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APM Applied Mathematics CoursesAPM233Y1 The application of mathematical techniques to economic analysis. Mathematical topics
include linear and matrix algebra, partial differentiation, optimization, Lagrange
multipliers, differential equations. Economic applications include consumer and producer
theory, theory of markets, macroeconomic models, models of economic growth. APM236H1 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. APM261H1 Formulation of problems in LP form, convexity and structure of LP constraint sets,
simplex algorithm, degeneracy, cycling and stalling, revised method, two-phase method,
duality, fundamental theorem, dual algorithm, integer programming, sensitivity analysis,
Karmarkar algorithm, network flows, transportation algorithm, two-person zero-sum games. APM346H1 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. APM351Y1 Diffusion and wave equations. Separation of variables. Fourier series. Laplace's
equation; Green's function. Schr”dinger 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). APM361H1 Topics selected from applied stochastic processes, queuing theory, inventory models,
scheduling theory and dynamic programming, decision methods, simulation. A project based
on a problem of current interest taken from course files or the student's own experience
is required. APM366H1 Convexity, fixed points, stable mappings optimization. Relations orderings and utility
functions; choice and decision making by individuals and groups. Non-cooperative and
cooperative games, core, Shapley value; market games. Decision making by economic agents:
consumers, producers, banks, investors, and financial intermediaries. APM421H1 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.
Schr”dinger and Heisenberg pictures, complete sets of observables, representations of
the canonical commutative relations, essential self-adjointness of Schr”dinger
operators, density operators, elements of scattering theory. APM426H1 Local and global geometries of Lorentz manifolds, stationary and statics space-times.
Einstein field equations, Schwarzchild, Kruskal and Kerr solutions. Mathematics of black
holes. Relativistic cosmology, big bang and inflationary models. Cauchy problem and
Petrov-Newman-Penrose classifications. APM436H1 Formulation of Navier-Stokes equations, exact solutions. Slow viscous flow. Boundary
layers and singular perturbations. Wave propagation, stratified fluids; stability.
Compressible flow. The main emphasis is the description of basic physical phenomena
determined from simple analytical solutions of the governing equations. APM441H1 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) APM446H1 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. APM456H1 Differential systems and reachable sets. Non-commutativity, controllability and the Lie
bracket. Optimality and the Maximum Principle. Applications to engineering, mechanics and
geometry. APM461H1 A selection of topics from such areas as graph theory, combinatorial algorithms,
enumeration, construction of combinatorial identities. APM466H1 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. APM496H1/497H1/498Y1/499Y1 Independent study under the direction of a faculty member. Topic must be outside
undergraduate offerings. |
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