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MAT Mathematics CoursesMAT123H/124H see below MAT 133Y 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. MAT125H/126H see below MAT 135Y 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. MAT133Y1 Mathematics of finance. Matrices and linear equations. Review of differential calculus;
applications. Integration and fundamental theorem; applications. Introduction to partial
differentiation; applications. MAT135Y1 Review of differential calculus; applications. Integration and fundamental theorem;
applications. Series. Introduction to differential equations. 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. 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. 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 42 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; 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. 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 Introduction to Real Analysis (replaces MAT338H) 39L 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-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. MAT354H1 (formerly part of MAT357Y) 39L, 13T MAT357H1 (formerly part of MAT357Y) 39L, 13T MAT363H1 Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces.
Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature. MAT365H1 Classical Geometries (replaces MAT325H) 39L 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 Independent study under the direction of a faculty member. Topic must be outside
undergraduate offerings. MAT395H1/396H1/397H1 Independent study under the direction of a faculty member. Topic must be outside
undergraduate offerings. MAT398H0/399Y0 An instructor-supervised group project in an off-campus setting. See page 42 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, Gröbner 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 (formerly MAT437H) 39L 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 (formerly MAT364H) 39L 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 (formerly MAT467H) TBA MAT495H1/496H1/497H1/498Y1/499Y1 Independent study under the direction of a faculty member. Topic must be outside
undergraduate offerings. |
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