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APM Applied Mathematics Courses

| Course Winter Timetable |

Applications of Linear 39L

Introduction to linear programming including a rapid review of linear algebra (row reduction, linear independence), the simplex method, the duality theorem, complementary slackness, and the dual simplex method. A selection of the following topics are covered: the revised simplex method, sensitivity analysis, integer programming, the transportation algorithm.
Prerequisite: MAT223H1/MAT240H1 (Note: no waivers of prerequisites will be granted)

Differential Equations 39L

Sturm-Liouville problems, Green’s functions, special functions (Bessel, Legendre), partial differential equations of second order, separation of variables, integral equations, Fourier transform, stationary phase method.
Prerequisite: MAT235Y1/MAT237Y1/MAT257Y1, MAT244H1

Partial Differential Equations 78L

Diffusion and wave equations. Separation of variables. Fourier series. Laplace’s equation; Green’s function. Schrödinger equations. Boundary problems in plane and space. General eigenvalue problems; minimum principle for eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction to nonlinear equations (shock waves, solitary waves).
Prerequisite: MAT267H1
Co-requisite: MAT334H1/MAT354H1

Nonlinear Optimization 39L

An introduction to first and second order conditions for finite and infinite dimensional optimization problems with mention of available software. Topics include Lagrange multipliers, Kuhn-Tucker conditions, convexity and calculus variations. Basic numerical search methods and software packages which implement them will be discussed.
Prerequisite: MAT224H1, MAT235Y1

Mathematical Foundations of 39L

The general formulation of non-relativistic quantum mechanics based on the theory of linear operators in a Hilbert space, self-adjoint operators, spectral measures and the statistical interpretation of quantum mechanics; functions of compatible observables. Schrödinger and Heisenberg pictures, complete sets of observables, representations of the canonical commutative relations; essential self-adjointedness of Schrödinger operators, density operators, elements of scattring theory.
Prerequisite: MAT337H1/MAT357H1

General Relativity 39L

Einstein’s theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manisfestation of spacetime curvature. Einstein’s equations. Cosmological implications: big bang and inflationary universe. Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from black hole dynamics and gravity waves.
Prerequisite: MAT363H1

Fluid Mechanics 39L

Boltzmann, Euler and Navier-Stokes equations. Viscous and non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propagation. Analysis of one dimensional gas flow.
Prerequisite: APM351Y1

Asymptotic and 39L

Perturbation Methods
Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained co-ordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering)
Prerequisite: APM346H1/APM351Y1, MAT334H1

Applied Nonlinear Equations 39L

Nonlinear partial differential equations and their physical origin. Fourier transform; Green’s function; variational methods; symmetries and conservation laws. Special solutions (steady states, solitary waves, travelling waves, self-similar solutions). Calculus of maps; bifurcations; stability, dynamics near equilibrium. Propogation of nonlinear waves; dispersion, modulation, optical bistability. Global behaviour solutions; asymptotics and blow-up.
Prerequisite: APM346H1/APM351Y1

Control Theory and Optimization 39L

Differential systems with controls and reachable sets. Non-commutativity, Lie bracket and controllability. Optimality and maximum principle. Hamiltonian formalism and symplectic geometry. Integrability. Applications to engineering, mechanics and geometry.
Prerequisite: MAT357H1 or MAT244H1/MAT267H1, MAT337H1

Combinatorial Methods 39L

A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisite: MAT224H1
Recommended preparation: MAT344H1/CSC238H1

Mathematical Theory of Finance 39L

Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.
Prerequisite: APM346H1, STA347H1
Co-requisite: STA457H1

Readings in Applied Mathematics TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Prerequisite: minimum GPA 3.5 for math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisor

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