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MAT Mathematics Courses APM233Y1 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. Schrodinger 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. Schrodinger and Heisenberg pictures, complete sets of observables,
representations of the canonical commutative relations; essential self-adjointedness of
Schrodinger operators, density operators, elements of scattring theory. APM426H1 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. APM436H1 Boltzmann, Euler and Navier-Stokes equations. Viscous and
non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propragation. Analysis
of one dimensional gas flow. 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 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. 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 TBA NOTE: In Prerequisites, Calc = Ontario Academic Course Calculus; A&G = Ontario Academic Course Algebra and Geometry; FM = Ontario Academic Course Finite Mathematics SCI199Y1 Undergraduate seminar that focuses on specific ideas, questions, phenomena or controversies, taught by a regular Faculty member deeply engaged in the discipline. Open only to newly admitted first year students. It may serve as a breadth requirement course; see page 44. JUM102H1 A study of the interaction of mathematics with other fields
of inquiry: how mathematics influences, and is influenced by, the evolution of science and
culture. Art, music, and literature, as well as the more traditionally related areas of
the natural and social sciences, are considered. (Offered every three years) JUM103H1 A study of games, puzzles and problems focusing on the deeper
principles they illustrate. Concentration is on problems arising out of number theory and
geometry, with emphasis on the process of mathematical reasoning. Technical requirements
are kept to a minimum. A foundation is provided for a continuing lay interest in
mathematics. (Offered every three years) JUM105H1 An in-depth study of the life, times and work of several
mathematicians who have been particularly influential. Examples may include Newton, Euler,
Gauss, Kowalewski, Hilbert, Hardy, Ramanujan, Godel, Erdos, Coxeter, Grothendieck.
(Offered every three years) MAT123H,124H: see below MAT 133Y MAT133Y1 Mathematics of finance. Matrices and linear equations. Review
of differential calculus; applications. Integration and fundamental theorem; applications.
Introduction to partial differentiation; applications. MAT133Y counts as a Social Science course MAT123H1 First term of MAT133Y.
Students in academic difficulty in MAT133Y who have
written two midterm examinations with a mark of at least 20% in the second may withdraw
from MAT133Y and enrol in MAT123H in the Spring Term. These students are informed of
this option by the beginning of the Spring Term. Classes begin in the second week of the
Spring Term; late enrolment is not permitted. Students not enrolled in MAT133Y in the Fall Term are not allowed to enrol in MAT123H. MAT123H
together with MAT124H is equivalent for program and
prerequisite purposes to MAT133Y. MAT124H1 Second Term content of MAT133Y;
the final examination includes topics covered in MAT123H.
Offered in the Summer Session only; students not enrolled in MAT123H in the preceding Spring Term will NOT be allowed
to enrol in MAT124H. MAT123H
together with MAT124H is equivalent for program and
prerequisite purposes to MAT133Y. MAT135Y1 Review of differential calculus; applications. Integration
and fundamental theorem; applications. Series. Introduction to differential equations. MAT125H1 First term of MAT135Y.
Students in academic difficulty in MAT135Y who have
written two midterm examinations with a mark of at least 20% in the second may withdraw
from MAT135Y and enrol in MAT125H in the Spring Term. These students are informed of
this option by the beginning of the Spring Term. Classes begin in the second week of the
Spring Term; late enrolment is not permitted. Students not enrolled in MAT135Y in the Fall Term will not be allowed to enrol in MAT125H. MAT125H
together with MAT126H is equivalent for program and
prerequisite purposes to MAT135Y. MAT126H1 Second Term content of MAT135Y;
the final examination includes topics covered in MAT125H.
Offered in the Summer Session only; students not enrolled in MAT125H in the preceding Spring Term will NOT be allowed
to enrol in MAT126H. MAT125H
together with MAT126H is equivalent for program and
prerequisite purposes to MAT135Y. MAT136Y1 Limited to out-of-province students interested in the
biological, physical, or computer sciences, whose high school mathematics preparation is
strong but does not include calculus. Develops the concepts of calculus at least at the
level of MAT135Y. May include background material on
functions, analytic geometry, and trigonometry, as well as on calculus. Note that this
course counts as full-course credit, although it involves double the number of lecture and
tutorial hours as MAT 135Y. MAT137Y1 A conceptual approach for students with a serious interest in
mathematics. Geometric and physical intuition are emphasized but some attention is also
given to the theoretical foundations of calculus. Material covers the basic concepts of
calculus: limits and continuity, the mean value and inverse function theorems, the
integral, the fundamental theorem, elementary transcendental functions, Taylor's theorem,
sequence and series, uniform convergence and power series. MAT123H, 124H, 125H, 126H, 133Y, 135Y, 136Y, 157Y MAT157Y1 A theoretical course in calculus; emphasizing proofs and
techniques, as well as geometric and physical understanding. Limits and continuity; least
upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse
function theorems. Integrals; fundamental theorem; elementary transcendental functions.
Taylor's theorem; sequences and series; uniform convergence and power series. JMB170Y1 Applications of mathematics to biological problems in
physiology, biomechanics, genetics, evolution, growth, population dynamics, cell biology,
ecology and behaviour. MAT223H1 Matrices, linear systems, elementary matrices and the inverse
of a matrix. Vector spaces over R, subspaces, basis and dimension. Real inner product
spaces, geometry in Rn, lines and hyperplanes. Linear transformation, kernel, range,
matrix representation, isomorphisms. The determinant, Cramer's rule, the adjoint matrix.
Eigenvalues, eigenvectors, similarity, diagonalization. Projections, Gram-Schmidt process,
orthogonal transformations and orthogonal diagonalization, isometries, quadratic forms,
conics, quadric surfaces. MAT224H1 Fields. Vector spaces over a field. Linear transformations,
dual spaces. Diagonalizability, direct sums. Invariant subspaces, Cayley-Hamilton theorem.
Complex inner product, orthogonality, the adjoint of a linear operator, the projection
matrix and the method of least squares. Normal, self-adjoint and unitary operators.
Spectral theorem. Conditioning and Rayleigh quotient. Jordan canonical form. MAT235Y1 Differential and integral calculus of functions of several
variables. Line and surface integrals, the divergence theorem, Stokes' theorem. Sequences
and series, including an introduction to Fourier series. Some partial differential
equations of Physics. MAT237Y1 Sequences and series. Uniform convergence. Convergence of
integrals. Elements of topology in R2 and R3. Differential and integral calculus of vector
valued functions of a vector variable, with emphasis on vectors in two and three
dimensional euclidean space. Extremal problems, Lagrange multipliers, line and surface
integrals, vector analysis, Stokes' theorem, Fourier series, calculus of variations. MAT240H1 A theoretical approach to: vector spaces over arbitrary
fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices,
change of basis, similarity, determinants. Polynomials over a field (including unique
factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial,
diagonalization. Minimal polynomial, Cayley-Hamilton theorem. MAT244H1 Ordinary differential equations of the first and second
order, existence and uniqueness; solutions by series and integrals; linear systems of
first order; non-linear equations; difference equations. MAT246Y1 Designed to introduce students to mathematical proofs and
abstract mathematical concepts. Topics may include modular arithmetic, prime numbers,
sizes of infinite sets, a proof that some angles cannot be trisected with straightedge and
compass, an introduction to group theory, or an introduction to topology. MAT247H1 A theoretical approach to real and complex inner product
spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint.
Hermitian and symmetric transformations. Spectral theorem for symmetric and normal
transformations. Polar representation theorem. Primary decomposition theorem. Rational and
Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear
forms, quadratic surfaces, multilinear algebra. Examples of symmetry groups and linear
groups, stochastic matrices, matrix functions. MAT257Y1 Topology of Rn; compactness, functions and continuity,
extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and
minima, Lagrange multipliers. Integrals; Fubini's theorem, partitions of unity, change of
variables. Differential forms. Manifolds in Rn; integration on manifolds; Stokes' theorem
for differential forms and classical versions. MAT267H1 First-order equations. Linear equations and first-order
systems. Non-linear first-order systems. Existence and uniqueness theorems for the Cauchy
problem. Method of power series. Elementary qualitative theory; stability, phase plane,
stationary points. Examples of applications in mechanics, physics, chemistry, biology and
economics. MAT299Y1 Credit course for supervised participation in faculty research project. See page 44 for details. MAT301H1 Congruences and fields. Permutations and permutation groups.
Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular
polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets,
Lagrange's theorem. Normal subgroups, quotient groups. Classification of finitely
generated abelian groups. Emphasis on examples and calculations. MAT302H1 Commutative rings; quotient rings. Construction of the
rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of
roots of a polynomial. Constructibility, trisection of angles, construction of regular
polygons. Galois groups of polynomials, in particular cubics, quartics. Solvable groups,
simple groups. Insolvability of quintics by radicals. MAT309H Predicate calculus. Relationship between truth and
provability; Godel's completeness theorem. First order arithmetic as an example of a
first-order system. Godel's incompleteness theorem; outline of its proof. Introduction to
recursive functions. MAT315H1 Elementary topics in number theory: arithmetic functions;
polynomials over the residue classes modulo m, characters on the residue classes modulo m;
quadratic reciprocity law, representation of numbers as sums of squares. MAT327H1 Metric spaces, topological spaces and continuous mappings;
separation, compactness, connectedness. Topology of function spaces. Fundamental group and
covering spaces. Cell complexes, topological and smooth manifolds, Brouwer fixed-point
theorem. MAT329Y1 The formation of mathematical concepts and techniques, and
their application to the everyday world. Nature of mathematics and mathematical
understanding. Role of observation, conjecture, analysis, structure, critical thinking and
logical argument. Numeration, arithmetic, geometry, counting techniques, recursion,
algorithms. MAT334H1 Theory of functions of one complex variable, analytic and
meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings,
introduction to analytic continuation and harmonic functions. MAT335H1 An elementary introduction to a modern and fast-developing
area of mathematics. One-dimensional dynamics: iterations of quadratic polynomials.
Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia
sets. History and applications. MAT337H1 Metric spaces; compactness and connectedness. Sequences and
series of functions, power series; modes of convergence. Interchange of limiting
processes; differentiation of integrals. Function spaces; Weierstrass approximation;
Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary
differential equations. Countability; Cantor set; Hausdorff dimension. MAT344H1 Basic counting principles, generating functions, permutations
with restrictions. Fundamentals of graph theory with algorithms; applications (including
network flows). Combinatorial structures including block designs and finite geometries. MAT347Y1 Groups, subgroups, quotient groups, Sylow theorems,
Jordan-Holder theorem, finitely generated abelian groups, solvable groups. Rings, ideals,
Chinese remainder theorem; Euclidean domains and principal ideal domains: unique
factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field
extensions, algebraic closure, straight-edge and compass constructions. Galois theory,
including insolvability of the quintic. MAT354H1 Complex numbers, the complex plane and Riemann sphere, Mobius
transformations, elementary functions and their mapping properties, conformal mapping,
holomorphic functions, Cauchy's theorem and integral formula. Taylor and Laurent series,
maximum modulus principle, Schwarz's lemma, residue theorem and residue calculus. MAT357H1 Function spaces; Arzela-Ascoli theorem, Weierstrass
approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces;
contraction mapping principle, fundamental existence and uniqueness theorem for ordinary
differential equations. Lebesgue integral; convergence theorems, comparison with Riemann
integral, Lp spaces. Applications to probability. MAT363H1 Geometry of curves and surfaces in 3-spaces. Curvature and
geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant
curvature. MAT365H1 Euclidean and non-Euclidean plane and space geometries. Real
and complex projective space. Models of the hyperbolic plane. Connections with the
geometry of surfaces. MAT390H1 Asurvey of ancient, medieval, and early modern mathematics
with emphasis on historical issues. (Offered in alternate years) MAT391H1 A survey of the development of mathematics from 1700 to the
present with emphasis on technical development. (Offered in alternate years) MAT393Y1/394Y1 TBA MAT395H1/396H1/397H1 TBA MAT398H0/399Y0 An instructor-supervised group project in an off-campus
setting. See page 44 for details. MAT409H1 Set theory and its relations with other branches of
mathematics. ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible
sets and the continuum hypothesis. Introduction to independence proofs. Topics from large
cardinals, infinitary combinatorics and descriptive set theory. MAT415H1 A selection from the following: finite fields; global and
local fields; valuation theory; ideals and divisors; differents and discriminants;
ramification and inertia; class numbers and units; cyclotomic fields; diophantine
equations. MAT417H1 A selection from the following: distribution of primes,
especially in arithmetic progressions and short intervals; exponential sums;
Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann
zeta-function; sieve methods, large and small; diophantine approximation, modular forms. MAT425H1 Smooth manifolds, Sard's theorem and transversality. Morse
theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector
fields and Euler characteristic. Hopf degree theorem. Additional topics may vary. MAT427H1 Introduction to homology theory: singular and simplicial
homology; homotopy invariance, long exact sequence, excision, Mayer-Vietoris sequence;
applications. Homology of CW complexes; Euler characteristic; examples. Singular
cohomology; products; cohomology ring. Topological manifolds; orientation; Poincare
duality. MAT443H1 Introduction to algebraic algorithms used in computer science
and computational mathematics. Topics may include: generating sequences of random numbers,
fast arithmetic, Euclidean algorithm, factorization of integers and polynomials, primality
tests, computation of Galois groups, Grobner bases. Symbolic maniputlators such as Maple
and Mathematica are used. MAT445H1 A selection of topics from: Representation theory of finite
groups, topological groups and compact groups. Group algebras. Character theory and
orthogonality relations. Weyl's character formula for compact semisimple Lie groups.
Induced representations. Structure theory and representations of semisimple Lie algebras.
Determination of the complex Lie algebras. MAT448H1 Basic notions of algebraic geometry, with emphasis on
commutative algebra or geometry according to the interests of the instructor. Algebraic
topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory,
power series rings and completion, dimension theory. Geometric topics: affine and
projective varieties, dimension and intersection theory, curves and surfaces, varieties
over the complex numbers. MAT449H1 Projective geometry. Curves and Riemann surfaces. Algebraic
methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic
curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of
singularities. MAT454H1 Harmonic functions, Harnack's principle, Poisson's integral
formula and Dirichlet's problem. Infinite products and the gamma function. Normal families
and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary
Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem. MAT457Y1 Measure theory and Lebesgue integration; convergence
theorems. Riesz representation theorem, Fubini's theorem, complex measures. Banach spaces;
Lp spaces, density of continuous functions. Hilbert spaces; weak and strong topologies;
self-adjoint, compact and projection operators. Hahn-Banach theorem, open mapping and
closed graph theorems. Inequalities. Schwartz space; introduction to distributions;
Fourier transforms on Rn (Schwartz space and L2). Spectral theorem for bounded normal
operators. MAT464H1 Riemannian metrics and connections. Geodesics. Exponential
map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scarlar
curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental
form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem.
Vector bundles. MAT468H1 Sturm-Liouville problem and oscillation theorems for
second-order linear equations. Qualitative theory; integral invariants, limit cycles.
Dynamical systems; invariant measures; bifurcations, chaos. Elements of the calculus of
variations. Hamiltonian systems. Analytic theory; singular points and series solution.
Laplace transform. MAT477H1 TBA MAT495H1/496H1/497H1/498Y1/499Y1 TBA |
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