Review and further study of vector spaces over arbitrary fields. General linear transformations. Kernel and range. Matrices of linear transformations. Complex vector spaces and inner product spaces.

## Undergraduate Calendar - Mathematics and Statistics

Unitary, normal, symmetric, skew-symmetric and Hermitian operators. Orthogonal projections and the spectral theorem.

Bilinear and quadratic forms. Jordan canonical form. Theory and applications of mathematical modelling and simulation. Topics may include discrete dynamical systems, Monte-Carlo methods, stochastic models, the stock market, epidemics, analysis of DNA, chaotic dynamical systems, cellular automata and predator-prey. Mathematical concepts and ideas in number systems; geometry and probability arising in the Primary and Junior school curriculum. Not open to students holding credit in any grade 12 or university mathematics course.

Mathematical models arising in finance and insurance. Compound interest, the time-value of money, annuities, mortgages, insurance, measures of risk. Introduction to stocks, bonds and options. Probability, events, algebra of sets, independence, conditional probability, Bayes' theorem; random variables and their univariate, multivariate, marginal and conditional distributions. Expected value of a random variable, the mean, variance and higher moments, moment generating function, Chebyshev's theorem. Some common discrete and continuous distributions: Binomial, Poisson, hypergeometric, normal, uniform and exponential.

Use of SAS, Maple or other statistical packages. Development of Euclidean and non-Euclidean geometry from Euclid to the 19th century. Deductive nature of plane Euclidean geometry as an axiomatic system, central role of the parallel postulate and general consideration of axiomatic systems for geometry in general and non-Euclidean geometry in particular. Introduction to transformation geometry. Use of Geometer's Sketchpad.

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Problems and methods in discrete optimization. Linear programming: problem formulation, the simplex method, software, and applications. Network models: assignment problems, max-flow problem.

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Directed graphs: topological sorting, dynamic programming and path problems, and the travelling salesman's problem. General graphs: Eulerian and Hamiltonian paths and circuits, and matchings. Counting, inclusion and exclusion, pigeonhole principle, permutations and combinations, derangements, binomial expansions. Introduction to discrete probability and graph theory, Eulerian graphs, Hamilton Cycles, colouring, planarity, and trees. Complex networks and their properties, random graphs, network formation models.

Webgraph: models, search engines, page-ranking algorithms. Community clustering, community structure. Opinion formation, on-line social networks. Scales and temperaments, history of the connections between mathematics and music, set theory in atonal music, group theory applied to composition and analysis, enumeration of rhythmic canons, measurement of melodic similarity using metrics, topics in mathematical music theory, applications of statistics to composition and analysis.

Solving mathematical problems using insight and creative thinking. Topics may include pigeonhole principle, finite and countable sets, probability theory, congruences and divisibility, polynomials, generating functions, inequalities, limits, geometry, and mathematical games. Note: recommended to students wishing to participate in mathematical problem solving competitions. Single-factor and factorial experimental design methods; nested-factorial experiments.

Simple and multiple linear regression methods, correlation analysis, indicator regression; regression model building and transformations. Contingency tables, binomial tests, nonparametric rank tests. Simple random and stratified sampling techniques, estimation of sample size and related topics. Approximation of functions by algebraic and trigonometric polynomials Taylor and Fourier series ; Weierstrass approximation theorem; Riemann integral of functions on R n , the Riemann-Stieltjes integral on R; improper integrals; Fourier transforms. Algebra and geometry of complex numbers, complex functions and their derivatives; analytic functions; harmonic functions; complex exponential and trigonometric functions and their inverses; contour integration; Cauchy's theorem and its consequences; Taylor and Laurent series; residues.

Vector fields, vector algebra, vector calculus; gradient, curl and divergence.

### Junior-to-Senior Level Courses

Polar, cylindrical and spherical coordinates. Green's, Stokes' and divergence theorems. Introduction to differential geometry of surfaces. Topics may include differential forms, exterior calculus, frames, Gauss-Bonnet theorem. Linear second-order differential equations and special functions. Introduction to Sturm-Liouville theory and series expansions by orthogonal functions.

Boundary value problems for the heat equation, wave equation and Laplace equation. Green's functions. Emphasis on applications to physical sciences. Survey of linear and nonlinear partial differential equations. Analytical solution methods. Existence and uniqueness theorems, variational principles, symmetries, and conservation laws. Group theory with applications. Topics include modular arithmetic, symmetry groups and the dihedral groups, subgroups, cyclic groups, permutation groups, group isomorphism, Burnside's theorem, cosets and Lagrange's theorem, direct products and cryptography, normal subgroups and factor groups.

Further topics in group theory: homomorphisms and isomorphism theorems, structure of finite abelian groups. Rings and ideals; polynomial rings; quotient rings.

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Division rings and fields; field extensions; finite fields; constructability. Triumphs in mathematical thinking emphasizing many cultures up to AD. Analytical understanding of mathematical problems from the past, referencing the stories and times behind the people who solved them.

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## Sparse Arrays: Linear Algebra

Matching wits with great mathematicians by solving problems and developing activities related to their discoveries. Concepts and programming of contemporary models and simulations used in mathematics and sciences. Techniques in the visual representation of mathematical data and the interactive presentation of mathematical ideas. Topics may include modelling and simulation, visualization of real world data, interactive learning environments, and interactive websites.

Blending mathematical concepts with computations and visualization in Maple. Modelling of physical flows, waves and vibrations. Animation of the heat equation and wave equation; applications including vibrations of rectangular and circular drums, heat flow and diffusion, sound waves. Eigenfunctions and convergence theorems for Fourier eigenfunction series. Approximations, Gibbs phenomena, and asymptotic error analysis using Maple.