## Applied Mathematics CoursesFor Distribution Requirement purposes, all APM courses are classified as Science courses.
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.
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.
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). Note: Some courses at the 400-level are cross-listed as graduate courses and may not be offered every year. Please see the Department’s graduate brochure for more details.
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 scattering theory.
Einstein’s theory of gravity. Special relativity and
the geometry of Lorentz manifolds. Gravity as a manifestation 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.
Boltzmann, Euler
and Navier-Stokes
equations.
Viscous and non-viscous
flow. Vorticity.
Exact solutions.
Boundary
layers. Wave
propagation. Analysis of
one dimensional gas flow.
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)
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.
Propagation of
nonlinear waves;
dispersion, modulation,
optical bistability.
Global
behaviour solutions; asymptotics and blow-up.
A selection of
topics from such
areas as
graph theory, combinatorial
algorithms, enumeration,
construction of
combinatorial identities.
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.
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.
Independent
study under
the direction
of a
faculty
member. Topic must
be
outside
undergraduate offerings. ## Mathematics CoursesFor Distribution Requirement purposes, all MAT courses except MAT123H1, 124H1 AND133Y1 are classified as SCIENCE courses (see page 26). High school prerequisites for students coming from outside the Ontario high school system: - MAT133Y1: high school level calculus and (algebra-geometry or finite math or discrete math)
- MAT135Y1: high school level calculus
- MAT137Y1: high school level calculus and algebra-geometry
- MAT157Y1: high school level calculus and algebra-geometry
- MAT223H1: high school level calculus and algebra-geometry
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 distribution requirement course; Details here.. NOTE: Transfer students who have received MAT1**H1 – Calculus with course exclusion to MAT133Y1/135Y1/136Y1 may take MAT137Y1/157Y1 without forfeiting the half credit in Calculus.
See below MAT 133Y1
See below MAT 135Y1
Mathematics of finance. Matrices and linear equations.
Review of differential calculus; applications.
Integration and fundamental
theorem; applications.
Introduction to partial differentiation; applications.
First term of MAT133Y1. Students in academic difficulty in MAT133Y1 who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT133Y1 and enrol in MAT123H1 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 MAT133Y1 in the Fall Term are not allowed to enrol in MAT123H1. MAT123H1 together with MAT124H1 is equivalent for program and Second Term content of MAT133Y1; the final examination includes topics
covered in MAT123H1. Offered in the Summer Session only; students not enrolled
in MAT123H1
in the preceding Spring Term will NOT be allowed to enrol in MAT124H1. MAT123H1
together with MAT124H1 is equivalent for program and prerequisite purposes
to MAT133Y1.
Review of trigonometric functions; trigonometric identities and trigonometric limits. Review of differential calculus; applications. Integration and fundamental theorem; applications. Series. Introduction to differential equations.
First term of MAT135Y1. Students in academic difficulty in MAT135Y1 who have written two midterm examinations with a mark of at least 20% in the second may withdraw from MAT135Y1 and enrol in MAT125H1 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 MAT135Y1 in the Fall Term will not be allowed to enrol in MAT125H1. MAT125H1 together with MAT126H1 is equivalent for program and
Second Term content of MAT135Y1; the final examination includes topics covered in MAT125H1. Offered in the Summer Session only; students not enrolled in MAT125H1 in the preceding Spring Term will NOT be allowed to enrol in MAT126H1. MAT125H1 together with MAT126H1 is equivalent for program and
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 the level of MAT135Y1. May include background material on functions, analytic geometry, and trigonometry, as well as on calculus.
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 first a review of trigonometric functions followed by discussion of trigonometric identities. 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.
A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. 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.
Applications of mathematics to biological problems in physiology, biomechanics, genetics, evolution, growth, population dynamics, cell biology, ecology and behaviour.
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 may be considered. (Offered every four years)
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)
An interdisciplinary exploration of creativity and imagination as they arise in the study of mathematics and poetry. The goal of the course is to guide each participant towards the experience of an independent discovery. Students with and without backgrounds in either subject are welcome. No calculus required. (Offered every three years)
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, Gödel, Erdös, Coxeter, Grothendieck. (Offered every four years)
Matrix arithmetic and linear systems. Rn subspaces, linear independence, bases, dimension; column spaces, null spaces, rank and dimension formula. Orthogonality orthonormal sets, Gram-Schmidt orthogonalization process; least square approximation. Linear transformations Rn —>Rm . The determinant, classical adjoint, Cramer’s Rule. Eigenvalues, eigenvectors, eigenspaces, diagonalization. Function spaces and application to a system of linear differential equations.
Abstract vector spaces: subspaces, dimension theory. Linear mappings: kernel, image, dimension theorem, isomorphisms, matrix of linear transformation. Changes of basis, invariant spaces, direct sums, cyclic subspaces, Cayley-Hamilton theorem. Inner product spaces, orthogonal transformations, orthogonal diagonalization, quadratic forms, positive definite matrices. Complex operators: Hermitian, unitary and normal. Spectral theorem. Isometries of R2 and R3 .
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.
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.
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.
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. Applications in life and physical sciences and economics.
Designed to introduce students to mathematical proofs and abstract mathematical concepts. Topics may include modular arithmetic, sizes of infinite sets, and a proof that some angles cannot be trisected with straightedge and compass.
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.
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.
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.
This breadth course is accessible to students with limited mathematical background. Various mathematical techniques will be illustrated with examples from humanities and social science disciplines. Some of the topics will incorporate user friendly computer explorations to give participants the feel of the subject without requiring skill at calculations.
Credit course for supervised participation in faculty research project. Details here.
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.
Predicate calculus. Relationship between truth and provability; Gödel’s completeness theorem. First order arithmetic as an example of a first-order system. Gödel’s incompleteness theorem; outline of its proof. Introduction to recursive functions.
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.
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. Students in the math specialist program wishing to take additional topology courses are advised to obtain permission to take MAT1300Y. Students must meet minimum GPA requirements as set by SGS and petition with their college.
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. This course is specifically addressed to students intending to become elementary school teachers and is strongly recommended by the Faculty of Education. Previous experience working with children is useful. The course content is considered in the context of elementary school teaching. In particular, the course may include a practicum in school classrooms. The course has an enrolment limit of 40, and students are required to ballot.
This course will explore the following topics: Graphs, Subgraphs, Isomorphism,
Trees, Connectivity, Euler and Hamiltonian Properties, Matchings, Vertex
and Edge Colourings, Planarity, Network Flows and Strongly Regular Graphs.
Participants will be encouraged to use these topics and execute applications
to such problems as timetabling, tournament scheduling, experimental design
and finite geometries. Students are invited to replace MAT344H1 with MAT332H1.
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.
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.
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.
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.
(Not offered in 2009-2010).
Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hölder 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.
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.
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.
Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.
A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)
A survey of the development of mathematics fROM 1700 to the present with emphasis on technical development. (Offered in alternate years)
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
An instructor-supervised group project in an off-campus setting. Details here.
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. Insolvability of quintics by radicals.
Euclidean and non-euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.
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.
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.
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.
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.
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.
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, Gröbner bases. Symbolic manipulators such as Maple and Mathematica are used.
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.
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.
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.
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.
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.
Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds. Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors. Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from: Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.
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.
This course addresses the question: “How do you attack a problem the
likes of which you’ve never seen before?” Students will apply Polya’s principles
of mathematical problem solving, draw upon their previous mathematical
knowledge,
and explore the creative side of mathematics in solving a variety of interesting
problems and explaining those solutions to others.
Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required)
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. |