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# Mathematics

### Faculty List

- S. Aretakis, B.Sc. (Patras), M.Sc., Ph.D. (Cambridge), Assistant Professor
- N. Breuss, B.Sc., M.Sc. (Kharkov), Ph.D. (Moscow), Associate Professor, Teaching Stream
- S. Chrysostomou, B.Sc., M.Sc. (Toronto), Associate Professor, Teaching Stream Emeritus
- J. Friedlander, B.Sc. (Toronto), M.A. (Waterloo), Ph.D. (Penn. State), F.R.S.C., University Professor
- M. Goldstein, B.A., M.Sc., Ph.D. (Tashkent), Professor
- R. Grinnell, B.Sc. (Toronto), M.A. (York), Ph.D. (Queen's), Associate Professor, Teaching Stream
- R. Haslhofer, B.Sc., M.Sc., Ph.D. (ETH Zurich), Assistant Professor
- L.C. Jeffrey, A.B. (Princeton), M.A. (Cambridge), D. Phil. (Oxford), F.R.S.C., Professor
- X. Jiang, B.Sc., M.Sc., Ph.D. (Glasgow), Associate Professor, Teaching Stream
- C. Karimian Pour, B.Sc. (Tehran), M.Sc., Ph.D. (Ottawa), CLTA Assistant Professor, Teaching Stream
- A. Kupers, B.Sc., M.Sc. (Utrecht), Ph.D. (Stanford), Assistant Professor
- E. Mendelsohn, B.Sc., M.Sc. (Manitoba), Ph.D. (McGill), Professor Emeritus
- E. Moore, Hon. B.A., B.Ed., M.A. (Memorial), Ph.D. (Toronto), Associate Professor, Teaching Stream Emeritus
- J. Scherk, B.Sc., M.Sc. (Toronto), D.Phil. (Oxford), Associate Professor Emeritus
- P. Selick, B.Sc., M.Sc., Ph.D. (Princeton), Professor Emeritus
- Z. Shahbazi, B.Sc. (Sharif), M.Sc., Ph.D. (Toronto), Associate Professor, Teaching Stream
- R.W. Sharpe, B.Sc., M.Sc. (Toronto), Ph.D. (Yale), Professor Emeritus
- K.Smith, Hon. B.Sc., M.Sc., Ph.D. (Toronto), Assistant Professor, Teaching Stream
- G. Tiozzo, B.Sc., M.Sc., (dePisa), Ph.D. (Scuola Normale Superiore), Assistant Professor
- B. Virag, B.A. (Harvard), M.A., Ph.D.(Berkeley), Professor
- W. Yu, B.A., B.Sc. (Indiana), M.Res., M.Phil. (Imperial College London), Ph.D. (MIT), Assistant Professor

Associate Chair: B. Virág (416-287-7261) Email: balint@math.toronto.edu

Our Mathematics began in the ancient Mesopotamian civilizations. The Babylonians already knew much of the mathematics taught traditionally in our schools. Their algebra and geometry were phrased in terms of crops and fields and money. Since the Renaissance, much of mathematics has come from problems in physics and astronomy; for example, calculus arose from problems in mechanics. In turn, mathematics has provided the theoretical framework and tools in the Physical Sciences. In the 19th century, some parts of mathematics appeared to develop away from their origins in the physical world. To the great surprise of many scientists and mathematicians, some of the "pure" mathematics has turned out to be essential in many aspects of 20th-century science. Differential geometry provides the language for general relativity and cosmology, and Hilbert space theory and group representations are the tools for quantum mechanics. Similarly, graph theory, combinatorics and number theory play a major role in computer science.** **

**Note on Admission to MAT programs**

Beginning in 2018-19 there are admissions criteria for the Specialist/Specialist (Co-op) and Major/Major (Co-op) Program in Mathematics. Details and information on how to apply for admission to these programs can be found in the program descriptions below.

**Combined Degree Programs, Honours Bachelor of Science/ Master of Teaching**

The Combined Degree Programs for UTSC Honours Bachelor of Science (HBSc) /Honours Bachelor of Arts (HBA) with the Master of Teaching (MT) offered by the Ontario Institute for Studies in Education provide students with a direct pathway to the completion, in 6 years, of their Undergraduate degree, Ontario Teacher’s Certificate of Qualifications, and Master’s degree. These Combined Degree Programs allow students to complete 1.0 credit in courses that may be counted towards both degrees.

*The Combined Degree Programs options are:*

- Mathematics (Specialist), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Major), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Major Co-op), Honours Bachelor of Science/ Master of Teaching

For more information, including Admission and Program requirements, see the Combined Degree Programs section of the *Calendar*.

**Experiential Learning and Outreach**

For a community-based experiential learning opportunity in your academic field of interest, consider the course CTLB03H3, which can be found in the Teaching and Learning section of the *Calendar*.

### Mathematics Programs

### COMBINED DEGREE PROGRAMS, HONOURS BACHELOR OF SCIENCE OR HONOURS BACHELOR OF ARTS / MASTER OF TEACHING

The Combined Degree Programs for UTSC Honours Bachelor of Science (HBSc)/ Honours Bachelor of Arts (HBA) with the Master of Teaching (MT) offered by the Ontario Institute for Studies in Education are designed for students who are interested in a career in Education. They allow exceptional students who are registered in one of the 50 identified Specialist and Major programs to gain early admission to the MT, which is a full-time professional program that leads to both a Master's degree and eligibility to become a certified teacher in Ontario. Students who successfully complete one of the Combined Degree Programs listed below will earn two University of Toronto degrees (HBA/ HBSc and MT), and be recommended to the Ontario College of Teachers for a Certificate of Qualifications as elementary or secondary school teachers.

**Contact Information:**

Marcelle DeFreitas (Combined Degree Programs Coordinator)

Email: mdefreitas@utsc.utoronto.ca

**The Combined Degree Programs options are:**

**Department of Anthropology**

- Evolutionary Anthropology (Specialist), Honours Bachelor of Science/ Master of Teaching
- Evolutionary Anthropology (Major), Honours Bachelor of Science/ Master of Teaching
- Socio-Cultural Anthropology (Specialist), Honours Bachelor of Arts/ Master of Teaching
- Socio-Cultural Anthropology (Major), Honours Bachelor of Arts/ Master of Teaching

**Department of Arts, Culture and Media**

- Theatre and Performance Studies (Major), Honours Bachelor of Arts/ Master of Teaching

**Department of Biological Sciences**

- Biology (Major), Honours Bachelor of Science/ Master of Teaching
- Conservation and Biodiversity (Specialist), Honours Bachelor of Science/ Master of Teaching
- Conservation and Biodiversity (Major), Honours Bachelor of Science/ Master of Teaching
- Human Biology (Specialist), Honours Bachelor of Science/ Master of Teaching
- Human Biology (Major), Honours Bachelor of Science/ Master of Teaching
- Integrative Biology (Specialist), Honours Bachelor of Science/ Master of Teaching
- Molecular Biology and Biotechnology (Specialist), Honours Bachelor of Science/ Master of Teaching
- Molecular Biology and Biotechnology (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Molecular Biology, Immunology and Disease (Major), Honours Bachelor of Science/ Master of Teaching
- Plant Biology (Major), Honours Bachelor of Science/ Master of Teaching

**Department of Computer and Mathematical Sciences**

- Mathematics (Specialist), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Major), Honours Bachelor of Science/ Master of Teaching
- Mathematics (Major Co-op), Honours Bachelor of Science/ Master of Teaching

**Department of English**

- English (Specialist), Honours Bachelor of Arts/ Master of Teaching
- English (Specialist Co-op), Honours Bachelor of Arts/ Master of Teaching
- English (Major), Honours Bachelor of Arts/ Master of Teaching
- English (Major Co-op), Honours Bachelor of Arts/ Master of Teaching

**Department of Language Studies**

- French (Specialist), Honours Bachelor of Arts/ Master of Teaching
- French (Specialist Co-op), Honours Bachelor of Arts/ Master of Teaching
- French (Major), Honours Bachelor of Arts/ Master of Teaching
- French (Major Co-op), Honours Bachelor of Arts/ Master of Teaching

**Department of Historical and Cultural Studies**

- History (Specialist), Honours Bachelor of Arts/ Master of Teaching
- History (Major), Honours Bachelor of Arts/ Master of Teaching

**Department of Human Geography**

- Human Geography (Specialist), Honours Bachelor of Arts/ Master of Teaching
- Human Geography (Major), Honours Bachelor of Arts/ Master of Teaching

**Department of Physical and Environmental Sciences**

- Biological Chemistry (Specialist), Honours Bachelor of Science/ Master of Teaching
- Biological Chemistry (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Biochemistry (Major), Honours Bachelor of Science/ Master of Teaching
- Biochemistry (Major Co-op), Honours Bachelor of Science/ Master of Teaching
- Chemistry (Specialist), Honours Bachelor of Science/ Master of Teaching
- Chemistry (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Chemistry (Major), Honours Bachelor of Science/ Master of Teaching
- Chemistry (Major Co-op), Honours Bachelor of Science/ Master of Teaching
- Environmental Biology (Specialist), Honours Bachelor of Science/ Master of Teaching
- Environmental Biology (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Environmental Chemistry (Specialist), Honours Bachelor of Science/ Master of Teaching
- Environmental Chemistry (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Environmental Physics (Specialist), Honours Bachelor of Science/ Master of Teaching
- Environmental Physics (Specialist Co-op), Honours Bachelor of Science/ Master of Teaching
- Physics and Astrophysics (Specialist), Honours Bachelor of Science/ Master of Teaching
- Physics and Astrophysics (Major), Honours Bachelor of Science/ Master of Teaching
- Physical and Mathematical Sciences (Specialist), Honours Bachelor of Science/ Master of Teaching

**Department of Sociology**

- Sociology (Specialist), Honours Bachelor of Arts/ Master of Teaching
- Sociology (Major), Honours Bachelor of Arts/ Master of Teaching

Students applying to the MT must have two teaching subjects regardless of the concentration they are applying to (Primary/Junior, Junior/Intermediate, or Intermediate/Senior), and must have completed at least 6.0 credits in their first teaching subject and at least 3.0 credits in their second teaching subject (note: both French as a Second Language and Science require at least 6.0 credits in university courses even when they are a second teaching subject). Each of the programs listed below includes a minimum of 6.0 credits in courses that can be applied towards the completion of the prerequisites for the identified OISE teaching subject(s).

**UTSC Programs Fit With OISE MT Teaching Subjects:**

UTSC Program | MT Teaching Subjects - Required Number of Courses/Credits Completed |
---|---|

- Specialist/ Specialist Co-op in Biological Chemistry | Science - Chemistry, or Science - Biology, or Science - General |

- Specialist/Specialist Co-op in Molecular Biology and Biotechnology | Science - Biology, or Science - General |

- Major/Major Co-op In Biochemistry - Major in Biology - Specialist in Conservation and Biodiversity - Major in Conservation and Biodiversity - Specialist in Human Biology - Major in Human Biology - Specialist in Integrative Biology - Major in Molecular Biology, Immunology and Disease - Major in Plant Biology - Specialist/Specialist Co-op in Environmental Biology |
Science - Biology |

- Specialist/Specialist Co-op in Chemistry - Major/Major Co-op in Chemistry - Specialist/Specialist Co-op in Environmental Chemistry |
Science - Chemistry |

- Specialist/Specialist Co-op in Environmental Physics - Specialist in Physics and Astrophysics - Major in Physics and Astrophysics - Specialist in Physical and Mathematical Sciences |
Science - Physics |

- Specialist/Specialist Co-op in Mathematics - Major/Major Co-op in Mathematics |
Mathematics |

- Specialist in Evolutionary Anthropology - Major in Evolutionary Anthropology - Specialist in Socio-Cultural Anthropology - Major in Socio-Cultural Anthropology - Specialist in Sociology - Major in Sociology |
Social Science - General |

- Major in Theatre and Performance Studies | Dramatic Arts |

- Specialist/Specialist Co-op in English - Major/Major Co-op in English |
English |

- Specialist/Specialist Co-op in French - Major/Major Co-op in French |
French (Second Language) |

- Specialist in History - Major in History |
History |

- Specialist in Human Geography - Major in Human Geography |
Geography |

**Application Process:**

- Applicants must apply to the Honours Bachelor of Arts (HBA)/ Honours Bachelor of Science (HBSc) program, the MT program and the CDP.
- Qualified students in Year 3 of their HBA/ HBSc degree program apply to the MT program; those accepted will receive a conditional offer to start the MT program upon completion of their HBA/ HBSc program and degree requirements.

**Minimum Admission Requirements:**

To be considered for **conditional admission to the MT program and the selected CDP,** applicants must meet the following admission requirements:

- Be admitted to the HBA/ HBSc degree and at least one of the above-listed undergraduate programs at UTSC.
- Meet the admission requirements of the School of Graduate Studies and the MT program.
- Be enrolled full-time and in good standing in the HBA/ HBSc program(s):
- have a B+ average or higher in Year 2;
- carry a full course load of 5.0 credits each year (i.e., complete 5.0 credits over the three academic sessions - Fall, Winter, Summer); where necessary, exceptions will be made for students in Co-op programs.

- Have completed at least half of the teaching subjects' prerequisite courses - i.e., 3.0 credits in the first teaching subject and at least 1.5 credits in the second teaching subject (or 3.0 credits if the second teaching subject is French as a Second Language or Science) - by the end of Year 3.
- Provide at least two letters of reference (see: http://www.oise.utoronto.ca/mt/Home.html).
- Provide a Statement of Intent indicating their preferred concentration (Primary/Junior, Junior Intermediate, or Intermediate/Senior) and describe three significant teaching and/or teaching-related experiences they have had, especially with groups of children; with reference to these experiences, applicants should identify insights gained about teaching and learning, and explain how, based on these insights, they might contribute to the education of students in today's schools. On their resumé, applicants must list, in chart form, the extent of their teaching experiences; the chart should include dates, location of the experience, applicants' role, and number of hours working with students.
- Meet other qualifications as specified by the MT program, including: a police record check, relevant teaching experiences, academic and professional references, and satisfying teaching subject prerequisites.

To be given **full, unconditional admission to the MT program**, applicants must meet the following admission requirements:

- Maintain a B+ average or higher in their final year of study in the HBA/ HBSc program, or over upper-level (C- and D-level) courses.
- Achieve at least a B+ average in 1.0 credit in graduate courses taken in Year 4.
- Regardless of the concentration to which they are applying (Primary/Junior, Junior/Intermediate, Intermediate/Senior), complete the prerequisites for both the first and second teaching subjects; students are encouraged to consult often with their HBA/HBSc Program Supervisor, as well as the Combined Degree Programs Coordinator.
- Be conferred with the HBA/ HBSc degree.

**Program Requirements and Path to Completion:**

- Year 1 to 4: HBA/ HBSc degree requirements:
- students must complete all of the HBA/ HBSc program and degree requirements;
- students are expected to carry a full course load of 5.0 credits over the three academic sessions (Fall, Winter, Summer) of each year;
- in Year 3, qualified students may apply to the MT and the CDP and may be offered conditional admission to the MT;
- by the end of Year 3 students must complete at least 3.0 credits required for the first teaching subject, and at least 1.5 credits for the second teaching subject (or 3.0 credits if the second teaching subject is French as a Second Language or Science);
- in Year 4, students who receive a conditional offer of admission to the CDP must complete any two of the graduate elective half courses recommended by OISE for CDP students; these courses (1.0 credit) are counted towards the completion of both the HBA/ HBSc degree and the MT program and degree; CDP students are graded as graduate students in these courses and are required to meet graduate expectations;
- by the end of Year 4, students must complete all HBA/ HBSc program requirements and degree requirements, including at least 6.0 credits required for the first teaching subject, and at least 3.0 credits for the second teaching subject (or 6.0 credits if the second teaching subject is French as a Second Language or Science).

- Year 5 and 6: Remaining MT program and degree requirements:
- students must complete 11.0 credits as identified by OISE.

### SPECIALIST PROGRAM IN MATHEMATICS (SCIENCE)

Supervisor of Studies: R. Grinnell (416-287-5655) Email: raymond.grinnell@utoronto.ca

**Program Objectives**

This program provides the student with a sound foundation in the main areas of mathematics, and some exposure to computer programming and statistics. It comprises three streams: Comprehensive, Statistics, and Teaching, each serving a more specific goal.

The **Comprehensive Stream** provides a broad and deep knowledge of mathematics at the undergraduate level. It is the recommended program for students who plan to pursue graduate study in mathematics, but it is also suitable for other career paths.

The **Statistics Stream** provides greater exposure to statistics, and the areas of mathematics most closely associated with it. This stream prepares students for careers in industry, or for graduate study in certain mathematically-oriented subjects, including statistics and financial mathematics.

The **Teaching Stream** is intended for students with a serious interest in mathematics but whose career objectives lie in mathematics education at the elementary or secondary level.

**Enrolment Requirements**

Enrolment in the Specialist Program in Mathematics (all streams) is limited.

Students may apply to enter the program after completing 4.0 credits, and must have passed all of the A-level MAT and CSC courses required in the program ([CSCA08H3 or CSCA20H3], CSCA67H3/MATA67H3, MATA22H3, MATA31H3, and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage.

**Program Requirements**

The Program requirements consist of a core 15 courses (7.5 credits), common to all streams, and additional requirements that depend on the stream, for a total of 26-27 courses (13.0-13.5 credits).

The structure of the programs allows for easy switching between streams until relatively late. Consequently, these programs should not be viewed as rigidly separated channels feeding students to different career paths, but as a flexible structure that provides guidance to students in their course selection based on their broad (but possibly fluid) interests.

**Core (7.5 credits)**

**1. Writing Requirement (0.5 credit)(*)**

*0.5 credits from the following:* ANTA01H3, ANTA02H3, (CLAA02H3), (CTLA19H3), CTLA01H3, ENGA10H3, ENGA11H3, ENGB06H3, ENGB07H3, ENGB08H3, ENGB09H3, ENGB17H3, ENGB19H3, ENGB50H3, (ENGB51H3), GGRA02H3, GGRA03H3, GGRB05H3, (GGRB06H3), (HISA01H3), (HLTA01H3), ACMA01H3, (HUMA01H3), (HUMA11H3), (HUMA17H3), (LGGA99H3), LINA01H3, PHLA10H3, WSTA01H3.

(*) It is recommended that this requirement be satisfied by the end of the second year.

**2. A-level courses (2.5 credits)**

[CSCA08H3 Introduction to Computer Science I or CSCA20H3 Introduction to Programming]

MATA22H3 Linear Algebra I for Mathematical Sciences

MATA31H3 Calculus I for Mathematical Sciences

MATA37H3 Calculus II for Mathematical Sciences

[MATA67H3 or CSCA67H3 Discrete Mathematics]

**3. B-level courses (3.5 credits)**

MATB24H3 Linear Algebra II

MATB41H3 Techniques of the Calculus of Several Variables I

MATB42H3 Techniques of the Calculus of Several Variables II

MATB43H3 Introduction to Analysis

MATB44H3 Differential Equations I

STAB52H3 Introduction to Probability (**)

STAB57H3 Introduction to Statistics (**)

(**) This course may be taken after the second year, except for the Statistics stream.

**4. C-level courses (1.0 credit)**

MATC01H3 Groups and Symmetry

MATC34H3 Complex Variables

**A. Comprehensive Stream**

This stream requires a total of 27 courses (13.5 credits) In addition to the core requirements 1-4 common to all streams, 12 other distinct courses must be chosen satisfying all of the following requirements:

**5. Additional courses in analysis and algebra (1.5 credits):**

1.5 credits from the following:

MATC37H3 Introduction to Real Analysis

MATC46H3 Differential Equations II

MATD01H3 Fields and Groups

MATD35H3 Introduction to Discrete Dynamical Systems

MATD46H3 Partial Differential Equations

*6. Courses in key areas of mathematics (1.0 credit):*

1.0 credit from the following:** **

MATC15H3 Introduction to Number Theory

MATC27H3 Introduction to Topology

MATC63H3 Differential Geometry

MATD02H3 Classical Plane Geometries and their Transformations

MATD34H3 Complex Variables II

*7. Mathematics of computation (1.0 credit):*

1.0 credit from the following:

CSCC37H3 Introduction to Numerical Algorithms for Computational Mathematics

CSCC63H3 Computability and Computational Complexity

CSCC73H3 Algorithm Design and Analysis

MATC09H3 Introduction to Mathematical Logic

MATC32H3 Graph Theory and Algorithms for its Applications

MATC44H3 Introduction to Combinatorics

MATD16H3 Coding Theory and Cryptography

MATD44H3 Topics in Combinatorics

*8. Electives (2.5 credits):*

2.5 credits from CSC/MAT/STA/PHY of which at least 1.5 must be at the C- or D-level MAT courses (excluding MATC90H3).

**B. Statistics Stream**

This stream requires a total of 26 courses (13.0 credits). In addition to the core requirements 1-4 common to all streams, 11 other distinct courses must be chosen, satisfying all of the following requirements (in choosing courses to satisfy requirements 7-9, students must select at least one D-level course).

*5. Algebra and Analysis (1.5 credits):*

MATB61H3 Linear Programming and Optimization

MATC46H3 Differential Equations II

MATD01H3 Fields and Groups

*6. Statistics (1.5 credits):*

STAC58H3 Statistical Inference

STAC62H3 Stochastic Processes

STAC67H3 Regression Analysis

*7. Discrete mathematics and geometry (0.5 credit):*

0.5 credit from the following:

MATC32H3 Graph Theory and Algorithms for its Applications

MATC44H3 Introduction to Combinatorics

MATD02H3 Classical Plane Geometries and their Transformations

MATD44H3 Topics in Combinatorics

MATD50H3 Mathematical Introduction to Game Theory

*8. Upper-level MAT electives (1.0 credit):*

1.0 credit from any C- or D-level MAT courses (*)

(*) For students wishing to pursue graduate studies in Mathematics or Statistics it is recommended that MATC37H3 be chosen as one of these two courses.

*9. Upper-level STA electives (1.0 credit):*

1.0 credit from the following:

(ACTB47H3) Introductory Life Contingencies

Any C- or D-level STA course, excluding STAC32H3, STAC53H3 and STAD29H3

**C. Teaching Stream**

This stream requires a total of 26 courses (13.0 credits). In addition to the core requirements 1-4 common to all streams, 11 other distinct courses must be chosen, satisfying all of the following requirements:

*5. Algebra, analysis, and geometry (1.5 credits):*

1.5 credits from the following:

MATC15H3 Introduction to Number Theory

MATD01H3 Fields and Groups

MATD02H3 Classical Plane Geometries and their Transformations

MATD35H3 Introduction to Discrete Dynamical Systems

MATD46H3 Partial Differential Equations

*6. Discrete mathematics (0.5 credit):*

0.5 credit from the following:

MATC32H3 Graph Theory and Algorithms for its Applications

MATC44H3 Introduction to Combinatorics

MATD44H3 Topics in Combinatorics

*7. MAT electives (1.5 credits):*

1.5 credits of any C- or D-level MAT courses

*8. MAT/STA/CSC electives (2.0 credits):*

2.0 credits of any C- or D-level MAT, STA, CSC courses, excluding STAC32H3, STAC53H3 and STAD29H3

It is recommended that students obtain a TA-ship within the Department of Computer and Mathematical Sciences.

### SPECIALIST (CO-OPERATIVE) PROGRAM IN MATHEMATICS (SCIENCE)

Supervisor of Studies: R. Grinnell, (416-287-5655) Email: raymond.grinnell@utoronto.ca

Co-op Contact: __askcoop@utoronto.ca__

The Specialist (Co-operative) Program in Mathematics is a Work Integrated Learning (WIL) program that combines academic studies with paid work terms in the public, private, and/or non-profit sectors. The program provides students with the opportunity to develop the academic and professional skills required to pursue employment in these areas, or to continue on to graduate training in an academic field related to Mathematics upon graduation.

In addition to their academic course requirements, students must successfully complete the additive Arts & Science Co-op Work Term Preparation courses and a minimum of three Co-op work terms.

**Enrolment Requirements**

Enrolment in the Specialist (Co-operative) Program in Mathematics is limited.

*Current Co-op Students:*Students admitted to a Co-op Degree POSt in their first year of study must request a Co-op Subject POSt on ACORN upon completion of 4.0 credits and must have passed all of the A-level CSC and MAT courses required in the program ([CSCA08H3 or CSCA20H3], CSCA67H3/MATA67H3, MATA22H3, MATA31H3, and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage. In addition, they must also have achieved a CGPA of at least 2.5 across all attempted courses.

*Prospective Co-op Students:*Prospective students (i.e., those not already admitted to a Co-op Degree POSt) may apply to the Co-op Program after completing 4.0 credits, and must have passed all of the A-level CSC and MAT courses required in the program ([CSCA08H3 or CSCA20H3], CSCA67H3/MATA67H3, MATA22H3, MATA31H3, and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage. In addition, they must also have achieved a CGPA of at least 2.5 across all attempted courses.

In addition to requesting the program on ACORN, prospective Co-op students (i.e., those not yet admitted to a Co-op Degree POSt) must also submit a Co-op Supplementary Application Form, which is available from the Arts & Science Co-op Office website. Submission deadlines follow the Limited Enrolment Program Application Deadlines set by the Office of the Registrar each year. Failure to submit both the Supplementary Application Form and the program request on ACORN will result in that student’s application not being considered.

**Program Requirements **Students must complete the program requirements as described in the Specialist Program in Mathematics.

**Co-op Work Term Requirements**Students must satisfactorily complete three Co-op work terms, each of four-months duration. To be eligible for their first work term, students must be enrolled in the Specialist (Co-op) Program in Mathematics and have completed at least 10.0 credits.

In addition to their academic program requirements, Co-op students complete up to five Co-op specific courses. These courses are designed to prepare students for their job search and work term experience, and to maximize the benefits of their Co-op work terms. They cover a variety of topics intended to assist students in developing the skills and tools required to secure work terms that are appropriate to their program of study, and to perform professionally in the workplace. These courses must be completed in sequence, and are taken in addition to a full course load. They are recorded on transcripts as credit/no credit (CR/NCR) and are considered to be additive credit to the 20.0 required degree credits. No additional course fee is assessed as registration is included in the Co-op Program fee.

*Co-op Preparation Course Requirements:*

1. COPB50H3/(COPD01H3) – Foundations for Success in Arts & Science Co-op

- Students entering Co-op from outside of UTSC (high school or other postsecondary) will complete this course in Fall or Winter of their first year at UTSC. Enrolment in each section is based on admission category: Typically, students in Computer Science, Mathematics and Statistics enroll in the Fall semester while all other Arts & Science Co-op admission categories enroll in the Winter semester however this may vary year to year.

- Current UTSC students entering Co-op in April/May will complete this course in the Summer semester.

- Current UTSC students entering Co-op in July/August will complete this course in the Fall semester.

2. COPB51H3/(COPD03H3) – Preparing to Compete for your Co-op Work Term

- This course will be completed eight months in advance of the first scheduled work term.

3. COPB52H3/(COPD11H3) – Managing your Work Term Search & Transition to Work

- This course will be completed four months in advance of the first work scheduled work term.

4. COPC98H3/(COPD12H3) – Integrating Your Work Term Experience Part I

- This course will be completed four months in advance of the second scheduled work term.

5. COPC99H3/(COPD13H3) – Integrating Your Work Term Experience Part II

- This course will be completed four months in advance of the third scheduled work term (for programs that require the completion of 3 work terms and/or four months in advance of any additional work terms that have been approved by the Arts and Science Co-op Office.

Students must be available for work terms in each of the Fall, Winter and Summer semesters and must complete at least one of their required work terms in either a Fall or Winter semester. This, in turn, requires that students take courses during at least one Summer semester.

For information on fees, status in Co-op programs, and certification of completion of Co-op programs, see Section 6B.5 or the Arts and Science Co-op section in the UTSC *Calendar*.

### MAJOR PROGRAM IN MATHEMATICS (SCIENCE)

Supervisor of Studies: N. Breuss (416-287-7226) Email: n.breuss@utoronto.ca

**Program Objectives**This program provides a solid foundation in basic areas of mathematics, especially those with applications in other disciplines. This program is intended to be combined with other programs, typically a major program in another discipline.

**Enrolment Requirements**

Enrolment in the Major Program in Mathematics is limited.

Students may apply to enter the program after completing 4.0 credits, and must have passed all of the A-level MAT and CSC courses required in the program (CSCA08H3, CSCA67H3/MATA67H3, MATA22H3, MATA31H3 and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage.

**Program Requirements**

This stream requires a total of 8.5 credits, chosen so as to satisfy all of the following requirements:

**1. Foundational courses - 5.5 credits from the following:**

[MATA67H3 or CSCA67H3 Discrete Mathematics]

MATA22H3 Linear Algebra I for Mathematical Sciences

MATA31H3 Calculus I for Mathematical Sciences

MATA37H3 Calculus II for Mathematical Sciences

CSCA08H3 Introduction to Computer Science I

MATB24H3 Linear Algebra II

MATB41H3 Techniques of the Calculus of Several Variables I

MATB42H3 Techniques of the Calculus of Several Variables II

MATB44H3 Differential Equations I

STAB52H3 Introduction to Probability

[MATC01H3 Groups and Symmetry OR MATC15H3 Introduction to Number Theory]

**2. Further analysis courses - 1.0 credit from the following:**

MATB43H3 Introduction to Analysis

MATC27H3 Introduction to Topology

MATC34H3 Complex Variables

MATC46H3 Differential Equations II

MATD35H3 Introduction to Discrete Dynamical Systems

MATD46H3 Partial Differential Equations

MATD67H3 - Differentiable Manifolds

**3. Further algebra, geometry, and discrete mathematics courses - 1.0 credit from the following:**

MATC01H3 Groups and Symmetry

MATC09H3 Introduction to Mathematical Logic

MATC15H3 Introduction to Number Theory

MATC32H3 Graph Theory and Algorithms for its Applications

MATC44H3 Introduction to Combinatorics

MATC63H3 Differential Geometry

MATD01H3 Fields and Groups

MATD02H3 Classical Plane Geometries and their Transformations

MATD44H3 Topics in Combinatorics

**4. Elective courses - 1.0 credit from the following:**

MATB61H3 Linear Programming and Optimization

STAB57H3 Introduction to Statistics

MATD50H3 Mathematical Introduction to Game Theory

Any C- or D-level MAT, STA, or CSC course, excluding STAC32H3, STAC53H3 and STAD29H3

**Recommended Writing Course**Students are urged to take a course from the following list of courses by the end of their second year.

ANTA01H3, ANTA02H3, (CLAA02H3), (CTLA19H3), CTLA01H3, ENGA10H3, ENGA11H3, ENGB06H3, ENGB07H3, ENGB08H3, ENGB09H3, ENGB17H3, ENGB19H3, ENGB50H3, (ENGB51H3), GGRA02H3, GGRA03H3, GGRB05H3, (GGRB06H3), (HISA01H3), (HLTA01H3), ACMA01H3, (HUMA01H3), (HUMA11H3), (HUMA17H3), (LGGA99H3), LINA01H3, PHLA10H3, PHLA11H3, WSTA01H3.

### MAJOR (CO-OPERATIVE) PROGRAM IN MATHEMATICS (SCIENCE)

Supervisor of Studies: N. Breuss (416-287-7226), n.breuss@utoronto.ca

Co-op Contact: __askcoop@utoronto.ca__

The Major (Co-op) Program in Mathematics is a Work Integrated Learning (WIL) program that combines academic studies with paid work terms in the public, private, and/or non-profit sectors. The program provides students with the opportunity to develop the academic and professional skills required to pursue employment in these areas, or to continue on to graduate training in an academic field related to Mathematics upon graduation.

In addition to their academic course requirements, students must successfully complete the additive Arts & Science Co-op Work Term Preparation courses and a minimum of three Co-op work terms.

**Enrolment Requirements **

Enrolment in the Major (Co-operative) Program in Mathematics is limited.

*Current Co-op Students: *Students admitted to a Co-op Degree POSt in their first year of study must request a Co-op Subject POSt on ACORN upon completion of 4.0 credits and must have passed all of the A-level CSC and MAT courses required in the program (CSCA08H3, CSCA67H3/MATA67H3, MATA22H3, MATA31H3 and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage. In addition, they must also have achieved a CGPA of at least 2.5 across all attempted courses.

*Prospective Co-op Students:*Prospective students (i.e., those not already admitted to a Co-op Degree POSt) may apply to the Co-op Program after completing 4.0 credits, and must have passed all of the A-level CSC and MAT courses required in the program (CSCA08H3, CSCA67H3/MATA67H3, MATA22H3, MATA31H3 and MATA37H3). Students are admitted on the basis of academic performance in program courses; for more information about the admission requirements, please visit the following CMS webpage. In addition, they must also have achieved a CGPA of at least 2.5 across all attempted courses.

In addition to requesting the program on ACORN, prospective Co-op students (i.e., those not yet admitted to a Co-op Degree POSt) must also submit a Co-op Supplementary Application Form, which is available on the Arts & Science Co-op Office website. Submission deadlines follow the Limited Enrolment Program Application Deadlines set by the Office of the Registrar’s Office each year. Failure to submit both the Supplementary Application Form and the program request on ACORN will result in that student’s application not being considered.

**Program Requirements**Students must complete the program requirements as described in the Major Program in Mathematics.

Co-op Work Term Requirements

Students must satisfactorily complete three Co-op work terms, each of four-months duration. To be eligible for their first work term, students must be enrolled in the Major (Co-op) Program in Mathematics and have completed at least 7.0 credits.

In addition to their academic program requirements, Co-op students complete up to five Co-op specific courses. These courses are designed to prepare students for their job search and work term experience, and to maximize the benefits of their Co-op work terms. They cover a variety of topics intended to assist students in developing the skills and tools required to secure work terms that are appropriate to their program of study, and to perform professionally in the workplace. These courses must be completed in sequence, and are taken in addition to a full course load. They are recorded on transcripts as credit/no credit (CR/NCR) and are considered to be additive credit to the 20.0 required degree credits. No additional course fee is assessed as registration is included in the Co-op Program fee.

*Co-op Preparation Course Requirements:*

1. COPB50H3/(COPD01H3) – Foundations for Success in Arts & Science Co-op

- Students entering Co-op from outside of UTSC (high school or other postsecondary) will complete this course in Fall or Winter of their first year at UTSC. Enrolment in each section is based on admission category: Typically, students in Computer Science, Mathematics and Statistics enroll in the Fall semester while all other Arts & Science Co-op admission categories enroll in the Winter semester however this may vary year to year.

- Current UTSC students entering Co-op in April/May will complete this course in the Summer semester.

- Current UTSC students entering Co-op in July/August will complete this course in the Fall semester.

2. COPB51H3/(COPD03H3) – Preparing to Compete for your Co-op Work Term

- This course will be completed eight months in advance of the first scheduled work term.

3. COPB52H3/(COPD11H3) – Managing your Work Term Search & Transition to Work

- This course will be completed four months in advance of the first work scheduled work term.

4. COPC98H3/(COPD12H3) – Integrating Your Work Term Experience Part I

- This course will be completed four months in advance of the second scheduled work term.

5. COPC99H3/(COPD13H3) – Integrating Your Work Term Experience Part II

- This course will be completed four months in advance of the third scheduled work term (for programs that require the completion of 3 work terms and/or four months in advance of any additional work terms that have been approved by the Arts and Science Co-op Office.

Students must be available for work terms in each of the Fall, Winter and Summer semesters and must complete at least one of their required work terms in either a Fall or Winter semester. This, in turn, requires that students take courses during at least one Summer semester.

For information on fees, status in Co-op programs, and certification of completion of Co-op programs, see Section 6B.5 or the Arts and Science Co-op section in the UTSC *Calendar*.

### Mathematics Courses

### MATA02H3 - The Magic of Numbers

A selection from the following topics: the number sense (neuroscience of numbers); numerical notation in different cultures; what is a number; Zeno’s paradox; divisibility, the fascination of prime numbers; prime numbers and encryption; perspective in art and geometry; Kepler and platonic solids; golden mean, Fibonacci sequence; elementary probability.

**Exclusion:**(MATA20H3), MATA23H3, MATA30H3, MATA31H3, MATA32H3, MAT102H, MAT123H, MAT125H, MAT133Y, MAT134Y, MAT135Y, MAT137Y, MAT157Y

**Breadth Requirements:**Quantitative Reasoning

### MATA22H3 - Linear Algebra I for Mathematical Sciences

A conceptual and rigorous approach to introductory linear algebra that focuses on mathematical proofs, the logical development of fundamental structures, and essential computational techniques. This course covers complex numbers, vectors in Euclidean n-space, systems of linear equations, matrices and matrix algebra, Gaussian reduction, structure theorems for solutions of linear systems, dependence and independence, rank equation, linear transformations of Euclidean n-space, determinants, Cramer's rule, eigenvalues and eigenvectors, characteristic polynomial, and diagonalization.

**Prerequisite:**Grade 12 Calculus and Vectors or [Grade 12 Advanced Functions and Introductory Calculus and Geometry and Discrete Mathematics]

**Exclusion:**MATA23H3, MAT223H, MAT240H

**Breadth Requirements:**Quantitative Reasoning

**Note:**

Students are cautioned that MAT223H cannot be used as a substitute for MATA22H3 in any courses for which MATA22H3 appears as a prerequisite.

### MATA23H3 - Linear Algebra I

Systems of linear equations, matrices, Gaussian elimination; basis, dimension; dot products; geometry to Rn; linear transformations; determinants, Cramer's rule; eigenvalues and eigenvectors, diagonalization.

**Prerequisite:**Grade 12 Calculus and Vectors or [Grade 12 Advanced Functions and Introductory Calculus and Geometry and Discrete Mathematics]

**Exclusion:**MATA22H3, MAT223H

**Breadth Requirements:**Quantitative Reasoning

### MATA29H3 - Calculus I for the Life Sciences

A course in differential calculus for the life sciences. Algebraic and transcendental functions; semi-log and log-log plots; limits of sequences and functions, continuity; extreme value and intermediate value theorems; approximation of discontinuous functions by continuous ones; derivatives; differentials; approximation and local linearity; applications of derivatives; antiderivatives and indefinite integrals.

**Prerequisite:**Grade 12 Calculus and Vectors

**Exclusion:**(MATA20H3), (MATA27H3), MATA30H3, MATA31H3, MATA32H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA30H3 - Calculus I for Physical Sciences

An introduction to the basic techniques of Calculus. Elementary functions: rational, trigonometric, root, exponential and logarithmic functions and their graphs. Basic calculus: limits, continuity, derivatives, derivatives of higher order, analysis of graphs, use of derivatives; integrals and their applications.

**Prerequisite:**Grade 12 Calculus and Vectors

**Exclusion:**(MATA20H3), (MATA27H3), MATA29H3, MATA31H3, MATA32H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA31H3 - Calculus I for Mathematical Sciences

A conceptual introduction to Differential Calculus of algebraic and transcendental functions of one variable; focus on logical reasoning and fundamental notions; first introduction into a rigorous mathematical theory with applications. Course covers: real numbers, set operations, supremum, infimum, limits, continuity, Intermediate Value Theorem, derivative, differentiability, related rates, Fermat's, Extreme Value, Rolle's and Mean Value Theorems, curve sketching, optimization, and antiderivatives.

**Prerequisite:**Grade 12 Calculus and Vectors

**Exclusion:**(MATA20H3), (MATA27H3), MATA29H3, MATA30H3, MATA32H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA32H3 - Calculus for Management I

This is a calculus course with most examples and applications of an economic nature. Topics to be covered: introduction to financial mathematics; continuous functions including exponential and logarithmic functions with applications to finance; differential calculus of one variable; marginal analysis; optimization of single variable functions; techniques of integration.

**Prerequisite:**Grade 12 Calculus and Vectors

**Exclusion:**(MATA20H3), (MATA27H3), MATA30H3, MATA31H3, MAT123H, MAT125H, MAT133Y, MAT135Y, MAT136Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA33H3 - Calculus for Management II

This course will introduce the students to multivariable calculus and linear algebra. Topics will include: linear programming (geometric); matrix algebra; multi-variable functions; contour maps; partial and total differentiation; optimization of multi-variable functions; optimization of constrained multi-variable functions; Lagrange multipliers.

**Prerequisite:**MATA32H3

**Exclusion:**(MATA21H3), (MATA27H3), MATA35H3, MATA36H3, MATA37H3, MAT124H, MAT126H, MAT133Y, MAT134Y, MAT135Y, MAT136Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA35H3 - Calculus II for Biological Sciences

A calculus course emphasizing examples and applications in the biological and environmental sciences. Discrete probability; basic statistics: hypothesis testing, distribution analysis. Basic calculus: extrema, growth rates, diffusion rates; techniques of integration; differential equations; population dynamics; vectors and matrices in 2 and 3 dimensions; genetics applications.

**Prerequisite:**MATA29H3

**Exclusion:**(MATA21H3), MATA33H3, MATA36H3, MATA37H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y,(MATA27H3)

**Breadth Requirements:**Quantitative Reasoning

**Note:**

This course will not satisfy the Mathematics requirements for any Program in Computer and Mathematical Sciences, nor will it normally serve as a prerequisite for further courses in Mathematics. Students who are not sure which Calculus II course they should choose are encouraged to consult with the supervisor(s) of Programs in their area(s) of interest.

### MATA36H3 - Calculus II for Physical Sciences

This course is intended to prepare students for the physical sciences. Topics to be covered include: techniques of integration, Newton's method, approximation of functions by Taylor polynomials, numerical methods of integration, complex numbers, sequences, series, Taylor series, differential equations.

**Prerequisite:**MATA30H3 or MATA31H3

**Exclusion:**(MATA21H3), MATA33H3, MATA35H3, MATA37H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA37H3 - Calculus II for Mathematical Sciences

A rigorous introduction to Integral Calculus of one variable and infinite series; strong emphasis on combining theory and applications; further developing of tools for mathematical analysis. Riemann Sum, definite integral, Fundamental Theorem of Calculus, techniques of integration, improper integrals, numerical integration, sequences and series, absolute and conditional convergence of series, convergence tests for series, Taylor polynomials and series, power series and applications.

**Prerequisite:**MATA31H3 and [MATA67H3 or CSCA67H3]

**Exclusion:**(MATA21H3), MATA33H3, MATA35H3, MATA36H3, MAT123H, MAT124H, MAT125H, MAT126H, MAT133Y, MAT135Y, MAT137Y, MAT157Y, JMB170Y

**Breadth Requirements:**Quantitative Reasoning

### MATA67H3 - Discrete Mathematics

Introduction to discrete mathematics: Elementary combinatorics; discrete probability including conditional probability and independence; graph theory including trees, planar graphs, searches and traversals, colouring. The course emphasizes topics of relevance to computer science, and exercises problem-solving skills and proof techniques such as well ordering, induction, contradiction, and counterexample.

Same as CSCA67H3

**Prerequisite:**Grade 12 Calculus and Vectors and one other Grade 12 mathematics course

**Exclusion:**CSCA67H3, (CSCA65H3), CSC165H, CSC240H, MAT102H

**Recommended Preparation:**CSCA08H3 or CSCA20H3

**Breadth Requirements:**Quantitative Reasoning

### MATB24H3 - Linear Algebra II

Fields, vector spaces over a field, linear transformations; inner product spaces, coordinatization and change of basis; diagonalizability, orthogonal transformations, invariant subspaces, Cayley-Hamilton theorem; hermitian inner product, normal, self-adjoint and unitary operations. Some applications such as the method of least squares and introduction to coding theory.

**Prerequisite:**MATA22H3 or MAT240H

**Exclusion:**MAT224H

**Breadth Requirements:**Quantitative Reasoning

**Note:**

Students are cautioned that MAT224H cannot be used as a substitute for MATB24H3 in any courses for which MATB24H3 appears as a prerequisite.

### MATB41H3 - Techniques of the Calculus of Several Variables I

Partial derivatives, gradient, tangent plane, Jacobian matrix and chain rule, Taylor series; extremal problems, extremal problems with constraints and Lagrange multipliers, multiple integrals, spherical and cylindrical coordinates, law of transformation of variables.

**Prerequisite:**[MATA22H3 or MATA23H3 or MAT223H] and [[MATA36H3 or MATA37H3] or MAT137Y or MAT157Y]]

**Exclusion:**MAT232H, MAT235Y, MAT237Y, MAT257Y

**Breadth Requirements:**Quantitative Reasoning

### MATB42H3 - Techniques of the Calculus of Several Variables II

Fourier series. Vector fields in Rn, Divergence and curl, curves, parametric representation of curves, path and line integrals, surfaces, parametric representations of surfaces, surface integrals. Green's, Gauss', and Stokes' theorems will also be covered. An introduction to differential forms, total derivative.

**Prerequisite:**MATB41H3

**Exclusion:**MAT235Y, MAT237Y, MAT257Y, MAT368H

**Breadth Requirements:**Quantitative Reasoning

### MATB43H3 - Introduction to Analysis

Generalities of sets and functions, countability. Topology and analysis on the real line: sequences, compactness, completeness, continuity, uniform continuity. Topics from topology and analysis in metric and Euclidean spaces. Sequences and series of functions, uniform convergence.

**Prerequisite:**[MATA37H3 or MAT137Y] and MATB24H3

**Exclusion:**MAT246Y

**Breadth Requirements:**Quantitative Reasoning

### MATB44H3 - Differential Equations I

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.

**Prerequisite:**[MATA36H3 or MATA37H3] and [MATA22H3 or MATA23H3]

**Corequisite:**MATB41H3

**Exclusion:**MAT244H, MAT267H

**Breadth Requirements:**Quantitative Reasoning

### MATB61H3 - Linear Programming and Optimization

Linear programming, simplex algorithm, duality theory, interior point method; quadratic and convex optimization, stochastic programming; applications to portfolio optimization and operations research.

**Prerequisite:**[MATA22H3 or MATA23H3] and MATB41H3

**Exclusion:**APM236H

**Breadth Requirements:**Quantitative Reasoning

### MATC01H3 - Groups and Symmetry

Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange's theorem. Normal subgroups, quotient groups. Emphasis on examples and calculations.

**Prerequisite:**[MATA36H3 or MATA37H3] and [MATB24H3 or MAT224H]

**Exclusion:**MAT301H, MAT347Y

**Breadth Requirements:**Quantitative Reasoning

### MATC09H3 - Introduction to Mathematical Logic

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.

**Prerequisite:**MATB24H3 and [MATB43H3 or CSCB36H3]

**Exclusion:**MAT309H, CSC438H

**Breadth Requirements:**Quantitative Reasoning

### MATC15H3 - Introduction to Number Theory

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:**MATB24H3 and MATB41H3

**Exclusion:**MAT315H

**Breadth Requirements:**Quantitative Reasoning

### MATC27H3 - Introduction to Topology

Fundamentals of set theory, topological spaces and continuous functions, connectedness, compactness, countability, separatability, metric spaces and normed spaces, function spaces, completeness, homotopy.

**Prerequisite:**MATB41H3 and MATB43H3

**Exclusion:**MAT327H

**Breadth Requirements:**Quantitative Reasoning

### MATC32H3 - Graph Theory and Algorithms for its Applications

Graphs, subgraphs, isomorphism, trees, connectivity, Euler and Hamiltonian properties, matchings, vertex and edge colourings, planarity, network flows and strongly regular graphs; applications to such problems as timetabling, personnel assignment, tank form scheduling, traveling salesmen, tournament scheduling, experimental design and finite geometries.

**Prerequisite:**[MATB24H3 or CSCB36H3] and at least one other B-level course in Mathematics or Computer Science

**Breadth Requirements:**Quantitative Reasoning

### MATC34H3 - Complex Variables

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.

**Prerequisite:**MATB42H3

**Exclusion:**MAT334H

**Breadth Requirements:**Quantitative Reasoning

### MATC37H3 - Introduction to Real Analysis

### MATC44H3 - Introduction to Combinatorics

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:**MATB24H3

**Exclusion:**MAT344H

**Breadth Requirements:**Quantitative Reasoning

### MATC46H3 - Differential Equations II

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.

**Prerequisite:**MATB44H3

**Corequisite:**MATB42H3

**Exclusion:**APM346H

**Breadth Requirements:**Quantitative Reasoning

### MATC58H3 - An Introduction to Mathematical Biology

Mathematical analysis of problems associated with biology, including models of population growth, cell biology, molecular evolution, infectious diseases, and other biological and medical disciplines. A review of mathematical topics: linear algebra (matrices, eigenvalues and eigenvectors), properties of ordinary differential equations and difference equations.

**Prerequisite:**MATB44H3

**Breadth Requirements:**Quantitative Reasoning

### MATC63H3 - Differential Geometry

Curves and surfaces in Euclidean 3-space. Serret-Frenet frames and the associated equations, the first and second fundamental forms and their integrability conditions, intrinsic geometry and parallelism, the Gauss-Bonnet theorem.

**Prerequisite:**MATB42H3 and MATB43H3

**Exclusion:**MAT363H

**Breadth Requirements:**Quantitative Reasoning

### MATC82H3 - Mathematics for Teachers

The course discusses the Mathematics curriculum (K-12) from the following aspects: the strands of the curriculum and their place in the world of Mathematics, the nature of proofs, the applications of Mathematics, and its connection to other subjects.

**Prerequisite:**[MATA67H3 or CSCA67H3 or (CSCA65H3)] and [MATA22H3 or MATA23H3] and [MATA37H3 or MATA36H3]

**Exclusion:**MAT382H

**Breadth Requirements:**Quantitative Reasoning

### MATC90H3 - Beginnings of Mathematics

Mathematical problems which have arisen repeatedly in different cultures, e.g. solution of quadratic equations, Pythagorean theorem; transmission of mathematics between civilizations; high points of ancient mathematics, e.g. study of incommensurability in Greece, Pell's equation in India.

**Prerequisite:**10.0 credits, including 2.0 credits in MAT courses [excluding MATA02H3], of which 0.5 credit must be at the B-level

**Exclusion:**MAT390H

**Breadth Requirements:**Quantitative Reasoning

### MATD01H3 - Fields and Groups

Abstract group theory: Sylow theorems, groups of small order, simple groups, classification of finite abelian groups. Fields and Galois theory: polynomials over a field, field extensions, constructibility; Galois groups of polynomials, in particular cubics; insolvability of quintics by radicals.

**Prerequisite:**MATC01H3

**Exclusion:**(MAT302H), MAT347Y, (MATC02H3)

**Recommended Preparation:**MATC34H3

**Breadth Requirements:**Quantitative Reasoning

### MATD02H3 - Classical Plane Geometries and their Transformations

An introduction to geometry with a selection of topics from the following: symmetry and symmetry groups, finite geometries and applications, non-Euclidean geometry.

**Prerequisite:**[MATA22H3 or MATA23H3]

**Corequisite:**MATC01H3

**Exclusion:**MAT402H, (MAT365H), (MATC25H3)

**Breadth Requirements:**Quantitative Reasoning

### MATD10H3 - Topics in Mathematics

### MATD11H3 - Topics in Mathematics

### MATD12H3 - Topics in Mathematics

### MATD16H3 - Coding Theory and Cryptography

The main problems of coding theory and cryptography are defined. Classic linear and non-linear codes. Error correcting and decoding properties. Cryptanalysis of classical ciphers from substitution to DES and various public key systems [e.g. RSA] and discrete logarithm based systems. Needed mathematical results from number theory, finite fields, and complexity theory are stated.

**Prerequisite:**MATC15H3 and STAB52H3

**Exclusion:**(MATC16H3)

**Breadth Requirements:**Quantitative Reasoning

### MATD26H3 - Geometric Analysis and Relativity

An intuitive and conceptual introduction to general relativity with emphasis on a rigorous treatment of relevant topics in geometric analysis. The course aims at presenting rigorous theorems giving insights into fundamental natural phenomena. Contents: Riemannian and Lorentzian geometry (parallelism, geodesics, curvature tensors, minimal surfaces), Hyperbolic differential equations (domain of dependence, global hyperbolicity). Relativity (causality, light cones, inertial observes, trapped surfaces, Penrose incompleteness theorem, black holes, gravitational waves).

**Prerequisite:**MATC63H3

**Exclusion:**APM426H1

**Breadth Requirements:**Quantitative Reasoning

### MATD34H3 - Complex Variables II

Applications of complex analysis to geometry, physics and number theory. Fractional linear transformations and the Lorentz group. Solution to the Dirichlet problem by conformal mapping and the Poisson kernel. The Riemann mapping theorem. The prime number theorem.

**Prerequisite:**MATB43H3 and MATC34H3

**Exclusion:**MAT354H, (MATC65H3)

**Breadth Requirements:**Quantitative Reasoning

### MATD35H3 - Introduction to Discrete Dynamical Systems

This course provides an introduction and exposure to dynamical systems, with particular emphasis on low-dimensional systems such as interval maps and maps of the plane. Through these simple models, students will become acquainted with the mathematical theory of chaos and will explore strange attractors, fractal geometry and the different notions of entropy. The course will focus mainly on examples rather than proofs; students will be encouraged to explore dynamical systems by programming their simulations in Mathematica.

**Prerequisite:**[[MATA37H3 or MATA36H3] with a grade of B+ or higher] and MATB41H3 and MATC34H3

**Breadth Requirements:**Quantitative Reasoning

### MATD44H3 - Topics in Combinatorics

### MATD46H3 - Partial Differential Equations

This course provides an introduction to partial differential equations as they arise in physics, engineering, finance, optimization and geometry. It requires only a basic background in multivariable calculus and ODEs, and is therefore designed to be accessible to most students. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' bag of tools.

**Prerequisite:**[[MATA37H3 or MATA36H]3 with grade of at least B+] and MATB41H3 and MATB44H3

**Breadth Requirements:**Quantitative Reasoning

### MATD50H3 - Mathematical Introduction to Game Theory

This course introduces students to combinatorial games, two-player (matrix) games, Nash equilibrium, cooperative games, and multi-player games. Possible additional topics include: repeated (stochastic) games, auctions, voting schemes and Arrow's paradox. Numerous examples will be analyzed in depth, to offer insight into the mathematical theory and its relation to real-life situations.

**Prerequisite:**MATB24H3 and STAB52H3

**Exclusion:**MAT406H

**Breadth Requirements:**Quantitative Reasoning

### MATD67H3 - Differentiable Manifolds

Manifolds, vector fields, tangent spaces, vector bundles, differential forms, integration on manifolds.

**Prerequisite:**MATB43H3

**Exclusion:**MAT367H1

**Breadth Requirements:**Quantitative Reasoning

### MATD92H3 - Mathematics Project

A significant project in any area of mathematics. The project may be undertaken individually or in small groups. This course is offered by arrangement with a mathematics faculty member. This course may be taken in any session and the project must be completed by the last day of classes in the session in which it is taken.

**Prerequisite:**[1.5 credits at the C-level in MAT courses] and [permission of the Supervisor of Studies] and [a CGPA of at least 3.0 or enrolment in a Mathematics Subject POSt]

**Breadth Requirements:**Quantitative Reasoning

**Note:**

Enrolment procedures: the project supervisor's note of agreement must be presented to the Supervisor of Studies who will issue permission for registration.

### MATD93H3 - Mathematics Project

A significant project in any area of mathematics. The project may be undertaken individually or in small groups. This course is offered by arrangement with a mathematics faculty member. This course may be taken in any session and the project must be completed by the last day of classes in the session in which it is taken.

**Prerequisite:**[1.5 credits at the C-level in MAT courses] and [permission of the Supervisor of Studies] and [a CGPA of at least 3.0 or enrolment in a Mathematics Subject POSt]

**Breadth Requirements:**Quantitative Reasoning

**Note:**

Enrolment procedures: the project supervisor's note of agreement must be presented to the Supervisor of Studies who will issue permission for registration.

### MATD94H3 - Readings in Mathematics

Independent study under direction of a faculty member.

**Prerequisite:**[1.5 credits at the C-level in MAT courses] and [permission of the Supervisor of Studies] and [a CGPA of at least 3.0 or enrolment in a Mathematics Subject POSt]

**Note:**

Enrolment procedures: the project supervisor's note of agreement must be presented to the Supervisor of Studies who will issue permission for registration.

### MATD95H3 - Readings in Mathematics

Independent study under direction of a faculty member.

**Prerequisite:**[1.5 credits at the C-level in MAT courses] and permission of the Supervisor of Studies] and [a CPGA of at least 3.0 or enrolment in a Mathematics Subject POSt]

**Note:**