Faculty of Arts & Science 2012-2013 Calendar

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Mathematics

Faculty

Mathematics teaches you to think, analytically and creatively. It is a foundation for advanced careers in a knowledge-based economy. Students who develop strong backgrounds in mathematics often have distinct advantages in other fields such as physics, computer science, economics, and finance.

The past century has been remarkable for 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 training-ground for Canadian mathematicians. A large proportion of our Mathematics Specialist graduates gain admission to the worlds 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 highly-motivated 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, and the physical sciences. 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.) In many cases it is possible to complete a Specialist program in Mathematics and its Applications with a given concentration along with a major in the other subject without taking many extra courses. You might even consider choosing your options to fulfill the requirements for a double Specialist degree, in both Mathematics and its Applications and in the other discipline.

The Specialist program in Mathematical Applications in Economics and Finance is recommended to students with career aspirations in any form of the financial sector. Furthermore, the program is an excellent preparation for an MBA and an MMF. The Professional Experience Year program (PEY: see index) is available to eligible, full-time Specialist students after their second year of study. The PEY program is an optional 12-16 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 non-credit course, PUMP, limited to students admitted to the University. It is designed for students who require additional pre-university mathematics background. Details can be found at www.math.toronto.edu/cms/pump.

Associate Chair for Undergraduate Studies: Bahen Building, 40 George Street, Room 6236
Student Counselling: Bahen Building, Room 6291 & NC64

Mathematics Aid Centre: Sidney Smith Hall, Room 1071

Departmental Office: Bahen Building, Room 6290 (416-978-3323)

Mathematics Programs

Enrolment in Mathematics programs requires completion of four courses; no minimum GPA is required.

Students with a good grade in MAT137Y1 (75%) or MAT135Y1/MAT135H1 & MAT136H1 (85%) may apply to the Mathematics Undergraduate Office for permission to enter a Mathematics program requiring MAT157Y1.

Enrolment in this program requires the completion of 4.0 courses.

(12.0 full courses or their equivalent, including at least 1.5 full courses at the 400-level)

The Specialist Program in Mathematics is directed toward students who hope to pursue mathematical research as a career.

First Year:
MAT157Y1, MAT240H1, MAT247H1
Second Year:
MAT257Y1, MAT267H1
Third and Fourth Years:
1. MAT327H1, MAT347Y1, MAT354H1, MAT357H1
2. One of: APM351Y1; MAT457Y1/(MAT457H1, MAT458H1)
3. Three of: APM461H1; MAT309H1, MAT363H1, ANY 400-level APM/MAT
4. 2.5 APM/MAT including at least 1.5 at the 400 level (these may include options above not already chosen)
5. MAT477Y1

NOTE:
1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and that CSC150H1 and STA257H1 be taken during the program. If you do not have a year course in programming from high school, the Department strongly recommends that you take CSC108H1 and then CSC148H1 instead of CSC150H1.
2. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
3. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Enrolment in this program requires the completion of 4.0 courses.

(13.5 full courses or their equivalent, including at least one full course at the 400-level)

The Specialist Program in Applied Mathematics is directed toward students who hope to pursue applied mathematical research as a career.

First Year:
MAT157Y1, MAT240H1, MAT247H1; (CSC108H1/CSC148H1)/CSC150H1
Second Year:
MAT257Y1, MAT267H1; CSC260H1; (STA257H1, STA261H1)
Third and Fourth Years:
1. APM351Y1; MAT327H1, MAT347Y1, MAT354H1, MAT357H1,MAT363H1; STA347H1
2. At least 1.5 full courses chosen from: MAT332H1, MAT344H1, MAT454H1, MAT457Y1/(MAT457H1, MAT458H1), MAT464H1; STA302H1, STA457H1; CSC350H1, CSC351H1, CSC446H1, CSC456H1
3. Two courses from: APM421H1, APM426H1, APM436H1, APM441H1, APM446H1, APM461H1, APM462H1, APM466H1
4. MAT477Y1

NOTE:
1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and STA257H1 be taken during the program. If you do not have a year course in programming from high school, the Department strongly recommends that you take (CSC108H1/CSC148H1) instead of CSC150H1.
2. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
3. Students planning to take specific fourth year courses should ensure that they have the necessary second and  third year prerequisites.

Enrolment in this program requires the completion of 4.0 courses.

(14-14.5 full courses or their equivalent, including at least one full course at the 400-level)

First Year:
MAT157Y1, MAT240H1, MAT247H1; PHY151H1, PHY152H1
Second Year:
MAT257Y1, MAT267H1; PHY224H1, PHY250H1, PHY252H1, PHY254H1, PHY256H1
Note: PHY252H1 and PHY324H1 may be taken in the 2nd or 3rd year.
Third Year:
1. APM351Y1; MAT334H1/MAT354H1, MAT357H1
2. One of: MAT327H1, MAT347Y1, MAT363H1
3. PHY324H1, PHY352H1, PHY354H1, PHY356H1
Fourth Year:
1. Two of: APM421H1, APM426H1, APM436H1; MAT446H1
2. Two of: PHY450H1, PHY452H1, PHY454H1, PHY456H1, PHY460H1
3. One of: MAT477Y1; PHY424H1, PHY478H1, PHY479Y1

NOTE:
1. Students who are intending to apply to graduate schools in mathematics would be well-advised to take MAT347Y1
2. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
3. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Enrolment in this program requires the completion of 4.0 courses.

Core Courses:

First Year:
CSC108H1/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Note:
CSC150H1 is required for the Computer Science concentration. If you do not have a year course in programming from high school, the Department strongly recommends that you take CSC108H1 and CSC148H1 in place of CSC150H1.
Second Year:
MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/MAT267H1;STA257H1 
Note:
1. MAT237Y1/MAT257Y1 is a direct or indirect prerequisite for many courses in each of the areas of concentration except the Teaching Concentration. Students are advised to take MAT237Y1/MAT257Y1 unless they have planned their program and course selection carefully and are certain that they will not need it.
Higher Years:
MAT301H1, MAT334H1
NOTE:
1. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/ PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Teaching Concentration:

It may be to students’ advantage to keep in mind that OISE requires students to have a second teachable subject.
1. MAT329Y1, HPS/MAT390H1, HPS/MAT391H1
2. Two of:MAT332H1/MAT344H1, MAT335H1, MAT337H1, MAT363H1
3. Two of: MAT309H1, MAT315H1; STA302H1/STA347H1
4. MAT401H1/MAT402H1 and one half course at the 400-level from MAT475H1, APM, STA

Enrolment in this program requires the completion of 4.0 courses.

Core Courses:

First Year:
CSC108H1/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Note:
CSC150H1 is required for the Computer Science concentration. If you do not have a year course in programming from high school, the Department strongly recommends that you take CSC108H1 and CSC148H1 in place of CSC150H1.
Second Year:
MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/MAT267H1;STA257H1
Note:
1. MAT237Y1/MAT257Y1 is a direct or indirect prerequisite for many courses in each of the areas of concentration except the Teaching Concentration. Students are advised to take MAT237Y1/MAT257Y1 unless they have planned their program and course selection carefully and are certain that they will not need it.
Higher Years:
MAT301H1, MAT334H1
NOTE:
1. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/ PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1

or similar courses in other departments.
2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Physical Sciences Concentration:

1. PHY151H1, PHY152H1; AST221H1
2. Three of: AST222H1; PHY250H1, PHY252H1, PHY254H1, PHY256H1
3. APM346H1/APM351Y1
4. Three of: AST320H1, AST325H1; MAT337H1, MAT363H1; PHY352H1, PHY354H1, PHY356H1, PHY357H1, PHY358H1
5. Two of: APM421H1, APM426H1, APM441H1, APM446H1; PHY407H1, PHY408H1, PHY456H1

This is a limited enrolment program.  All students who request the program and obtain at least the specified mark(s) in the required course(s) will be eligible to enrol. For more information, consult the department.

Core Courses:

First Year:
CSC108H1/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Note:
CSC150H1 is required for the Computer Science concentration. If you do not have a year course in programming from high school, the Department strongly recommends that you take CSC108H1 and CSC148H1 in place of CSC150H1.
Second Year:
MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/MAT267H1;STA257H1
Note:
1. MAT237Y1/MAT257Y1 is a direct or indirect prerequisite for many courses in each of the areas of concentration except the Teaching Concentration. Students are advised to take MAT237Y1/MAT257Y1 unless they have planned their program and course selection carefully and are certain that they will not need it.
Higher Years:
MAT301H1, MAT334H1
NOTE:
1. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Design-Your-Own Concentration:

Eleven half-courses of which at least six must be at the 300+ level including at least 2 at the 400 level. Choice of courses in program must be approved by the Department no later than the beginning of the third year or it will not be allowed. It is understood that the remaining 5 half-courses may be in the departments pertaining to the area of concentration.

This is a limited enrolment program.  All students who request the program and obtain at least the specified mark(s) in the required course(s) will be eligible to enrol.

Core Courses:

First Year:
CSC108H1/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Note:
CSC150H1 is required for the Computer Science concentration. If you do not have a year course in programming from high school, the Department strongly recommends that you take CSC108H1 and CSC148H1 in place of CSC150H1.
Second Year:
MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/MAT267H1;STA257H1
Note:
1. MAT237Y1/MAT257Y1 is a direct or indirect prerequisite for many courses in each of the areas of concentration except the Teaching Concentration. Students are advised to take MAT237Y1/MAT257Y1 unless they have planned their program and course selection carefully and are certain that they will not need it.
Higher Years:
MAT301H1, MAT334H1
NOTE:
1. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL201H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281H1/PHL295H1 or similar courses in other departments.
2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Probability/Statistics Concentration:

1. APM346H1/APM351Y1/APM462H1; MAT337H1; STA261H1, STA302H1, STA347H1, STA352Y1
2. One additional full credit at 300+level from APM/MAT/STA
3. Two of: STA437H1, STA438H1, STA442H1, STA447H1, STA457H1

Enrolment in this program requires the completion of 4.0 courses.

(12-13 full courses or their equivalent including at least 1.5 full courses at the 400-level)

First Year:
ECO100Y1 (70% or more); MAT137Y1 (55%)/MAT157Y1 (55%), MAT223H1, MAT224H1
Second Year:
ECO206Y1; MAT237Y1, MAT244H1, MAT246H1 (waived for students takING157Y1); STA257H1, STA261H1
PHL295H1 (Business Ethics): This course may be taken in second, third, or fourth year.
Third Year:
1. APM346H1; ECO358H1; ECO359H1; MAT337H1; (STA302H1, ECO375H1)/ECO327Y1; STA347H1
2. One of: MAT332H1, MAT344H1, MAT334H1, MAT475H1
Fourth Year:
APM462H1, APM466H1; STA457H1
NOTE:
1. Students who do not include PHL295H1 (Business Ethics) as part of their degree are expected to take another Arts and Science course with a significant emphasis on ethics and social responsibility.

2. Students planning to take specific fourth year courses should ensure that they have the necessary  third year prerequisites.

Consult the Undergraduate Coordinators of the Departments of Mathematics and Philosophy.

Specialist program:

(12 full courses or their equivalent including one full course at the 400-level)

First Year:
MAT157Y1, MAT240H1, MAT247H1; PHL245H1
Higher Years:
1. MAT257Y1, MAT327H1, MAT347Y1, MAT354H1/MAT357H1
2. PHL345H1/H5, MAT309H1/PHL348H1/5
3. Four of: PHL246H1/H5, PHL346H1/5, PHL347H1/H5, PHL349H1, PHL451H1/H5, PHL480H1
4. One full course from PHL200Y1/PHL205H1/PHL206H1/PHL210Y1
5. PHL265H1/PHL275H1
6. 2 additional PHL or MAT courses to a total of 12.0

NOTE:
The logic component of this program is offered jointly with the Department of Philosophy at the University of Toronto Mississauga. Students enrolling in this program must be prepared to travel to the UTM campus in order to complete program requirements with an H5 designation.

Enrolment in this program requires the completion of 4.0 courses.

(7.5 full courses or their equivalent including at least 2.5 full courses at the 300+ level and at least .5 courses at the 400 level).

First Year:
MAT135Y1/MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Second Year:
MAT224H1/ MAT247H1, MAT235Y1/ MAT237Y1, MAT244H1, MAT246H1; PHL275H1, or PHL265H1/PHL268H1/PHL271/ PHL273H1

NOTE:
1. MAT224H1 may be taken in first year
2. PHL275H1, or PHL265H1/PHL268H1/PHL271H1/ PHL273H1 may be taken in any year.
Higher Years:
1. MAT301H1, MAT309H1/MAT315H1, MAT334H1
2. One half course at the 200+ level from: ACT240H1; APM236H1; MAT309H1/MAT315H1/MAT335H1/ MAT337H1; STA247H1/STA250H1/STA257H1
3. One additional half course at the 300+level from: APM346H1, APM462H1; MAT309H1, MAT315H1, MAT332H1/MAT344H1, MAT335H1, MAT337H1, MAT475H1; HPS390H1, HPS391H1; PSL432H1
4. MAT401H1/MAT402H1
NOTES:
1. In the major program, higher level courses within the same topic are acceptable substitutions.  With a judicious choice of courses, usually including introductory computer science, students can fulfill the requirements for a double major in mathematics and one of several other disciplines.
2. Students are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL20H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1/PHL275H1/PHL281/PHL295H1 or similar courses in other departments.
4. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Enrolment in this program requires the completion of 4.0 courses.

(4 full courses or their equivalent)

1. MAT135Y1/MAT137Y1
2. MAT223H1, MAT235Y1/MAT237Y1, MAT224H1/MAT244H1/APM236H1Note: MAT223H1 can be taken in first year
3. one 300+level full course or combination from: any APM; MAT; HPS390H1, HPS391H1; PSL432H1
NOTE:
1. In the minor program, higher level courses within the same topic are acceptable substitutions.
2. Students planning to take specific third and fourth year courses should ensure that they have the necessary first, second and third year prerequisites.

Computer Science and Mathematics,  see Computer Science

Economics and Mathematics,  see Economics

Statistics and Mathematics, see Statistics

Mathematics Courses

The 199Y1 and 199H1 seminars are designed to provide the opportunity to work closely with an instructor in a class of no more than twenty-four students. These interactive seminars are intended to stimulate the students’ curiosity and provide an opportunity to get to know a member of the professorial staff in a seminar environment during the first year of study. Details here.

Applications of mathematics to biological problems in physiology, genetics, evolution, growth, population dynamics, cell biology, ecology, and behaviour. Mathematical topics include: power functions and regression; exponential and logistic functions; binomial theorem and probability; calculus, including derivatives, max/min, integration, areas, integration by parts, substitution; differential equations, including linear constant coefficient systems; dynamic programming; Markov processes; and chaos. This course is intended for students in Life Sciences.

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 three years)

 JUM202H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

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)

 JUM203H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

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 three years)

 JUM205H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

Introduction to linear programming including a rapid review of linear algebra (row reduction, matrix inversion, linear independence), the simplex method with applications, the duality theorem, complementary slackness, the dual simplex method and the revised simplex method.

Sturm-Liouville problems, Greens 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. Laplaces equation; Greens function. Schrdinger 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. Schrdinger and Heisenberg pictures, complete sets of observables, representations of the canonical commutative relations; essential self-adjointedness of Schrdinger operators, density operators, elements of scattering theory.

Einsteins theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einsteins 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; Greens 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.

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.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings.

NOTE: Transfer students who have received MAT1**H1 – Calculus with course exclusion to MAT133Y1/MAT135Y1 may take MAT137Y1/MAT157Y1 without forfeiting the half credit in Calculus.

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

MAT123H1, MAT124H1

See below MAT133Y1

MAT125H1, MAT126H1

See below MAT135Y1

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 MAT133Y1 may be deregistered from this course.

Note that for Rotman Commerce students there is no Breadth Requirement status for this course (and courses deemed equivalent in the program requirements in the calendar).

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 prerequisite purposes to MAT133Y1.

NOTE: students who enrol in MAT133Y1 after completing MAT123H1 but not MAT124H1 do not receive degree credit for MAT133Y1; it is counted ONLY as an “Extra Course.”

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. Functions, limits, continuity. Derivatives, rules of differentiation and implicit differentiation, related rates, higher derivatives of logarithms, exponentials. Trigonometric and inverse trigonometric functions, linear approximations. Mean value theorem, graphing, min-max problems, l’Hôpital’s rule; anti- derivatives.  Examples from life science and physical science applications.

Definite Integrals, Fundamental theorem of Calculus, Areas, Averages, Volumes. Techniques: Substitutions, integration by parts, partial fractions, improper integrals. Differential Equations: Solutions and applications. Sequences, Series, Taylor Series. Examples from life science and physical science applications.

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, Taylors 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. Taylors theorem; sequences and series; uniform convergence and power series.

System of linear equations, matrix algebra, real vector spaces, suspaces, span, linear dependence and independence, bases, rank, inner products, orthogonality, orthogonal complements, Gram-Schmidt linear transformations, determinants, Cramers rule, eigenvalues, eigenvector, eigenspaces, diagonalization.

Fields, complex numbers, vector spaces over a field, linear transformations, matrix of a linear transfromation, kernel, range, dimension theorem, isomorphisms, change of basis, eigenvalues, & eigenvectors, diagonalizability, real and complex inner products, spectral theorem, adjoint/self-adjoint/normal linear operators, triangular form, nilpotent mappings =, Jordan cannonical form.

Parametric equations and polar coordinates. Vectors, vector functions and space curves. Differential and integral calculus of functions of several variables. Line integrals and surface integrals and classic vector calculus theorems. Examples from life sciences and physical science applications.

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; Fubinis 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, Lagranges theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.

Predicate calculus. Relationship between truth and provability; Gdels completeness theorem. First order arithmetic as an example of a first-order system. Gdels 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 and development of fundamental reasoning and learning skills required for a future teacher to understand the nature of mathematics, their concepts and elementary techniques, and their application to the everyday world. Role of observation, conjecture, analysis, structure, critical thinking, logical argument and problem solving methods. Set and logic numbers, numeration and basic number theory, geometry, counting techniques, recursion and algorithms. This course, specifically addressed to students intending to become elementary school teachers and is strongly recommended by the Faculty of Education and may include a practicum in school classrooms. The course content is considered in the interestsof elementary school teachers. The enrolment has a limit of 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. Cauchys 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.

This course provides the foundations of analysis and rigorous calculus for students who will take subsequent courses where these mathematical concepts are central of applications, but who have only taken courses with limited proofs. Topics include topology of Rⁿ, implicit and inverse function theorems and rigorous integration theory.

Construction of Real Numbers. 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.

Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hlder 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, Cauchys theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarzs 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.

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.

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.

An instructor-supervised group project in an off-campus setting. Details here.

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.

Some courses at the 400-level are cross-listed as graduate courses and may not be offered every year. Please see the Departments graduate brochure for more details.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, Sards 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.

The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced).  The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now.  Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books).

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classification of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications

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, Grbner 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. Weyls 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 Hilberts 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; Bezouts theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities.

Harmonic functions, Harnacks principle, Poissons integral formula and Dirichlets 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.

Lebesque measure and integration; convergence theorems, Fubinis theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem. Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, Lp spaces, Holder and Minkowski inequalities.

Fourier series and transform, convergence results, Fourier inversion theorem, L2 theory, estimates, convolutions. Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.

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 youve never seen before? Students will apply Polyas 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.

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.

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.