|
|||
|
|
|
|
|
|
|||
|
|
|
|
|
MAT Mathematics CoursesMAT123H1
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. 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. 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; Stoke's 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
Proof theory: formal logic and G”del's incompleteness theorems. Introduction to the theory of 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-H”lder 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 thoerem, 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 geodisics. 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
A survey 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) MAT395H1/396H1/397H1
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. MAT398Y1/399Y1
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. 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. 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. MAT454H1
Harmonic functions, Harnack's principle, Poission'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
Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required) MAT495H1/496H1/497H1/498Y1/499Y1
Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. |
Calendar Home ~
Calendar Contents~
Contact Us ~
Arts and Science Home
Copyright © 1999, University of Toronto