²ÝÝ®ÎÛÊÓÆµµ¼º½ : Applied Mathematics AMAT

Applied Mathematics AMAT

AMAT 501 503 505 507 509 581 583 601 605 613 617 621 643 651 653 671 673 677 681 683

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

Department Head - M. Lamoureux

Note: For listings of related courses, see Actuarial Science, Mathematics, Pure Mathematics and Statistics.

Applied Mathematics 501

Seminar in Applied Mathematics

Topics will be chosen according to the interests of instructors and students and could include analysis of optimization algorithms, approximation theory, control theory, differential equations, mathematical physics.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
MAY BE REPEATED FOR CREDIT

Applied Mathematics 503

The Mathematics of Wavelets, Signal and Image Processing

Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 491 or Computer Science 491.

Applied Mathematics 505

Calculus on Manifolds

Integral and differential calculus on manifolds including tensor fields, covariant differentiation, Lie differentiation, differential forms, Frobenius' theorem, Stokes' theorem, flows of vector fields.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 445 or 545; and one of Applied Mathematics 311 or 307; or consent of the Division.

Applied Mathematics 507

Introduction to Relativity Theory

Mathematical theories of space and time. Special Relativity. Electro-dynamics. General Relativity.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 505 or consent of the Division.

Applied Mathematics 509

Analytical Dynamics

Symplectic geometry, Hamilton's equation, Hamilton-Jacobi theory, constraints and reduction.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 505 or consent of the Division.

Applied Mathematics 581

Stochastic Calculus for Finance

Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Ito formula, stochastic differential equations, Girsanovâ€™s theorem, Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481.
Antirequisite(s): Credit for both Applied Mathematics 581 and 681 will not be allowed.

Applied Mathematics 583

Computational Finance

Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481 and 491.
Antirequisite(s): Credit for both Applied Mathematics 583 and 683 will not be allowed.

Graduate Courses

In addition to the prerequisites listed below, consent of the Applied Mathematics Division is a prerequisite for all graduate courses in Applied Mathematics.

Applied Mathematics 601

Topics in Applied Mathematics

Topics will be chosen according to the interests of instructors and students.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.
MAY BE REPEATED FOR CREDIT

Applied Mathematics 605

Differential Equations III

Linear systems, classification. Nonlinear systems: Existence and uniqueness. Flow and one parameter groups of transformations. Stability theory. Hyperbolicity, Unstable/Stable/Center manifold theorems. Poincare-Bendixson.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 411 and Pure Mathematics 445 or 545 or equivalents.

Applied Mathematics 613

Partial Differential Equations II

Fundamental solutions, integral equations, eigenvalue problems, non-linear problems.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Division.

Applied Mathematics 617

Analysis IV

Analysis in abstract spaces. Function spaces.
Course Hours: H(3-0)
Prerequisite(s): Pure Mathematics 545.
Antirequisite(s): Credit for Applied Mathematics 617 and Pure Mathematics 617 will not be allowed.

Applied Mathematics 621

Research Seminar

A professional skills course, focusing on the development of technical proficiencies that are essential for students to succeed in their future careers 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: Q(2S-0)
MAY BE REPEATED FOR CREDIT NOT INCLUDED IN GPA

Applied Mathematics 643

Perturbation Theory

Perturbation problems for ordinary differential equations, matrices and more general operators. Applications. Methods will be motivated by discussion of physical problems.
Course Hours: H(3-0)
Prerequisite(s): Familiarity with complex variables, linear algebra and differential equations.

Applied Mathematics 651

Monte Carlo Methods for Quantitative Finance

Fundamental concepts of Monte Carlo methods; review of quantitative finance; random number generation; simulating stochastic differential equations; variance reduction; quasi-Monte Carlo methods; computing sensitivities; early exercise options; Levy processes and other price models; applications to risk management.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Department.

Applied Mathematics 653

Introduction to Levy Processes with Applications

Infinite divisibility, Levy processes (LP), the Levy-Khintchine formula; examples of LP; Poisson integration, the Levy-Ito decomposition, subordinators; Markov processes, semi-groups and generators of LP; Ito-formula for LP, quadratic variation; LP as time-changed Brownian motion, change of measure (Girsanov theorem); stochastic differential equations driven by LP; Feynman-Kac formula and martingale problem for LP; applications of LP; simulation of LPs.
Course Hours: H(3-0)
Prerequisite(s): Consent of the Department.

Applied Mathematics 671

Numerical Linear Algebra

Iterative and elimination methods for linear systems of equations, determination of eigenvalues, linear and convex programming.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 441 and Applied Mathematics 491.

Applied Mathematics 673

Approximation Theory

Existence, uniqueness of minimal solutions, Haar systems, characterization by alternation, Remez algorithm, monotone operators, spline approximation.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 491; and Pure Mathematics 435 or 455.

Applied Mathematics 677

Numerical Solution of Partial Differential Equations

Explicit and implicit methods for PDE, difference equations.
Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 311 and 491.

Applied Mathematics 681

Stochastic Calculus for Finance

Applied Mathematics 683

Computational Finance

Review of financial asset price and option valuation models; model calibration; tree-based methods; finite-difference methods; Monte Carlo simulation; Fourier methods.

Course Hours: H(3-0)
Prerequisite(s): Applied Mathematics 481 and 491.
Antirequisite(s): Credit for both Applied Mathematics 683 and 583 will not be allowed.

In addition to the numbered and titled courses shown above, the department offers a selection of advanced level graduate courses specifically designed to meet the needs of individuals or small groups of students at the advanced doctoral level. These courses are numbered in the series 800.01 to 899.99. Such offerings are, of course, conditional upon the availability of staff resources.

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