mascheroni constructions, Principle of Duality, conic sections. Prerequisite: MATH 241.

#### MATH 347 - Set Theory View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Axioms for set theory, transfinite induction, cardinal and ordinal numbers, applications. Primarily intended for students enrolled in an honors or specialization program in mathematics. Prerequisite: One of MATH 215, 217, 228, 328.

#### MATH 348 - Differential Geometry of Curves and Surfaces View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Frenet-Seret theory of curves in the plane and in 3-space, examples; local theory of surfaces in 3-space: first and second fundamental forms, Gauss map and Gauss curvature, geodesics and parallel transport, theorema egregium, mean curvature and minimal surfaces. Prerequisites: One of MATH 102, 125 or 127 and one of MATH 209, 215 or 217.

#### MATH 356 - Introduction to Mathematical Finance I View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Simple Market Model: one-step binomial model, basic notions and assumptions. Risk-Free Assets: simple interest, zero-coupon bonds, money market account. Risky Assets: dynamic of stock prices, binomial tree model, trinomial tree model. Discrete time market model: stock and money market model, extended models. Portfolio management: risk, two securities, capital asset pricing model. Prerequisite: MATH 253 and STAT 265 or consent of the Department.

#### MATH 357 - Introduction to Mathematical Finance II View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Forward and futures contracts: forward and futures prices, hedging with futures. Options: put-call parity, bounds on option prices, time value of options. Option pricing: European and American options in the binomial tree model, Black-Scholes formula. Financial engineering: hedging option positions, hedging business risk. Variable interest rates: maturity-independent yields, general term structure. Stochastic interest rates: arbitrage pricing of bonds, interest rate derivative securities. Prerequisite: MATH 356 or consent of the Department.

#### MATH 371 - Mathematical Modelling in the Life Sciences View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Model development, computation, and analysis for problems in the life sciences. Models include differential equations, difference equations and stochastic formulations. Model evaluation and prediction. Applications are chosen from epidemiology, ecology, population biology, physiology and medicine. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Note: No previous computing experience is needed.

#### MATH 372 - Mathematical Modelling I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

This course is designed to develop the students' problem-solving abilities along heuristic lines and to illustrate the processes of Applied Mathematics. Students will be encouraged to recognize and formulate problems in mathematical terms, solve the resulting mathematical problems and interpret the solution in real world terms. Typical problems considered include nonlinear programming, optimization problems, diffusion models. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217.

#### MATH 373 - Mathematical Programming and Optimization I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Introduction to optimization. Problem formulation. Linear programming. The simplex method and its variants (revised Simplex method, dual simplex method). Extreme points of polyhedral sets. Theory of linear inequalities (Farkas Lemma). Complementary slackness and duality. Post-optimality analysis. Interior point methods. Applications (elementary games, transportation problems, networks, etc.). Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217.

#### MATH 381 - Numerical Methods I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-1)**

Approximation of functions by Taylor series, Newton's formulae, Lagrange and Hermite interpolation. Splines. Orthogonal polynomials and least-squares approximation of functions. Direct and iterative methods for solving linear systems. Methods for solving non-linear equations and systems of non-linear equations. Introduction to computer programming. Prerequisites: One of MATH 102, 125 or 127, and one of MATH 209, 214 or 217. Notes: (1) Credit can be obtained in at most one of MATH 280, 381 or CMPUT 340. (2) Extra classes may be held for students lacking a background in one of the major programming languages such as Fortran, C, C++ or Matlab.

#### MATH 408 - Computational Finance View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Principles of Monte Carlo methods. Essentials of stochastics. Introduction to financial derivatives pricing. Generating random variables. Simulating stochastic differential equations. Application to financial derivatives pricing and interest rate models. Variance reduction techniques. Prerequisite: STAT 471, or E E 387 and consent of the Department.

#### MATH 411 - Honors Complex Variables View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Complex number system. Analytic functions. Cauchy's Integral theorem and formula. Applications including the maximum modulus principle, Taylor expansion and Laurent expansion. Harmonic functions. The residue theorem with applications; calculus of residues, argument principle, and Rouche's theorem. Basics of analytic continuation. Additional topics at the instructor's discretion such as: Normal families, The Riemann mapping Theorem, Picard's Theorem. Prerequisite: MATH 314 or 317. Notes: (1) This course is primarily for Honors students in Mathematics or Physics. (2) Offered in alternate years. It may be offered in intervening years if demand is sufficient.

#### MATH 412 - Algebraic Number Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Valuations and their extensions, ramifications; integral dependence, algebraic number fields, ideals and divisors, class number. Prerequisite: MATH 326 or equivalent.

#### MATH 414 - Analysis II View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Differentiation of maps in Rn, implicit function and mapping theorems, sequences of functions, Riemann-Stielties integration, additional topics at the discretion of the instructor. Prerequisite: MATH 314. Note: This course may not be taken for credit if credit has already been obtained in MATH 317.

#### MATH 415 - Mathematical Finance I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of probability tools for discrete financial analysis; Conditional probabilities/expectations. Filtrations, adapted and predictable processes. Martingales, submartingales and supermartingales in discrete-time. Doob decomposition for supermartingales. Predictable representation. Discrete- time financial modes: Arbitrage, complete and incomplete markets. Self-financing property, value and gain processes. Valuation of contingent claims. Binomial model: Model specifications, Perfect hedging. Utility functions and consumption/ investment problems. European and American options in discrete time. Futures and forward contracts in discrete time. Transition to the continuous-time framework. Corequisite: STAT 471 or consent of the Department.

#### MATH 417 - Honors Real Variables I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Brief review of set operations and countable sets. Measure theory, integration theory, Lebesgue measure and integrals on R^n, product measure, Tonelli-Fubini theorem. Functions of bounded variation, absolutely continuous functions. Prerequisite: MATH 317 or 414.

#### MATH 418 - Honors Real Variables II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

#### MATH 421 - Combinatorics View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Permutations and combinations, Binomial Theorem, Principle of Inclusion-Exclusion, recurrence relations, generating functions, orthogonal Latin squares, balanced incomplete block designs, Steiner triple systems, perfect difference sets, Boolean algebra and Finite State Machines. Prerequisites: Either MATH 326 or one of MATH 111 or 228 and a 300-level MATH course (MATH 322 recommended).

#### MATH 422 - Coding Theory View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Elements of group theory, cosets, Lagrange's theorem, binary group codes, polynomials, finite field theory, error correcting codes. Prerequisites: either (1) MATH 227 or (2) MATH 228 and a 300-level MATH course.

#### MATH 424 - Algebra: Groups and Fields View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Field extensions. Groups of automorphisms of fields. Galois theory. Finite fields and applications. Solvable groups, the insolvability of the quintic equation. Ruler and compass construction. Prerequisites: MATH 326 (or MATH 228 by consent of the Department) and MATH 328. Note: This course cannot be taken for credit if credit has already been obtained in MATH 427 or 329.

#### MATH 428 - Algebra: Advanced Ring Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Introduction to commutative algebra, algebraic geometry, and homological algebra. Additional topics at the discretion of the instructor. Prerequisite: MATH 326 or consent of Department.

#### MATH 429 - Algebra: Advanced Group Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory. Prerequisite: MATH 328 or consent of the Department.

#### MATH 432 - Intermediate Differential Equations View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Elementary existence and uniqueness theorems. Systems of equations, stability, perturbation theory. Introduction to numerical methods. Introduction to phase plane analysis. Prerequisite: One of MATH 201, 334 or 336.

#### MATH 436 - Intermediate Partial Differential Equations I View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Partial differential equations as physical models. Introduction to basic generalized functions. Theory of linear and quasi-linear first-order equations: general solution, initial value problem, generalized solutions and propagation of singularities, characteristic surfaces, shock formation. Theory of fully nonlinear first order equations: complete solution and the initial value problem. Hamilton-Jacobi equation and its applications. Second order linear equations in n dimensions: classification, canonical form, characteristic surfaces and shock formation, initial and boundary value problem. Prerequisite: MATH 337.

#### MATH 438 - Intermediate Partial Differential Equations II View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Second order equations in n dimension: classification, canonical form, characteristic surfaces. Laplace equation as a representative of the elliptic equation: the mean value theorem, fundamental solutions and Green functions, the boundary value problems. Wave equation as a representative of hyperbolic equations: initial value problems, the d'Alambert formula, the method of descent, propagation of singularities, Duhamel's principle. Heat equation as a representative of parabolic equations: initial value problems. Introduction to integral transforms: Fourier, Laplace, Hankel transforms. Prerequisite: MATH 337.

#### MATH 447 - Elementary Topology View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

General point-set topology. Compactness, Tychonoff's tbeorem, connectedness. Metric spaces, completeness, Baire's theorem. Urysohn's lemma. Topological manifolds. Homotopy theory, fundamental group, covering spaces. Prerequisite : MATH 216 or 217. Corequisites: MATH 328 or MA PH 464. Offered in alternate years. It may be offered in intervening years if demand is sufficient.

#### MATH 448 - Introduction to Differential Geometry View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Riemannian geometry of n-space, metric tensors, various curvature concepts and their relationships, covariant differentiation, geodesics, parallel transport. Additional topics at the discretion of the instructor. Prerequisite: MATH 348, or MATH 217 and one of MATH 225 or 227. Note: Offered in alternate years. It may be offered in intervening years if demand is sufficient.

#### MATH 496 - Honors Seminar View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

This course is intended to give students experience with independent reading, and to improve their ability to present and explain mathematical ideas. The course is compulsory for all fourth year Honors students in BSc and BA Mathematics and BSc Applied Mathematics. Normally offered in alternate years. Prerequisite: MATH 317.

#### MATH 497 - Reading in Mathematics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

This course is designed to give credit to mature and able students for reading in areas not covered by courses, under the supervision of a staff member. A student, or group of students, wishing to use this course should find a staff member willing to supervise the proposed reading program. A detailed description of the material to be covered should be submitted to the Chair of the Department Honors Committee. (This should include a description of testing methods to be used.) The program will require the approval of both the Honors Committee, and the Chair of the Department. The students' mastery of the material of the course will be tested by a written or oral examination. This course may be taken in Fall or Winter and may be taken any number of times, subject always to the approval mentioned above. Prerequisite: Any 300-level MATH course.

#### MATH 499 - Research Project View Available Classes

**★ 3 (fi 6)(EITHER, 0-1S-6)**

This course provides students in Specialization and Honors programs an opportunity to pursue research in mathematics under the direction of a member of the Department. Course requirements include at least one oral presentation and a written final report. Students interested in taking this course should contact the course coordinator two months in advance. Credit for this course may be obtained more than once. Prerequisites: a 300-level MATH course and consent of the course coordinator.

#### MATH 505 - Stochastic Analysis I View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Discrete-time stochastic analysis: Stochastic basis, filtration, stochastic sequences. Absolute continuity of probability measures and conditional expectations. Martingale-like and predictable stochastic sequences. Doob's decomposition. Stopping times and related properties. Uniformly integrable stochastic sequences. Transition from discrete-time to continuous-time stochastic analysis. Introduction to stochastic integration with respect to Brownian motion. Prerequisites: STAT 471 or consent of the Department.

#### MATH 506 - Complex Variables View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

A review and some extensions of single variable complex analysis. Complex linearity and holomorphicity in several variables, Hartog's theorem, Weierstrass preparation theorem, Riemann extension theorem, Weierstrass division theorem, analytic Nullstellensatz, implicit and inverse function theorems, complex manifolds and analytic subvarieties, meromorphic maps. Prerequisite: MATH 411.

#### MATH 508 - Computational Finance View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Principles of Monte Carlo methods. Essentials of stochastics. Introduction to financial derivatives pricing. Generating random variables. Simulating stochastic differential equations. Application to financial derivatives pricing and interest rate models. Variance reduction techniques. Prerequisite: STAT 471 or FIN 654 or ECON 598 or consent of the Department. Note: This course may not be taken for credit if credit has already been obtained in MATH 408.

#### MATH 509 - Data Structures and Platforms View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Basic data analysis with R, SAS, and Python. Program development with Jupyter notebooks. Cloud computing, collaborative software development, docker containers, kubernets. Internet security, privacy and ethics. Technologies will be updated as new developments arise. Prerequisites: No programming skills are needed.

#### MATH 510 - Stochastic Analysis II View Available Classes

**★ 3 (fi 6)(SECOND, 3-0-0)**

Continuous semimartingales and quadratic variation. Stochastic integrals for continuous semimartingales. Ito's formula. Change of probability measure (Girsanov transformation). Martingale representation theorem for Brownian filtrations. Stochastic differential equations, diffusions. Introduction to discontinuous semimartingales with emphasis on Poisson processes. Prerequisites: MATH 505 or consent of the Department.

#### MATH 512 - Algebraic Number Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Valuations and their extensions, ramifications; integral dependence, algebraic number fields, ideals and divisors, class number. Prerequisite: MATH 326 or equivalent.

#### MATH 514 - Measure Theory I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Brief review of set operations and countable sets. Measure theory, integration theory, Lebesgue measure and integrals on R^n, product measure, Tonelli-Fubini theorem. Functions of bounded variation, absolutely continuous functions. Prerequisites: Math 317.

#### MATH 515 - Mathematical Finance I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of probability tools for discrete financial analysis; Conditional probabilities/expectations. Filtrations, adapted and predictable processes. Martingales, submartingales and supermartingales in discrete-time. Doob decomposition for supermartingales. Predictable representation. Discrete-time financial modes: Arbitrage, complete and incomplete markets. Self-financing property, value and gain processes. Valuation of contingent claims. Binomial model: Model specifications, Perfect hedging. Utility functions and consumption/investment problems. European and American options in discrete time. Futures and forward contracts in discrete time. Transition to the continuous-time framework. Prerequisite: STAT 471 or consent of the Department. Note: This course may not be taken for credit if credit has already been obtained in MATH 415.

#### MATH 516 - Linear Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

#### MATH 518 - Functional Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Locally convex spaces, weak topologies and duality in Banach spaces, weak compactness in Banach spaces, structure of classical Banach spaces, local structures, infinite-dimensional geometry of Banach spaces and applications. Prerequisite: MATH 516. Corequisite: MATH 447 or consent of Department.

#### MATH 519 - Introduction to Operator Algebras View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Banach algebras and spectral theory, compact and Fredholm operators, the spectral theorem for bounded normal operators, operator algebras, representations of C+-algebras, elementary von Neumann algebra theory, and other topics. Prerequisite: MATH 516. Corequisite: MATH 447 or consent of Department.

#### MATH 520 - Mathematical Finance II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Financial markets in continuous-time: Arbitrage, completeness, self-financing strategies. Black Scholes model. Option pricing and hedging: European, American and exotic options. Consumption-investment problem: Utility maximization, optimal portfolio and optimal consumption. Prerequisite: MATH 515. Corequisite: MATH 510 or consent of the Department.

#### MATH 521 - Differential Manifolds View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Finite dimensional manifolds/submanifolds; tangent bundle, differential, inverse, and implicit function theorems, partitions of unity; imbeddings, immersions, submersions; vector fields and associated flows; Lie derivative, Lie bracket; tensor analysis, differential forms, orientation, integration, Stokes' theorem; basics of smooth bundle theory, Riemannian metrics; notion of a Lie group with basic examples, smooth Lie group actions, principal bundles. Prerequisite: MATH 446 or 448.

#### MATH 524 - Ordinary Differential Equations IIA View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Existence theorems, uniqueness theorems; linear systems (basic theory); stability (basic theory); nonlinear systems (local theory); nonlinear systems (global theory); bifurcations. Prerequisite: MATH 334 or 336, or equivalent.

#### MATH 525 - Ordinary Differential Equations IIB View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Asymptotics; boundary value problems; Poincare-Bendixson theory. Additional material will be chosen from among the following topics at the option of the instructor: separation; dichotomies; comparison and oscillation theory; bifurcation theory; nonautonomous systems; dynamical systems; functional differential equations; contingent equations; differential equations in Banach spaces. Prerequisite: MATH 524 or equivalent.

#### MATH 527 - Intermediate Partial Differential Equations View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Notions; Elliptic PDE's; Parabolic PDE's; Hyperbolic PDE's; Nonlinear Integrable PDE's. Prerequisite: MATH 436 or equivalent; corequisite: MATH 516.

#### MATH 530 - Algebraic Topology View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Particular background from point set topology (pasting and quotienting constructions); homotopy relation between maps and spaces; fundamental group; Seifert VanKampen theorem; covering spaces. Additional topics at the discretion of the instructor. Prerequisites: MATH 227, 317 and 447 or consent of Department.

#### MATH 535 - Numerical Methods I View Available Classes

**★ 3 (fi 6)(FIRST, 3-0-0)**

Direct and iterative methods for solving linear systems, iterative methods for nonlinear systems, polynomial and spline interpolations, least square approximation, numerical differentiation and integration, initial value problems for ODE's (one-step, multistep methods, stiff ODE's). Prerequisite: 400-level MATH course. Students are required to have knowledge of advanced Calculus and introductory knowledge in Analysis and Linear Algebra and some computer programming. Note 1: Restricted to graduate students only. Note 2: May not be taken for credit if credit has already been obtained in MATH 381, 481 or 486 or equivalent.

#### MATH 536 - Numerical Solutions of Partial Differential Equations I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Finite difference and finite element methods for boundary-value problems of elliptic equations. Numerical algorithms for large systems of linear algebraic equations: direct, classical relaxation, multigrid and preconditioned conjugate gradient methods. Algorithms for vector/parallel computers and the domain decomposition method. Prerequisites: MATH 337, 436 or equivalent and some computer programming.

#### MATH 538 - Techniques of Applied Mathematics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Asymptotic analysis of integrals: Laplace, stationary phase, and steepest descent methods. Regular and singular perturbations: trained coordinates, multiple scales, asymptotic matching, renormalization techniques, WKB theory, Hamiltonian perturbation theory, center manifolds and stability. Singularities in differential equations. Applications to algebraic, ordinary and partial differential equations. Prerequisite: MATH 438 or equivalent.

#### MATH 539 - Applied Functional Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Linear part:structure of function spaces, Sobolev spaces, embeddings, topologies, linear operators, adjoint and inverse operators, spectra, distributions, semigroup theory, integral equations, well-posedness and the notion of a solution. Nonlinear part: inequalities, Frechet and Gateaux derivatives, fixed point theorems. Applications from mechanics, reaction-diffusion equations, the Navier-Stokes equations, nonlinear Schrödinger equation. Prerequisite: MATH 438 or equivalent.

#### MATH 542 - Fourier Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review, theory and extension of Fourier series for square integrable functions; orthonormal systems, Bessel's inequality, completeness, Parseval's identity, Riesz-Fischer Theorem. Extension to Fourier series for functions in other Lebesgue classes; Fejer means, conjugate series, Dirichlet, Fejer and Poisson kernels. Norm convergence; remarks on pointwise convergence. Fourier transforms and series in several dimensions; inverse transform, Plancherel formula, Poisson Formula, maximal functions, Riesz-Thorin Theorem and applications. Elementary distribution theory; D, D', S, S' and some elementary results, Fourier transforms of tempered distributions. Examination of some earlier results with tempered distributions instead of functions and getting familiar with basic concepts. Prerequisite: MATH 418.

#### MATH 543 - Measure Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of basic measure and integration theory. Signed and complex measures. Hahn and Jordan decompositions. The Radon-Nikodyn theorem. Lebesgue decomposition. The Lebesgue-Stieltjes integral. Measure theory over locally compact Hausdorff spaces, in particular, Riesz' representation theorem and Haar measures. Hausdorff measure. Introduction to matringales. Prerequisite: MATH 417 and MATH 447 or equivalent.

**Starting: 2022-09-01**

#### MATH 543 - Measure Theory II

**★ 3 (fi 6)(EITHER, 3-0-0)**

Review of basic measure and integration theory. Signed and complex measures. Hahn and Jordan decompositions. The Radon-Nikodym theorem. Lebesgue decomposition. The Lebesgue-Stieltjes integral. Measure theory over locally compact Hausdorff spaces, in particular, the Riesz representation theorem and Haar measures. Hausdorff measure. Introduction to martingales. Prerequisite: One of MATH 417 or MATH 514, and MATH 447 or equivalent.

#### MATH 556 - Introduction to Fluid Mechanics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Fundamentals including continuum hypothesis surface tension, classical thermodynamics, and transport phenomena. Introduction to Cartesian tensors. Kinematics of flow including Lagrangian and Eulerian descriptions, streamline, path line, streak line, vorticity and circulation. Derivation of the conservation laws for mass, momentum, and energy and a detailed description of the Boussinesq approximation. Conservation laws in a rotating frame. Vortex lines and tubes, role of viscosity in vortices, Kelvin's circulation theorem, the vorticity equation in nonrotating and rotating frames. Irrotational flow including its relevance, velocity potential, sources and sinks, and flow past various shapes. Gravity waves in deep and shallow water with and without surface tension in both the linear and nonlinear contexts. Dynamic similarity and Buckingham's Pi Theorem. Prerequisites: One of MATH 311, 411 and MATH 436 or consent of Instructor.

#### MATH 570 - Mathematical Biology View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Mathematical modeling in the biological and medical sciences. Students will learn how to apply mathematical methods and theory to a variety of different biological problems. Topics will be taken from: (i) continuous and discrete dynamical systems describing interacting and structured populations, resource management, biological control, reaction kinetics, biological oscillators and switches, the dynamics of infectious diseases and genetics and (ii) models of spatial processes in biology including random walks, pattern formation in morphogenesis and ecology, applications of traveling waves to population dynamics, epidemiology, chemical reactions, and models for neural patterns. Prerequisites: MATH 524 and a 400 or 500 level course on Partial Differential Equations or consent of Instructor.

#### MATH 572 - Mathematical Modelling in Industry, Government, and Sciences View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Developing mathematical models to solve real-world problems, model analysis, fitting model to data, model validation and selection, and interpretation of model outcomes. Types of models include difference equation models, differential equation models, network models, and stochastic models. Prerequisites: Linear algebra and differential equations or consent of the instructor.

#### MATH 581 - Group Theory View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory. Additional topics at the discretion of the instructor. Prerequisite: MATH 328 or consent of the Department.

#### MATH 582 - Rings and Modules View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Introduction to commutative algebra, algebraic geometry, and homological algebra. Additional topics at the discretion of the instructor. Prerequisite: MATH 326 or consent of the Department.

#### MATH 600 - Reading in Mathematics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Students registered in this course are supervised by individual staff members in areas of interest of the staff members. Students will be allowed to take this course only in exceptional circumstances and with the permission of the Chairman of the Department. This course shall not be counted against the minimum course requirement for graduate students.

#### MATH 601 - Graduate Colloquium View Available Classes

**★ 1 (fi 2)(EITHER, 0-2S-0)**

Credit for this course can be obtained twice.

#### MATH 617 - Topics in Functional Analysis I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 623 - Topics in Differential Geometry and Mechanics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 625 - Advanced Mathematical Finance View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Topics among: Incomplete markets; Models with imperfection (markets with transaction costs, constraints or defaults); Risk measures; Interplay between finance and insurance; Mathematical models for the term structure of interest rates. Prerequisites: MATH 520 or consent of the Department.

#### MATH 642 - Abstract Harmonic Analysis View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

Prerequisite: MATH 519.

#### MATH 653 - Seminar in Functional Analysis View Available Classes

**★ 1 (fi 2)(EITHER, 0-2S-0)**

Credit for this course may be obtained more than once.

#### MATH 655 - Topics in Fluid Dynamics View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 663 - Topics in Applied Mathematics I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 664 - Topics in Applied Mathematics II View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 667 - Topics in Differential Equations I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 676 - Topics in Geometry I View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 681 - Topics in Algebra View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 682 - Topics in Algebra View Available Classes

**★ 3 (fi 6)(EITHER, 3-0-0)**

#### MATH 900A - Directed Research Project View Available Classes

**★ 3 (fi 12)(VAR, UNASSIGNED)**

Open only to students taking the MSc non-thesis option in mathematics.

#### MATH 900B - Directed Research Project View Available Classes

**★ 3 (fi 12)(VAR, UNASSIGNED)**

Open only to students taking the MSc non-thesis option in mathematics.