Faculty of Arts & Science 2016-2017 Calendar

Mathematics

• Mathematics Specialist | Mathematics Major | Mathematics Minor
• Applied Mathematics Specialist
• Mathematics and Physics Specialist
• Mathematics and Philosophy
• Mathematics & Its Applications Specialist (Teaching)
• Mathematics & Its Applications Specialist (Physical Science)
• Mathematics & Its Applications Specialist (Probability/Statistics)
• Mathematical Applications in Economics and Finance Specialist
• Combined Degree Program (CDP) in Science and Education: Mathematics (Major), Honours Bachelor of Science/Master of Teaching
• Mathematics Courses

Faculty

Mathematics is the study of shape, quantity, pattern and structure. It serves as a tool for our scientific understanding of the world. Knowledge of mathematics opens gateways to many different professions such as economics, finance, computing, engineering, and the natural sciences. Aside from practical considerations, mathematics can be a highly satisfying intellectual pursuit, with career opportunities in teaching and research.

The department counts many of Canada's leading research mathematicians among its faculty. Our mathematics programs are flexible, allowing students to select courses based on specialization and interest. Contents range from calculus and linear algebra in the non-specialist programs to more advanced topics such as real and complex analysis, ordinary and partial differential equations, differential geometry, topology, commutative algebra, graph theory, mathematical logic, number theory, and functional analysis.

The department offers eight specialist programs in addition to the major and minor programs.

In the Mathematics, Applied Mathematics, Mathematics and Physics, and Mathematics and Philosophy specialist programs, students acquire an in-depth knowledge and expertise in mathematical reasoning and the language of mathematics, with its emphasis on rigor and precision. These programs are designed for students wishing to pursue graduate studies; most of the graduates of these programs continue on to graduate school with some of them gaining admission to the world’s best graduate schools.

The Mathematical Applications in Economics and Finance specialist program is designed to prepare students for direct entry into the world of finance. It can also serve as a gateway to an MBA or a Master of Finance degree, possibly followed by an eventual doctorate.

The Mathematics and its Applications specialist programs offer three areas of concentration: teaching, physical science, and probability/statistics. These specialist programs are designed as `enhanced double majors.' The required courses for these concentrations are almost identical for the first two years, but they diverge in the upper years. Students in these programs can also continue on to graduate studies.

The Major and Minor programs are intended for students who want to combine mathematical skills with work in other subjects. These programs require less coursework than the specialist programs, but still require the completion of some upper year mathematics courses. 

Students interested in becoming K-12 teachers should consider applying to the combined degree program --- a six-year program that leads to an Honours Bachelor of Science (HBSc) from the University of Toronto and a Master of Teaching (MT) from the Ontario Institute for Studies in Education (OISE).  The HBSc part of this program involves completing a Math Major, a Minor in Education and Society (offered by Victoria College) and a Minor in an area that would lead to a second "teachable" subject.  Please see the Victoria College website for more information.

The Professional Experience Year program (PEY: see http://engineeringcareers.utoronto.ca/students/pey/ ) is available to eligible full-time Specialist and Major students after their second or third year of study. The PEY program is an optional 12-16 month work term providing industrial experience. It gives students an opportunity to apply their skills in the context of a paid internship.

The Department of Mathematics offers introductory courses for incoming students to foster the development of mathematics skills.

PUMP (Preparing for University Mathematics Program) is a non-credit course that equips students with the necessary background knowledge required to succeed in first year mathematics courses.  It is designed for students who have not taken the appropriate high school mathematics prerequisites for university calculus and linear algebra. It is also useful for students who wish to close any existing gap between high school math and University level math courses or anyone who wishes to review high school math before attempting University level math or other science courses.

MAT138H1 (Introduction to Proofs) has been introduced into the curriculum as a preparation for MAT157Y1, MAT240H1, MAT247H1, MAT237Y1, and other proof-oriented advanced courses.  The course covers the reading and comprehension of mathematical statements, analyzing definitions and properties, formulation of arguments, and strategies for proofs.

Visit http://www.math.toronto.edu/cms/potential-students-ug/ for up-to-date information on the availability of PUMP and MAT138H1.

Some of the more advanced first- and second-year courses have "change dates" during the first few weeks of the academic year.  The "change date" occurs after the general "add date" for courses and before the "drop date" for courses.  For example, a student enrolled in Mat157Y1 can change their enrolment to MAT137Y1 or MAT135H1 at any time on or before the change date. For deadlines and further details, see http://www.math.toronto.edu/cms/change-dates

Associate Chair for Undergraduate Studies: Professor M. Pugh

Enquiries and student counseling: Bahen Centre, Room 6291

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

Websites: http://www.math.toronto.edu/cms/potential-students-ug/

http://www.math.toronto.edu/cms/current-students-ug/

Mathematics Programs

Students with a good grade in MAT223H1 (80%) and MAT224H1 (80%) and MAT137Y1 (75%) (or (MAT135H1, MAT136H1) (85%)) may apply to the Mathematics Undergraduate Office for permission to enter a Mathematics program requiring MAT157Y1. For such students, MAT138H1 is highly recommended.

(12.5 FCE, including at least 3.0 FCE 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

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note: Students may use the CR/NCR option with this H course and have it count toward the Mathematics Specialist program. Students in the VIC program may also use VIC172Y1.

2. MAT327H1

Third and Fourth Years:
1. MAT347Y1, MAT354H1, MAT357H1, MAT363H1/MAT367H1 (MAT363H1 can be taken in the second year, if desired)
2. 2.0 FCE of: MAT309H1, MAT351Y1, ANY 400-level APM/MAT
3. 3.0 FCE of APM/MAT at the 300+ level, including at least 2.0 FCE at the 400 level (these may include options above not already chosen)
4. MAT477H1

NOTE:
1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and that CSC148H1 and STA257H1 be taken during the program. If you do not have a year-long course in programming from high school, the Department strongly recommends that you take CSC108H1 prior to CSC148H1.

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

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

(13.0-13.5 FCE, including at least 1.5 FCE 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; (STA257H1, STA261H1)
Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.
Third and Fourth Years:
1. MAT351Y1; MAT327H1, MAT347Y1, MAT354H1, MAT357H1, MAT363H1/MAT367H1 (MAT363H1 can be taken in the second year, if desired); STA347H1
2. At least 1.5 FCE chosen from: MAT332H1, MAT344H1, MAT454H1, MAT457H1, MAT458H1, MAT464H1; STA302H1, STA457H1; CSC336H1, CSC436H1, CSC446H1, CSC456H1
3. 1.0 FCE from: APM421H1, APM426H1, APM441H1, APM446H1, APM461H1, APM462H1, APM466H1
4. MAT477H1

NOTE:
1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and that CSC148H1 and STA257H1 be taken during the program. If you do not have a year-long course in programming from high school, the Department strongly recommends that you take CSC108H1 prior to CSC148H1.

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

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

(14.5-15.5 FCE, including at least 1.0 FCE at the 400-level)

First Year:
MAT157Y1, MAT240H1, MAT247H1; PHY151H1, PHY152H1
Second Year:
MAT257Y1, MAT267H1; PHY224H1, PHY250H1, PHY252H1, PHY254H1, PHY256H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172.

2. Note: PHY252H1 and PHY324H1 may be taken in the 2nd or 3rd year.

Third Year:
1. MAT351Y1; MAT334H1/MAT354H1, MAT357H1
2. One of: MAT327H1, MAT347Y1, MAT363H1/MAT367H1 (MAT363H1 can be taken in the second year, if desired)
3. PHY324H1, PHY350H1, PHY354H1, PHY356H1
Fourth Year:
1. Two of: APM421H1, APM426H1, APM446H1, APM441H1
2. Two of: PHY450H1, PHY452H1, PHY454H1, PHY456H1, PHY460H1
3. One of: MAT477H1; 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 planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

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

(12.0 FCE including at least 1.0 FCE at the 400-level)

First Year:
MAT157Y1, MAT240H1, MAT247H1; PHL232H1 or PHL233H1
Higher Years:
1. MAT257Y1, MAT327H1, MAT347Y1, MAT354H1/MAT357H1
2. PHL345H1, MAT309H1/PHL348H1
3. Four of: PHL325H1, PHL331H1, PHL332H1, PHL346H1, PHL347H1, PHL349H1, PHL355H1, PHL451H1, PHL480H1
4. 1.0 FCE from PHL200Y1/PHL205H1/PHL206H1/PHL210Y1
5. PHL265H1/PHL275H1
6. Additional 2.0 FCE from PHL or MAT to a total of 12.0 FCE

NOTE: Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

(11.5 - 12 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:
(CSC108H1,CSC148H1)/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
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.
Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Higher Years:
MAT301H1, MAT334H1
NOTE:
1. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

Teaching Concentration:

For course selection, note that OISE requires students to have a second teachable subject.
1. MAT329Y1, HPS/MAT390H1, HPS/MAT391H1
2. Two of:MAT332H1/MAT344H1, MAT335H1, MAT337H1, MAT363H1/MAT367H1
3. Two of: MAT309H1, MAT315H1; STA302H1/STA347H1
4. MAT401H1/MAT402H1 and 0.5 FCE at the 400-level from MAT, APM, STA

(12.5-13.5 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:
(CSC108H1,CSC148H1)/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
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.
Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172.

Higher Years:
MAT301H1, MAT334H1
NOTE:
1. 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/MAT351Y1
4. Three of: AST320H1, AST325H1; MAT337H1, MAT363H1/MAT367H1; PHY350H1, PHY354H1, PHY356H1, PHY357H1, PHY358H1
5. Two of: APM421H1, APM426H1, APM441H1, APM446H1; PHY407H1, PHY408H1, PHY456H1

(11.5-13.0 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:
(CSC108H1,CSC148H1)/CSC150H1; MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
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.
Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Higher Years:
MAT301H1, MAT334H1
NOTE:
1. 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/MAT351Y1/APM462H1; MAT337H1; STA261H1, STA302H1, STA347H1, STA352Y1/(STA452H1, STA453H1)
2. Additional 1.0 FCE at the 300+level from APM/MAT/STA
3. Two of: STA437H1, STA442H1, STA447H1, STA457H1

(12-12.5 FCE, including at least 1.5 FCE at the 400-level)

First Year:
ECO100Y1; MAT137Y1/MAT157Y1, MAT223H1, MAT224H1
(Please check the requirements for ECO260Y1 to ensure that you pass these first year courses with grades that allow registration in ECO206Y1)
Second Year:
ECO206Y1; MAT237Y1, MAT244H1, MAT246H1 (waived for students taking157Y1); STA257H1, STA261H1
Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ETH201H1/ETH210H1/ ETH220H1/HPS200H1/IMC200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Third Year:
1. APM346H1; ECO358H1; ECO359H1; MAT337H1; STA302H1/ECO375H1; STA347H1
2. One of: MAT332H1, MAT344H1, MAT334H1, MAT475H1
Fourth Year:
APM462H1, APM466H1; STA457H1
NOTE:
1. Students planning to take specific fourth year courses should ensure that they have the necessary  third year prerequisites.

(7.5 full courses or their equivalent. These must include at least 2.5 full course equivalent (FCE) at the 300+ level. Of those 2.5 FCE,at least 0.5 FCE must be at the 400 level).

First Year:
(MAT135H1, MAT136H1)/MAT137Y1/MAT157Y1, MAT223H1/MAT240H1
Second Year:
MAT224H1/ MAT247H1, MAT235Y1/ MAT237Y1/MAT257Y1, MAT244H1, MAT246H1

NOTE:
1. MAT224H1 may be taken in first year

Second and Higher Years:
1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ETH210H1/ ETH220H1/HPS200H1/JPH441H1/PHL265H1/PHL273H1/PHL275H1/PHL281H1 or another H course approved by the Department.  Note:  Students may use the CR/NCR option with this H course and have it count toward the program.
Higher Years:
1. MAT301H1, MAT309H1/MAT315H1, MAT334H1
2. Additional 0.5 FCE  at the 200+ level from: ACT240H1/ACT230H1; APM236H1; MAT309H1/MAT315H1/MAT335H1/ MAT337H1; STA247H1/STA257H1
3. Additional 0.5 FCE  at the 300+level from: APM346H1, APM462H1; MAT309H1, MAT315H1, MAT332H1/MAT344H1, MAT335H1, MAT337H1, MAT363H1, MAT475H1; HPS390H1, HPS391H1; PSL432H1
4. MAT401H1/MAT402H1 or any other MAT/APM 400-level course

NOTES:

1. Students using MAT157Y1 towards the first year program requirements must replace the exclusion course MAT246H1 with a different H level MAT/APM course at the 200+ level. 

2. 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.

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

4. Students interested in becoming K-12 teachers should consider applying to the combined degree program --- a six-year program that leads to an Honours Bachelor of Science (H B Sc) from the University of Toronto and a Master of Teaching (M T) from the Ontario Institute for Studies in Education (OISE). The HBSc part of this program involves completing a Math Major, a Minor in Education and Society (offered by Victoria College) and a Minor in an area that would lead to a second "teachable" subject. Please see the Victoria College website for more information.

(4.0 FCE)

1. (MAT135H1, MAT136H1)/MAT137Y1
2. MAT221H1(80%+)/MAT223H1, MAT235Y1/MAT237Y1, MAT224H1/MAT244H1/MAT246H1/APM236H1 Note: MAT221H1/MAT223H1 should be taken in first year
3. Additional 1.0 FCE  at the 300+ level from 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

Combined Degree Program: STG, Honours Bachelor of Science, Major in Mathematics/Master
of Teaching

Combined Degree Program (CDP) in Science and Education: Mathematics (Major), Honours Bachelor of Science/Master of Teaching

The Combined Degree Program in Arts/Science and Education is designed for students interested in studying the intersections of teaching subjects and Education, coupled with professional teacher preparation. Students earn an Honours Bachelor’s degree from the Faculty of Arts and Science (St. George) and an accredited professional Master of Teaching (MT) degree from the Ontario Institute for Studies in Education (OISE). They will be recommended to the Ontario College of Teachers for an Ontario Teacher’s Certificate of Qualifications as elementary or secondary school teachers. The CDP permits the completion of both degrees in six years with 1.0 FCE that may be counted towards both the undergraduate and graduate degrees.

Program requirements:
1. Minor in Education and Society, Victoria College
2. Major in Mathematics (first teaching subject)
3. Minor in an area corresponding to the second teaching subject as determined by OISE (see http://pepper.oise.utoronto.ca/~jhewitt/mtresources/intermediate_senior_teaching_subject_prerequisites_2016-17.pdf)

See here for additional information on the CDP, including admission, path to completion and contact information.

Mathematics Courses

Applied Mathematics/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 can be found at www.artsci.utoronto.ca/current/course/fyh-1/.

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.  Note: this course does not involve computer programming (despite the course's name).

This course examines the relationship between legal reasoning and mathematical logic; provides a mathematical perspective on the legal treatment of interest and actuarial present value; critiques ethical issues; analyzes how search engine techniques on massive databases transform legal research and considers the impact of statistical analysis and game theory on litigation strategies.

Sturm-Liouville problems, Green's functions, special functions (Bessel, Legendre), partial differential equations of second order, separation of variables, integral equations, Fourier transform, stationary phase method.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Similar workload to a 36L course.  Not eligible for CR/NCR option.

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.

Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.

Einstein's theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein's equations. Cosmological implications: big bang and inflationary universe. Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from black hole dynamics and gravitational waves. The Penrose singularity theorem.

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)

Partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. Nonlinear evolution equations. Existence and long-time behaviour of solutions. Existence of static, traveling wave, self-similar, topological and localized solutions. Stability. Formation of singularities and pattern formation. Fixed point theorems, spectral analysis, bifurcation theory. Equations considered in this course may include: Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics), Cahn-Hilliard (material science, biology), nonlinear Schroedinger equation (quantum and plasma physics, water waves, etc). mean curvature flow (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), and Chern-Simmons equations (particle and condensed matter physics).

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 current undergraduate offerings. Similar workload to a course that has 36 lecture hours.  Not eligible for CR/NCR option.

NOTES:

1.Transfer students who have received MAT1**H1 – Calculus with course exclusion to MAT133Y1/MAT135H1 may take MAT137Y1 without forfeiting the half credit in Calculus. Students who have credit for MAT135H1 may take MAT137Y1, but only by forfeiting the half credit. 

2. Courses no longer in the calendar:  For the purpose of course requirements, prerequisites, or exclusions,

• (125H1,126H1)/135Y1 will be considered as 135H1,136H1,
• 123H1,124H1 will be considered as MAT133Y1.

For other courses no longer in the calendar, students should consult with the undergraduate advisor. 

3.  MAT133Y1, MAT135H1, MAT137Y1, MAT138H1, MAT157Y1, MAT221H1, MAT223H1, MAT240H1 require high school level calculus.

4. The Mathematics Department enforces prerequisites for APM236H1, MAT136H1, MAT224H1, MAT235Y1, MAT237Y1, MAT244H1, MAT337H1, and others.

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 --- see the Rotman Commerce entry in the calendar for how that program addresses breadth requirements. Also, MAT133Y is not a valid prerequisite for a number of more advanced quantitative courses.
A student who took MAT133Y may need to subsequently take MAT135H and MAT136H as "extra" or take MAT137Y or MAT157Y in order to proceed in non-Commerce PoSts.

Review of trigonometric functions, trigonometric identities and trigonometric limits. Functions, limits, continuity. Derivatives, rules of differentiation and implicit differentiation, related rates, higher derivatives, 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. Attention is given to computational aspects as well as theoretical foundations and problem solving techniques. Review of Trigonometry. Limits and continuity, mean value theorem, inverse function theorem, differentiation, integration,  fundamental theorem of calculus, elementary transcendental functions, Taylor's theorem, sequence and series, power series. Applications.

The reading and understanding mathematical statements, analyzing definitions and properties, formulating conjectures and generalizations, providing and writing reasonable and precise arguments, modelling and solving proofs. This course is an excellent preparation for MAT157Y1, MAT237Y1, MAT240H1, and other proof-oriented courses.

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. Techniques of integration. Taylor's theorem; sequences and series; uniform convergence and power series.

An application-oriented approach to linear algebra, based on calculations in standard Euclidean space. Systems of linear equations, matrices, Gauss-Jordan elimination, subspaces, bases, orthogonal vectors and projections. Matrix inverses, kernel and range, rank-nullity theorem. Determinants, eigenvalues and eigenvectors, Cramer's rule, diagonalization. This course has strong  emphasis on building computational skills in the area of algebra. Applications to curve fitting, economics, Markov chains and cryptography.

Systems of linear equations, matrix algebra, real vector spaces, subspaces, span, linear dependence and independence, bases, rank, inner products, orthogonality, orthogonal complements, Gram-Schmidt, linear transformations, determinants, Cramer's rule, eigenvalues, eigenvectors, 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 canonical 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 R^2 and R^3. 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 and Z_p. 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.

First order ordinary differential equations: Direction fields, integrating factors, separable equations, homogeneous equations, exact equations, autonomous equations, modeling. Existence and uniqueness theorem. Higher order equations: Constant coefficient equations, reduction of order, Wronskian, method of undetermined coefficients, variation of parameters. Solutions by series and integrals. First order linear systems, fundamental matrices. Non-linear equations, phase plane, stability. 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. 

Topology of R^n; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integration; Fubini's theorem, partitions of unity, change of variables. Differential forms. Manifolds in R^n; integration on manifolds; Stokes' theorem for differential forms and classical versions.

A theoretical course on Ordinary Differential Equations. First-order equations: separable equations, exact equations, integrating factors. Variational problems, Euler-Lagrange equations. Linear equations and first-order systems. Fundamental matrices, Wronskians. Non-linear equations. Existence and uniqueness theorems. Method of power series. Elementary qualitative theory; stability, phase plane, stationary points. Oscillation theorem, Sturm comparison. 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.

Note: This course cannot be used to satisfy requirements of program in the math department.

A course in mathematics on a topic outside the current undergraduate offerings.  For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

Independent study under the direction of a faculty member.  Topic must be outside undergraduate offerings.  Similar workload to a 36L course.  Not eligible for CR/NCR option.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

Independent research under the direction of a faculty member. Similar workload to a 72L course.  Not eligible for CR/NCR option. 

Credit course for supervised participation in faculty research project. Details at http://www.artsci.utoronto.ca/current/course/rop. Not eligible for CR/NCR option.

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

Predicate calculus. Relationship between truth and provability; Gödel's completeness theorem. First order arithmetic as an example of a first-order system. Gödel's incompleteness theorem; outline of its proof. Introduction to recursive functions.

Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.

Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Fundamental group and covering spaces. Brouwer fixed-point theorem. Students in the math specialist program wishing to take additional topology courses are advised to obtain permission to take MAT1300H,MAT1301H.

This course is aimed at students intending to become elementary school teachers. Emphasis is placed on the formation and development of fundamental reasoning and learning skills required to understand and to teach mathematics at the elementary level. Topics may include: Problem Solving and Strategies, Sets and Elementary Logic, Numbers and Elements of Number Theory, Introductory Probability and Fundamentals of Geometry. 

The course may include an optional practicum in school classrooms. 

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.

Theory of functions of one complex variable, analytic and meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.

An elementary introduction to a modern and fast-developing area of mathematics. One-dimensional dynamics: iterations of quadratic polynomials. Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia sets. History and applications.

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-Hölder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic.

Diffusion and wave equations. Separation of variables. Fourier series. Laplace's equation; Green's function. Schrödinger equations. Boundary problems in plane and space. General eigenvalue problems; minimum principle for eigenvalues. Distributions and Fourier transforms. Laplace transforms. Differential equations of physics (electromagnetism, fluids, acoustic waves, scattering). Introduction to nonlinear equations (shock waves, solitary waves).

Complex numbers, the complex plane and Riemann sphere, Möbius 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' 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, L^p spaces. Applications to probability.

Curves and surfaces in 3-spaces. Frenet formulas. Curvature and geodesics. Gauss map. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.

Manifolds, partitions of unity, submersions and immersions, vector fields, vector bundles, tangent and cotangent bundles, foliations and Frobenius’ theorem, multillinear algebra, differential forms, Stokes’ theorem, Poincare-Hopf theorem

A course in mathematics on a topic outside the current undergraduate offerings.  For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

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 reading under the direction of a faculty member.  Topic must be outside current undergraduate offerings.  Similar workload to a 36L course. Not eligible for CR/NCR option.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Similar workload to a 36L course.  Not eligible for CR/NCR option.

Independent research under the direction of a faculty member.  Workload similar to a 72L course.  Not eligible for CR/NCR option.

An instructor-supervised group project in an off-campus setting. Details at http://www.artsci.utoronto.ca/current/course/399. Not eligible for CR/NCR option.

An instructor-supervised group project in an off-campus setting. Details at http://www.artsci.utoronto.ca/current/course/399. Not eligible for CR/NCR option.

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.

Commutative rings; quotient rings. Construction of the rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of roots of a polynomial. Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Insolvability of quintics by radicals.

Euclidean and non-euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.

Set theory and its relations with other branches of mathematics. ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible sets and the continuum hypothesis. Introduction to independence proofs. Topics from large cardinals, infinitary combinatorics and descriptive set theory.

A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; diophantine equations.

A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; diophantine approximation, modular forms.

Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

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.

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.

Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers. This course will be offered in alternating years.

Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities. This course will be offered in alternating years.

Harmonic functions, Harnack's principle, Poisson's integral formula and Dirichlet's problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.

Lebesque measure and integration; convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem. Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p spaces, Hölder and Minkowski inequalities.

Fourier series and transform, convergence results, Fourier inversion theorem, L^2 theory, estimates, convolutions. Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.

Riemannian metrics. Levi-Civita connection. Geodesics. Exponential map. Second fundamental form. Complete manifolds and Hopf-Rinow theorem. Curvature tensors. Ricci curvature and scalar curvature. Spaces of constant curvature.

This course addresses the question: How do you attack a problem the likes of which you have never seen before? Students will apply Polya's principles of mathematical problem solving, draw upon their previous mathematical knowledge, and explore the creative side of mathematics in solving a variety of interesting problems and explaining those solutions to others.

Seminar in an advanced topic. Content will generally vary from semester to semester. Student presentations are required.

A course in mathematics on a topic outside the current undergraduate offerings.  For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

Independent research under the direction of a faculty member. Not eligible for CR/NCR option.  Similar workload to a 72L course.

Prerequisite: Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor
Distribution Requirement Status: Science
Breadth Requirement: The Physical and Mathematical Universes (5)

MAT498Y1    Readings in Mathematics[TBA]

Offered for the last time as MAT498Y1 in Summer 2016. Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Not eligible for CR/NCR option.