Faculty of Arts & Science
2011-2012 Calendar |
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Professors Emeriti

M.A. Akcoglu, M Sc, Ph D, FRSC

E.J. Barbeau, MA Ph D (U)

B. Brainerd, MS, Ph D

H.C. Davis, MA, Ph D (N)

E.W. Ellers, Dr Rer Nat

P.C. Greiner, MA, Ph D, FRSC

S. Halperin, M Sc, Ph D, FRSC

W. Haque, MA, Ph D FRSC

V. Jurdjevic, MS, PhD

I. Kupka, AM, Ph D, Dr s Sc M

D.R. Masson, M Sc, Ph D (U)

J. McCool, B Sc, Ph D

K. Murasugi, MA, D Sc, FRSC

P.G. Rooney, B Sc, Ph D, FRSC

P. Rosenthal, MA, Ph D, LLB

D.K. Sen, M Sc, Dr s Sc

R.W. Sharpe, MA, Ph D (UTSC)

F.A. Sherk, M Sc, Ph D (U)

S.H. Smith, B Sc, Ph D

Associate Professors Emeriti

N.A. Derzko, B Sc, Ph D

M.P. Heble, M Sc, Ph D

Professor and Chair of the Department

K. Murty, B Sc, Ph D, FRSC

Professors and Associate Chairs

J. Colliander, BA, Ph D

C. Sulem, M Sc, Dr D'Etat

University Professors

J.G. Arthur, MA, Ph D, FRSC, FRS

J. Friedlander, MA, Ph D, FRSC (UTSC)

I.M. Sigal, BA, Ph D, FRSC

Professors

D. Bar-Natan, B Sc, Ph D

E. Bierstone, MA, Ph D, FRSC

J. Bland, M Sc, Ph D

T. Bloom, MA, Ph D, FRSC

R.O. Buchweitz, Dipl Maths, Dr Rer Nat (UTSC)

A. Burchard, B Sc, Ph D

M.D. Choi, MA, Ph D, FRSC

A. del Junco, M Sc, Ph D

G. Elliott, B Sc, Ph D, FRSC

M. Goldstein, B Sc, Ph D (UTSC)

I.R. Graham, B Sc, Ph D (UTM)

V. Ivrii, MA, Ph D, Dr Math, FRSC

L. Jeffrey, AB, Ph D (UTSC)

R. Jerrard, M Sc, Ph D (U)

Y. Karshon, B Sc, Ph D (UTM)

K. Khanin, M Sc, Ph D (UTM)

B. Khesin, M Sc, Ph D

A. Khovanskii, M Sc, Ph D

H. Kim, B Sc, Ph D

S. Kudla, B A, MA, Ph D

J.W. Lorimer, M Sc, Ph D (U)

R. McCann BSc, Ph D

E. Meinrenken, B Sc, Ph D

E. Mendelsohn, M Sc, Ph D (UTSC)

P. Milman, Dipl Maths, Ph D, FRSC

F. Murnaghan, M Sc, Ph D

A. Nabutovsky, M Sc, Ph D

A. Nachman, B Sc, Ph D

J. Quastel MSc, Ph D

J. Repka, B Sc, Ph D (U)

L. Seco, BA, Ph D (UTM)

P. Selick, B Sc, MA, Ph D (UTSC)

F.D. Tall, AB, Ph D (UTM)

S. Todorcevic, B Sc, Ph D

W.A.R. Weiss, M Sc, Ph D (UTM)

M. Yampolsky, B Sc, Ph D (UTM)

Associate Professors

I. Binder, B Sc, M Sc, Ph D (UTM)

V. Kapovitch, B Sc, Ph D

M. Pugh, B Sc, Ph D

R. Rotman BA, Ph D

J. Scherk, D Phil (UTSC)

S.M. Tanny, B Sc, Ph D (UTM)

B. Virag, BA, Ph D (UTSC)

Assistant Professors

S. Alexakis, BA, Ph D

S. Arkhipov, B Sc, Ph D

M. Braverman, BA, MSc, Ph D

M. Gualtieri, B Sc, Ph D

L. Guth, B Sc, Ph D

F. Herzig, BA, Ph D

J. Kamnitzer, B Sc, Ph D

B. Szegedy, B Sc, Ph D (UTSC)

Senior Lecturers

D. Burbulla, B Sc, B Ed, MA

A. Igelfeld, M Sc

A. Lam, M Sc

F. Recio, M Sc, Ph D

Lecturers

S. Homayouni, B Sc, Ph D

N. Jung, BA, MSc, Ph D

P. Kergin, M Sc, Ph D

E.A.P. LeBlanc, MA, Ph D

J. Tate, B Sc, B Ed

S. Uppal, M Sc

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)

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.

Mathematics Specialist (Science program)

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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 or similar courses in other departments.

3. Students planning to take specific fourth year courses should ensure that they have the necessary 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; 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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 or similar courses in other departments.

3. Students planning to take specific fourth year courses should ensure that they have the necessary 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, PHY350H1, 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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 or similar courses in other departments.

3. Students planning to take specific fourth year courses should ensure that they have the necessary 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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 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

Computer Science Concentration:

1. CSC148H1/CSC150H1, CSC165H1, CSC236H1/ CSC240H1, CSC209H1

2. CSC207H1, CSC236H1/CSC240H1, CSC209H1

3. MAT332H1/MAT344H1 and three of MAT309H1; CSC320H1, CSC350H1, CSC351H1, CSC373H1

4. Two of: APM461H1; CSC446H1, CSC456H1, CSC465H1, CSC487H1

NOTE:

1. In order to take the Computer Science concentration, you will be required to register also for a Computer Science Major. (The latter is a restricted enrolment program and has certain admission requirements and much higher fees; please see the Computer Science program description).

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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 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; PHY350H1, 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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 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.

Mathematics & Its Applications Specialist (Probability/Statistics)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: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 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/ECO327Y1/(ECO375H1, ECO376H1), 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.

Mathematics and Philosophy (Science program)

Consult the Associate Chairs for Undergraduate Studies, Department of Mathematics and Department of 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. One full course from PHL200Y1/(PHL205H1, PHL206H1)/ PHL210Y1

3. PHL232H1, HPS250H1/ PHL246H5, PHL265H1/ PHL275H1

4. MAT309H1; PHL345H1

5. Two of: PHL331H1, PHL342H1, PHL351H1, PHL355H1/PHL356H1

6. Two of : MAT409H1; PHL404H1, PHL411H1, PHL451H1, PHL480H1, PHL481H1, PHL482H1

7. One additional full course credit in PHL or MAT courses to a total of 12 full courses.

NOTE:

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

2. If a course number ends in H5, the course is offered only at the University of Toronto Missisauga

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. MAT 224H1 may be taken in first year

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 are required, as part of their degree, to take a course with a significant emphasis on ethics and social responsibility such as: PHL275H1/PHL265H1/PHL268H1/PHL271H1/PHL273H1 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**

Applied Mathematics/Mathematics Courses

First Year Seminars

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.

JMB170Y1 Biology, Models, and Mathematics [48L/24T]

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.

Corequisite: BIO120H1Exclusion: Exclusion: MAT135H1/135Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

JUM202H1 Mathematics as an Interdisciplinary Pursuit (formerly JUM102H1) [24L/12T]

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.

Exclusion: JUM102H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

JUM203H1 Mathematics as a Recreation (formerly JUM103H1)[24L/12T]

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.

Exclusion: JUM103H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

JUM205H1 Mathematical Personalities (formerly JUM105H1) [24L/12T]

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.

Exclusion: JUM105H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

Applied Mathematics Courses

APM236H1 Applications of Linear Programming[36L]

Introduction to linear programming including a rapid review of linear algebra (row reduction, linear independence), the simplex method, the duality theorem, complementary slackness, and the dual simplex method. A selection of the following topics are covered: the revised simplex method, sensitivity analysis, integer programming, the transportation algorithm.

Prerequisite: MAT223H1/MAT240H1 (Note: no waivers of Prerequisites will be granted)Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM346H1 Partial Differential Equations[36L]

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.

Prerequisite: MAT235Y1/MAT237Y1/MAT257Y1, MAT244H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM351Y1 Partial Differential Equations[72L]

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).

Prerequisite: MAT267H1Corequisite: MAT334H1/MAT354H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

400-SERIES COURSES

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.

APM421H1 Mathematical Foundations of Quantum Mechanics[36L]

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.

Prerequisite: (MAT224H1, MAT337H1)/MAT357H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM426H1 General Relativity[36L]

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.

Prerequisite: MAT363H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM436H1 Fluid Mechanics[36L]

Boltzmann, Euler and Navier-Stokes equations. Viscous and non-viscous flow. Vorticity. Exact solutions. Boundary layers. Wave propagation. Analysis of one dimensional gas flow.

Prerequisite: APM351Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM441H1 Asymptotic and Perturbation Methods[36L]

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)

Prerequisite: APM346H1/APM351Y1, MAT334H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM446H1 Applied Nonlinear Equations[36L]

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.

Prerequisite: APM346H1/APM351Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM461H1 Combinatorial Methods[36L]

A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.

Prerequisite: MAT224H1Recommended Preparation: MAT344H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM462H1 Nonlinear Optimization[36L]

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.

Prerequisite: MAT223H1, MAT235Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM466H1 Mathematical Theory of Finance[36L]

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.

Prerequisite: APM346H1, STA347H1Corequisite: STA457H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM496H1 Readings in Applied Mathematics[TBA]

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

Prerequisite: minimum GPA 3.5 for math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisorDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM497H1 Readings in Applied Mathematics[TBA]

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

Prerequisite: minimum GPA 3.5 for math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisorDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM498Y1 Readings in Applied Mathematics[TBA]

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

Prerequisite: minimum GPA 3.5 for math courses. Permission of the Associate Chair for Undergraduate Studies and prospective supervisorDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

APM499Y1 Readings in Applied Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

Mathematics Courses

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

MAT123H1, MAT124H1

See below MAT133Y1

MAT125H1, MAT126H1

MAT125H1, MAT126H1

See below MAT135Y1

MAT133Y1 Calculus and Linear Algebra for Commerce[72L]

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.

Prerequisite: MCV4U, MHF4UExclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT135Y1, MAT137Y1, MAT157Y1

Distribution Requirement Status: This is a Social Science course

Breadth Requirement: None

MAT123H1 Calculus and Linear Algebra for Commerce (A)[36L]

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.”

Prerequisite: Enrolment in MAT133Y1, and withdrawal from MAT133Y1 after two midterms, with a mark of at least 20% in the second midterm.Exclusion: MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1

Distribution Requirement Status: This is a Social Science course

Breadth Requirement: None

MAT124H1 Calculus and Linear Algebra for Commerce (B)[36L]

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.

Prerequisite:
MAT123H1 successfully completed in the preceding Spring Term

Exclusion:
MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1

Distribution Requirement Status: This is a Social Science course

Breadth Requirement: None

**MAT135Y1 -- see MAT135H1, MAT136H1 below.
MAT135H1 Calculus 1(A)[36L/12T]**

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.

Prerequisite: MCV4U, MHF4UExclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT136H1 Calculus 1(B)[36L/12T]

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.

Prerequisite: MAT135H1Exclusion: MAT123H1, MAT124H1, MAT125H1, MAT126H1, MAT133Y1, MAT135Y1, MAT137Y1, MAT157Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT137Y1 Calculus[72L/24T]

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.

Prerequisite: MCV4U, MHF4UExclusion: MAT125H1, MAT126H1, MAT135Y1, MAT157Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT157Y1 Analysis I[72L/48T]

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.

Prerequisite: MCV4U, MHF4UExclusion: MAT137Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT223H1 Linear Algebra I[36L/12T]

Matrix arithmetic and linear systems. Rn subspaces, linear independence, bases, dimension; column spaces, null spaces, rank and dimension formula. Orthogonality orthonormal sets, Gram-Schmidt orthogonalization process; least square approximation. Linear transformations Rn\>Rm. The determinant, classical adjoint, Cramers Rule. Eigenvalues, eigenvectors, eigenspaces, diagonalization. Function spaces and application to a system of linear differential equations.

Prerequisite: MCV4U, MHF4UExclusion: MAT240H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT224H1 Linear Algebra II[36L/12T]

Abstract vector spaces: subspaces, dimension theory. Linear mappings: kernel, image, dimension theorem, isomorphisms, matrix of linear transformation. Changes of basis, invariant spaces, direct sums, cyclic subspaces, Cayley-Hamilton theorem. Inner product spaces, orthogonal transformations, orthogonal diagonalization, quadratic forms, positive definite matrices. Complex operators: Hermitian, unitary and normal. Spectral theorem. Isometries of R2 and R3.

Prerequisite: MAT223H1/MAT240H1Exclusion: MAT247H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT235Y1 Calculus II[72L]

Differential and integral calculus of functions of several variables. Line and surface integrals, the divergence theorem, Stokes theorem. Sequences and series, including an introduction to Fourier series. Some partial differential equations of Physics.

Prerequisite: MAT135Y1/MAT137Y1/MAT157Y1Exclusion: MAT237Y1, MAT257Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT237Y1 Multivariable Calculus[72L]

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.

Prerequisite: MAT137Y1/MAT157Y1/MAT135Y1(90%),MAT223H1/MAT240H1Exclusion: MAT235Y1, MAT257Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT240H1 Algebra I[36L/24T]

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.

Prerequisite: MCV4U, MHF4UCorequisite: MAT157Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT244H1 Introduction to Ordinary Differential Equations[36L]

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.

Prerequisite: MAT135Y1/MAT137Y1/MAT157Y1, MAT223H1/MAT240H1Corequisite: MAT235Y1/MAT237Y1

Exclusion: MAT267H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT246H1 Concepts in Abstract Mathematics (formerly MAT246Y1)[36L]

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.

Prerequisite: MAT133Y1/MAT135Y1/MAT137Y1,MAT223H1Exclusion: MAT157Y1, MAT246Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT247H1 Algebra II[36L]

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.

Prerequisite: MAT240H1Corequisite: MAT157Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT257Y1 Analysis II[72L/48T]

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.

Prerequisite: MAT157Y1, MAT240H1, MAT247H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT267H1 Advanced Ordinary Differential Equations I[36L/12T]

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.

Prerequisite: MAT157Y1, MAT247H1Corequisite: MAT257Y1

Exclusion: MAT244H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT271H1 Insights from Mathematics[36L/6T]

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.

Distribution Requirement Status: This is a Science courseBreadth Requirement: The Physical and Mathematical Universes (5)

MAT299Y1 Research Opportunity Program

Credit course for supervised participation in faculty research project. Details here.

Distribution Requirement Status: This is a Science courseBreadth Requirement: None

300-Series Courses

MAT301H1 Groups and Symmetries[36L]

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.

Prerequisite: MAT224H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1. (These Prerequisites will be waived for students who have MAT257Y1)Exclusion: MAT347Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT309H1 Introduction to Mathematical Logic[36L]

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.

Prerequisite: MAT223H1/MAT240H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1 (These Prerequisites will be waived for students who have MAT257Y1)Exclusion: CSC438H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT315H1 Introduction to Number Theory[36L]

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.

Prerequisite: MAT223H1/MAT240H1, MAT235Y1/MAT237Y1, MAT246H1/CSC236H1/CSC240H1 (These Prerequisites will be waived for students who have MAT257Y1)Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT327H1 Introduction to Topology[36L]

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.

Prerequisite: MAT257Y1/(MAT224H1, MAT237Y1, MAT246H1 and permission of the instructor)Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT329Y1 Concepts in Elementary Mathematics[72L]

The formation of mathematical concepts and techniques, and their application to the everyday world. Nature of mathematics and mathematical understanding. Role of observation, conjecture, analysis, structure, critical thinking and logical argument. Numeration, arithmetic, geometry, counting techniques, recursion, algorithms. This course is specifically addressed to students intending to become elementary school teachers and is strongly recommended by the Faculty of Education. Previous experience working with children is useful. The course content is considered in the context of elementary school teaching. In particular, the course may include a practicum in school classrooms. The course has an enrolment limit of 40, and students are required to ballot.

Prerequisite: Any 7 full courses with a CGPA of at least 2.5Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT332H1 Introduction to Graph Theory[36L]

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.

Prerequisite: MAT224H1/MAT247H1Corequisite: Recommended Corequisite: MAT301H1/MAT347Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT334H1 Complex Variables[36L]

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

Prerequisite: MAT223H1, MAT235Y1/MAT237Y1Exclusion: MAT354H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT335H1 Chaos, Fractals and Dynamics[36L]

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.

Prerequisite:
MAT137Y1/200-level calculus, MAT223H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

**MAT336H1 Elements of Analysis [36L, 12T]**

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 R^{n}, implicit and inverse function theorems and rigorous integration theory.

Prerequisite: MAT223H1, MAT235Y1

Exclusion: MAT257Y1, MAT337H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT337H1 Introduction to Real Analysis[36L]

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.

Prerequisite: MAT224H1, MAT235Y1/MAT237Y1,MAT246H1; NOTE: These Prerequisites will be waived for students who have MAT257Y1Exclusion: MAT357H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT344H1 Introduction to Combinatorics[36L]

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.

Prerequisite: MAT223H1/MAT240H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT347Y1 Groups, Rings and Fields[72L/24T]

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.

Prerequisite: MAT257Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT354H1 Complex Analysis I[36L]

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.

Prerequisite: MAT257Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT357H1 Real Analysis I[36L]

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.

Prerequisite: MAT257Y1/(MAT327H1 and permission of instructor)Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT363H1 Introduction to Differential Geometry[36L]

Geometry of curves and surfaces in 3-spaces. Curvature and geodesics. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.

Prerequisite: MAT224H1, MAT237Y1/MAT257Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT390H1 History of Mathematics up to 1700[36L]

A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)

Prerequisite: at least one full MAT 200-level courseExclusion: HPS309H1, HPS310Y1, HPS390H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT391H1 History of Mathematics after 1700[24L/12T]

A survey of the development of mathematics from 1700 to the present with emphasis on technical development. (Offered in alternate years)

Prerequisite: At least one full 200-level MAT courseExclusion: HPS309H1, HPS310H1, HPS391H1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT393Y1 Independent Work in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT394Y1 Independent Work in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT395H1 Independent Work in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT396H1 Independent Work in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT397H1 Independent Work in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT398H0 Independent Experiential Study Project

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

Distribution Requirement Status: This is a Science courseBreadth Requirement: None

MAT399Y0 Independent Experiential Study Project

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

Distribution Requirement Status: This is a Science courseBreadth Requirement: None

400-Series Courses

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.

MAT401H1 Polynomial Equations and Fields[36L]

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.

Prerequisite: MAT301H1Exclusion: MAT347Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT402H1 Classical Geometries[36L]

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

Prerequisite: MAT301H1/MAT347Y1, MAT235Y1/MAT237Y1/MAT257Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT409H1 Set Theory[36L]

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.

Prerequisite: MAT357H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT415H1 Topics in Algebraic Number Theory[36L]

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.

Prerequisite: MAT347Y1 or permission of instructorDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT417H1 Topics in Analytic Number Theory[36L]

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.

Prerequisite: MAT334H1/MAT354H1/permission of instructorDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT425H1 Differential Topology[36L]

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.

Prerequisite: MAT257Y1, MAT327H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT427H1 Algebraic Topology[36L]

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.

Prerequisite: MAT327H1, MAT347Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT443H1 Computer Algebra[36L]

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.

Prerequisite: MAT347Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT445H1 Representation Theory[36L]

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.

Prerequisite: MAT347Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT448H1 Introduction to Commutative Algebra and Algebraic Geometry[36L]

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.

Prerequisite: MAT347Y1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT449H1 Algebraic Curves[36L]

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.

Prerequisite: MAT347Y1, MAT354H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT454H1 Complex Analysis II[36L]

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.

Prerequisite: MAT354H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT457H1 Real Analysis I (formerly MAT457Y1)[36L]

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.

Prerequisite: MAT357H1Exclusion: MAT457Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT458H1 Real Analysis II (formerly MAT457Y1)[36L]

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.

Prerequisite: MAT457H1Exclusion: MAT457Y1

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT464H1 Differential Geometry[36L]

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.

Prerequisite: MAT363H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT468H1 Ordinary Differential Equations II[36L]

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.

Prerequisite: MAT267H1, MAT354H1, MAT357H1Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT475H1 Problem Solving Seminar[TBA]

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.

Prerequisite: MAT224H1/MAT247H1, MAT235Y1/MAT237Y1/MAT257Y1, and at least ONE 300-level MAT or APM courseDistribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT477Y1 Seminar in Mathematics[TBA]

Seminar in an advanced topic. Content will generally vary from year to year. (Student presentations will be required)

Prerequisite: MAT347Y1, MAT354H1, MAT357H1; or permission of instructor.Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT495H1 Readings in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT496H1 Readings in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT497H1 Readings in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT498Y1 Readings in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)

MAT499Y1 Readings in Mathematics[TBA]

Distribution Requirement Status: This is a Science course

Breadth Requirement: The Physical and Mathematical Universes (5)