MAT MATHEMATICSOn this page: Introduction  Faculty Members  Programs  Courses See also: Course Summer Timetable  Course Winter Timetable  Secondary School Information  More on Department IntroductionMathematics teaches you to think, analytically and creatively. It is a foundation for advanced careers in a knowledgebased economy. Students who develop strong backgrounds in mathematics often have distinct advantages in other fields such as physics, computer science, economics, and finance. This has been a century of remarkable discovery in mathematics. From space and number to stability and chaos, mathematical ideas evolve in the domain of pure thought. But the relationship between abstract thought and the real world is itself a source of mathematical inspiration. Problems in computer science, economics and physics have opened new fields of mathematical inquiry. And discoveries at the most abstract level lead to breakthroughs in applied areas, sometimes long afterwards. The University of Toronto has the top mathematics department in Canada, and hosts the nearby Fields Institute (an international centre for research in mathematics). The Department offers students excellent opportunities to study the subject and glimpse current research frontiers. The Department offers three mathematical Specialist programs — Mathematics, Applied Mathematics, Mathematics and its Applications — as well as Major and Minor programs, and several joint Specialist programs with other disciplines (for example, with Computer Science, Economics, Philosophy, Physics, and Statistics). The Specialist program in Mathematics is for students who want a deep knowledge of the subject. This program has been the main trainingground for Canadian mathematicians. A large proportion of our Mathematics Specialist graduates gain admission to the world's best graduate schools. The Specialist program in Applied Mathematics is for students interested in the fundamental ideas in areas of mathematics that are directed towards applications. The mathematics course requirements in the first two years are the same as in the Mathematics Specialist program; a strong student can take the courses needed to get a degree in both Specialist programs. These programs are challenging, but small classes with excellent professors and highlymotivated students provide a stimulating and friendly learning environment. The Specialist program in Mathematics and its Applications is recommended to students with strong interests in mathematics and with career goals in areas such as teaching, computer science, the physical sciences and finance. The program is flexible; there is a core of courses in mathematics and related disciplines, but you can choose among several areas of concentration. The mathematics courses required for the program are essentially the same as those required for a Major in Mathematics. (They are less intense than the courses required for the Specialist programs above.) If you are interested in mathematics and are contemplating a double Major in Mathematics and in another discipline (let us take Computer Science, as an example), you should consider the advantages of fulfilling the requirements for a Specialist degree in Mathematics and its Applications with a computer science concentration. In this way, you can also get a Major in Computer Science; the difference in course requirements with a double major is that, among the courses you can choose for a Computer Science Major, you will be required to take some of a more mathematical nature. You might even consider choosing your options to fulfil the requirements for a double Specialist degree, in both Mathematics and its Applications and in the other discipline. The Professional Experience Year program ("PEY": see Study Elsewhere Program Options) is available to eligible, fulltime Specialist students after their second year of study. The PEY program is an optional 16 month work term providing industrial experience; its length often allows students to have the rewarding experience of initiating and completing a major project. The Department operates a noncredit summer course, PUMP, limited to students admitted to the University. It is designed for students from outside Ontario who require additional preuniversity mathematics background. Subject to available space, some Ontario students may also be accepted provided that they already have an OAC calculus credit.
Associate Chair for Undergraduate Studies: Professor E. Bierstone, Sidney Smith Hall, 100 St. George Street, Room 4072 (9785164) Student Counselling: Sidney Smith Hall, Room 4072 Mathematics Aid Centres: Sidney Smith Hall, Room 1071; University College, Room UC48 Departmental Office: Sidney Smith Hall, Room 4072 (9783323) Faculty Members
NOTE: In Prerequisites, Calc = Ontario Academic Course Calculus; A&G = Ontario Academic Course Algebra and Geometry; FM = Ontario Academic Course Finite Mathematics MATHEMATICS PROGRAMSEnrolment in the Mathematics programs requires completion of four courses; no minimum GPA is required. APPLIED MATHEMATICS (Hon.B.Sc.)Consult Professor E. Bierstone, Associate Chair, Department of MathematicsSpecialist program: S20531 (12 full courses or their equivalent, including at least one 400series course)
Third or Fourth Years:
ACT 335H/CSC 350H; APM 361H, 426H, 436H, 441H, 456H, 461H, 466H; CSC 351H, 446H, 456H; MAT 344H, 364H, 437H, 467H; STA 347H, 348H, 352Y, 450H, 457H Note: The Department recommends that students acquire a reading knowledge of French. MATHEMATICS (B.Sc.)Consult Professor E. Bierstone, Associate Chair, Department of MathematicsSpecialist program (Hon.B.Sc.): S11651 (11 full courses or their equivalent, including at least one 400series course)
Recommended choices: APM 351Y, MAT 364H NOTE: The Department recommends that PHY 140Y be taken in First Year, that CSC 148H and STA 347Y/352Y be taken during the program, and that students acquire a reading knowledge of French Major program Major program: M11651 (at least 7.5 full courses or their equivalent)
NOTE: MAT 223H, 224H, 240H may be taken in first year
Minor program Minor program: R11651 (4 full courses or their equivalent)
NOTE: MAT 329Y may not be counted towards the requirements of this program MATHEMATICS AND ITS APPLICATIONS (Hon.B.Sc.)Consult Professor E. Bierstone, Associate Chair, Department of MathematicsSpecialist program: (1112full courses or their equivalent including one full course at the 400level) The program requirements are the core courses below, together with the courses in one of the following areas of concentration. If you get a specialist degree in Mathematics and its Applications, your transcript and degree will indicate also your area of concentration (except in the case of a DesignYourOwn concentration). CORE COURSES:First Year: CSC 148H/260H; MAT 135Y/137Y/157Y (MAT 137Y strongly recommended), 223H First or Second Year: STA 107H/250H (waived for students taking MAT 257Y)
MAT 246Y/(CSC 238H, PHL 245H), MAT 244H/267H
AREAS OF CONCENTRATION: Teaching Concentration (S15801):
Computer Science Concentration (S15951):
Note: In order to take the Computer Science concentration, you will be required to register also for a Computer Science Major. (The latter is a restricted enrolment program and has certain admission requirements; please see the Computer Science program description.) Finance Concentration (S17001): ACT 451H, 466H; APM 346H, 466H; CSC 350H, 446H; MAT 338H; STA 347H, 348H, 457H Note: Students concentrating in Finance are encouraged also to take appropriate courses in Economics and Management. Physical Sciences Concentration (S17581):
Probability/Statistics Concentration (S18901):
DesignYourOwn Concentration (S20001): Nine halfcourses of which at least six must be at the 300/400level, to be approved by the Department no later than the beginning of your third year. MATHEMATICS AND COMPUTER SCIENCE — See COMPUTER SCIENCE MATHEMATICS AND ECONOMICS — See ECONOMICS MATHEMATICS AND PHILOSOPHY (Hon.B.Sc.)Consult Departments of Mathematics and Philosophy.Specialist program: S13611 (13.5 full courses or their equivalent including one full course at the 400level)
MATHEMATICS AND PHYSICS (Hon.B.Sc.) Consult Professor E. Bierstone, Associate Chair, Department of Mathematics, and the Associate Chair, Department of Physics. Specialist program: S03971 (13 full courses or their equivalent, including at least one 400series course)
MATHEMATICS AND STATISTICS — See STATISTICS APPLIED MATHEMATICS COURSES(see Section 4 for Key to Course Descriptions)For Distribution Requirement purposes, all APM courses are classified as Science courses.
APM233Y 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.
APM236H 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.
APM261H Formulation of problems in LP form, convexity and structure of LP constraint sets, simplex algorithm, degeneracy, cycling and stalling, revised method, twophase method, duality, fundamental theorem, dual algorithm, integer programming, sensitivity analysis, Karmarkar algorithm, network flows, transportation algorithm, twoperson zerosum games.
APM346H SturmLiouville problems, Green's functions, special functions (Bessel, Legendre), partial differential equations of second order, separation of variables, integral equations, Fourier transform, stationary phase method.
APM351Y Distributions, elliptic, parabolic and hyperbolic partial differential equations in their physical contexts, transforms, conformal mapping. Green's functions, selfadjoint operators and eigenfunction expansions, variational methods.
APM361H 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.
APM366H Convexity, fixed points, stable mappings optimization. Relations orderings and utility functions; choice and decision making by individuals and groups. Noncooperative and cooperative games, core, Shapley value; market games. Decision making by economic agents: consumers, producers, banks, investors, and financial intermediaries.
APM421H The general formulation of nonrelativistic quantum mechanics based on the theory of linear operators in a Hilbert space, selfadjoint 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 selfadjointness of Schrödinger operators, density operators, elements of scattering theory.
APM426H Local and global geometries of Lorentz manifolds, stationary and statics spacetimes. Einstein field equations, Schwarzchild, Kruskal and Kerr solutions. Mathematics of black holes. Relativistic cosmology, big bang and inflationary models. Cauchy problem and PetrovNewmanPenrose classifications.
APM436H Formulation of NavierStokes equations, exact solutions. Slow viscous flow. Boundary layers and singular perturbations. Wave propagation, stratified fluids; stability. Compressible flow. The main emphasis is the description of basic physical phenomena determined from simple analytical solutions of the governing equations.
APM441H 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 coordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering)
APM446H A survey of methods used for the analysis of nonlinear differential equations in physics and engineering. Nonlinear dynamics: limit cycles, stability, bifurcations, chaos. Solitons.
APM456H Theory of extrema for constrained problems; nonlinear programming, the calculus of variations, optimal control theory and application to engineering and management science.
APM461H A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
APM466H Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, singleperiod finance, financial derivatives (treeapproximation and BlackScholes model for equity derivatives, American derivatives, numerical methods, lattice models for interestrate derivatives), value at risk, credit risk, portfolio theory.
APM496H/497H/498Y/499Y Readings in Applied Mathematics Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
MATHEMATICS COURSES(see Section 4 for Key to Course Descriptions)For Distribution Requirement purposes, all MAT courses except MAT 123H, 124H and 133Y are classified as SCIENCE courses.
SCI199Y 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 First Year Seminars: 199Y.
JUM102H 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)
JUM102H
JUM103H 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)
JUM103H
JUM105H An indepth 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 three years)
JUM105H
MAT123H,124H: see below MAT 133Y MAT125H,126H: see below MAT 135Y
MAT133Y Mathematics of finance. Matrices and linear equations. Review of differential calculus; applications. Integration and fundamental theorem; applications. Introduction to partial differentiation; applications. NOTE: please note prerequisites listed below. Students without the proper prerequisites for MAT133Y may be deregistered from this course.
MAT133Y
MAT123H 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.
NOTE: students who enrol in MAT133Y after completing MAT123H but not MAT124H do not receive degree credit for MAT133Y; it is counted ONLY as an "Extra Course."
MAT123H
MAT124H 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.
MAT124H
MAT135Y Review of differential calculus; applications. Integration and fundamental theorem; applications. Series. Introduction to differential equations.
MAT125H 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.
NOTE: students who enrol in MAT135Y after completing MAT125H but not MAT126H do not receive degree credit for MAT135Y; it is counted ONLY as an "Extra Course."
MAT126H 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.
MAT137Y 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.
MAT157Y A complete theoretical foundation for the concepts in calculus of one variable, with emphasis on mathematical rigour. Covers all standard techniques of calculus: limits and continuity, intermediate and extreme value theorems, derivative, mean value and inverse function theorems, the integral, fundamental theorem, elementary transcendental functions, Taylor's theorem, sequence and series, uniform convergence and power series.
JMB170Y Applications of mathematics to biological problems in physiology, biomechanics, genetics, evolution, growth, population dynamics, cell biology, ecology and behaviour.
MAT223H 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, GramSchmidt process, orthogonal transformations and orthogonal diagonalization, isometries, quadratic forms, conics, quadric surfaces.
MAT224H Fields. Vector spaces over a field. Linear transformations, dual spaces. Diagonalizability, direct sums. Invariant subspaces, CayleyHamilton theorem. Complex inner product, orthogonality, the adjoint of a linear operator, the projection matrix and the method of least squares. Normal, selfadjoint and unitary operators. Spectral theorem. Conditioning and Rayleigh quotient. Jordan canonical form.
MAT235Y 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.
MAT237Y 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.
MAT240H A theoretical approach to: vector spaces over arbitrary fields (including the integer modulo a prime number). Subspaces, bases and dimension. Linear transformations, matrices, change of basis. Determinants. Eigenvalues, eigenvectors, diagonalization.
MAT244H Ordinary differential equations of the first and second order, existence and uniqueness; solutions by series and integrals; linear systems of first order; nonlinear equations; difference equations.
MAT246Y Transition to abstract mathematics. Foundations of mathematics through sets and structures. This is a transitional course from the more manipulative mathematics to abstraction and rigorous proof. Topics include construction of number systems, groups and symmetry, topology and sums of squares via quaternions.
MAT247H A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices. The spectral theorems for symmetric and normal operators; Jordan canonical forms; introduction to group theory: subgroups, symmetric groups, normal subgroups and quotient groups, cyclic groups, homomorphism and isomorphism theorems, direct products.
MAT257Y Topology of Rn: compactness, connectedness, completeness. Continuous functions on Rn: extreme and intermediate value theorems, uniform continuity. Differential calculus in Rn: differentiability and directional derivatives, maxima and minima, implicit and inverse function theorems, Lagrange multipliers. Uniform convergence and Arzelà's theorem. Multiple integrals: existence, properties, transformation and evaluation of integrals. Line and surface integrals. Theorems of Green and Stokes and the divergence theorem.
MAT267H First order and linear ordinary differential equations, first order linear systems. Laplace transform; existence and uniqueness of solutions; series solutions. Elementary qualitative theory.
MAT299Y Credit course for supervised participation in faculty research project. See Research Opportunity Program for details.
MAT301H An introduction to rings and fields, covering the standard topics of integral domains; ideals, quotient rings and homomorphisms; polynomial rings and factorization; divisibility in domains (unique factorization domains and euclidean domains); extension fields. These concepts are then applied to geometric constructions (i.e. impossibility of constructing angle trisectors by ruler and compass), finite fields and algebraic coding theory.
MAT302H An introduction to the rudiments of group theory illustrated extensively via the notion of euclidean geometry. Thus, throughout the presentation, emphasis is placed on concrete examples, often geometrical in nature, so that finite rotation groups of the platonic solids, the seven frieze groups and the 17 wallpaper groups are examined in detail along side theoretical considerations such as subgroups and Langrange's theorem, quotient groups and homomorphisms, the Sylow theorems, finite simple groups and finite abelian groups.
MAT309H Proof theory: formal logic and Gödel's incompleteness theorems. Introduction to the theory of recursive functions.
MAT314H Elementary set theory. Elementary topology, metric spaces, convexity in linear spaces, fixed point theorems.
MAT315H 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.
MAT323H Affine and projective spaces derived from vector spaces; semilinear mappings and collineations; Desargues' and Pappus' theorems; projectivities, projective collineations and cross ratio; dualities; polarities and conics.
MAT325H Motions, dilatations and Desargues' theorems, similarities and collineations in the euclidean plane; inversions and the conformal plane; models of the hyperbolic and real projective planes and their transformations.
MAT329Y 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 is taught jointly by the Department of Mathematics and the Faculty of Education. 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.
MAT334H 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.
MAT335H An elementary introduction to a modern and fastdeveloping area of mathematics. Onedimensional dynamics: iterations of quadratic polynomials. Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia sets. History and applications.
MAT338H Metric spaces, completeness, uniform convergence. Topics in measure theory: the Lebesgue integral, RiemannStieltjes integral, Lp spaces, Fourier series.
MAT344H 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.
MAT347Y Groups: definitions, Lagrange's theorem, factor groups, isomorphism theorems, conjugacy, Burnside's lemma, Cayley representation. Sylow theorems, JordanHölderSchreier theorem, solvability and nilpotence. Rings and modules: euclidean algorithm, unique factorization domains, principal ideal domains, Chinese remainder theorem, Gaussian integers, symmetric functions, location of roots, free and finitely generated modules, Noetherian rings and modules, Hilbert basis theorem, abelian groups as Zmodules, tensor, symmetric and skewsymmetric algebras.
MAT357Y Functions of bounded variation, Fourier series, Lebesgue integration in Rn. Complex numbers, the complex plane and Riemann sphere, holomorphic functions, conformal mapping, Mobius transformations, elementary functions and their mapping properties, Cauchy's theorem and integral formula. Taylor and Laurent series, uniqueness theorem, maximum modulus theorem, Schwarz's lemma, residue theorem and residue calculus, argument principle, Rouche's theorem.
MAT363H Curves and surfaces in Euclidean 3space. SerretFrenet frames and the associated equations, the first and second fundamental forms and their integrability conditions, intrinsic geometry and parallelism, the GaussBonnet theorem.
MAT364H This course treats generalizations of the concepts introduced in MAT363H although it is logically independent. Smooth manifolds; tensor fields; calculus on manifolds, riemannian structures and the associated geodesics, parallelism and curvature. Submanifolds and immersions. Lie groups.
MAT390H A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)
MAT391H A survey of the development of mathematics from 1700 to the present with emphasis on technical development. (Offered in alternate years)
MAT395H/396H/397H Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
MAT398Y/399Y Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
MAT437H Analytic continuation, the monodromy theorem, harmonic functions, the Dirichlet problem, Harnack's principle, normal families, Picard theorem, infinite products, gamma function, the Riemann mapping theorem, boundary behaviour, reflection principle.
MAT447H Algebraic and transcendental extensions of fields, existence of algebraic closures, Galois theory with applications to classical problems, finite fields, Kummer theory.
MAT457Y Abstract measure spaces, properties of measures. Extension theorems. Lebesgue measure. Definition and properties of the integral, convergence theorems, RadonNikodym theorem. Riesz representation theorem. Product spaces, Fubini's theorem. Banach spaces, Lp spaces, dual spaces, HahnBanach theorem. Linear operators, spectrum, openmapping and closed graph theorems. Hilbert spaces, normal operators, orthogonal projections.
MAT467H Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required)
MAT495H/496H/497H/498Y/499Y Readings in Mathematics Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.
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