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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
- 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), Professor
- X. Jiang, B.Sc., M.Sc., Ph.D. (Glasgow), Associate Professor, Teaching Stream
- 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
- J. Scherk, B.Sc., M.Sc. (Toronto), D.Phil. (Oxford), Associate Professor
- 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), CLTA 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

Associate Chair: M. Evans evans@utsc.utoronto.ca (416-287-7274)

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 was 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 Mathematics Specialist and Major programs (Non Co-op and Co-op). Details and information on how to apply for admission to these programs is found in the program descriptions below.

### Combined Degree Programs, Honours Bachelor of Science (various) or Honours Bachelor of Arts (various)/ Master of Teaching

The Combined Degree Programs for Honours Bachelor of Science/Honours Bachelor of Arts programs at UTSC (various) and the Master of Teaching (MT) offered by the Ontario Institute for Studies in Education are designed for students interested in studying the intersections of the Physical Sciences, Mathematical Sciences, or French, and Education coupled with professional teacher preparation. They allow exceptional students who are registered in specified Specialist and Major programs to apply during their third year of studies, and be considered, for admission to the MT.

**The Combined Degree Programs options include:**

- 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 this *Calendar*.

**Service Learning and Outreach**

For an experiential learning opportunity that also serves others, consider the course CTLB03H3 (Introduction to Service Learning), which can be found in the "Teaching and Learning” section of the *Calendar*.

### Mathematics Programs

### SPECIALIST PROGRAM IN MATHEMATICS (SCIENCE)

Supervisor of Studies: R. Grinnell (416-287-5655) Email: grinnell@utsc.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 with a CGPA of at least 2.5 across the core A-level courses are guaranteed admission.

**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 Probablity (**)

STAB57H3 Introduction to Statistics (**)

(**) This course may be taken after 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):**

MATC37H3 Introduction to Real Analysis

MATC46H3 Differential Equations II

MATD01H3 Fields and Groups

*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

*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. Regression Analysis (0.5 credit):*

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

*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 (2.0 credits):*

2.0 credits from the following:

(ACTB47H3) Introductory Life Contingencies

Any C- or D-level STA course, excluding 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):*

MATC15H3 Introduction to Number Theory

MATD01H3 Fields and Groups

MATD02H3 Classical Plane Geometries and their Transformations

*6. Discrete mathematics (0.5 credit):*

0.5 credit from the following:

MATC32H3 Graph Theory and Algorithms for its Applications

MATC44H3 Introduction to 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 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: grinnell@utsc.utoronto.ca

Co-op Contact: askcoop@utsc.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 with a CGPA of 2.5 or greater across the core A-level courses required in the program, as well as a CGPA of at least 2.5 across all attempted courses are guaranteed admission.

*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). Only students with a CGPA of at least 2.5 across the core A-level courses, as well as a CGPA of at least 2.5 across all attempted courses, will be considered for admission to the Co-op Program.

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 (http://www.utsc.utoronto.ca/askcoop/future-co-op-students). 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. 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 of their first year at UTSC

- Current UTSC students entering Co-op in April/May will complete this course in the summer term

- Current UTSC students entering Co-op in July/August will complete this course in the fall term

2. COPD03H3 – Preparing to Compete for your Co-op Work Term

- Prerequisite: COPD01H3

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

3. COPD11H3 – Managing your Work Term Search & Transition to Work

- Prerequisite: COPD03H3

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

4. COPD12H3 – Integrating Your Work Term Experience Part I

- Prerequisite: COPD11H3 and one Co-op work term

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

5. COPD13H3 – Integrating Your Work Term Experience Part II

- Prerequisite: COPD12H3 and two Co-op work terms

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

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

For information on fees, status in Co-op programs, and certification of completion of Co-op programs, see Section 6B.5 of 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, [MATA30H3 or MATA31H3], and [MATA36H3 or MATA37H3]). Students with a CGPA of at least 2.0 across the core A-level courses are guaranteed admission.

**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

[MATA30H3 Calculus I for Physical Sciences OR MATA31H3 Calculus I for Mathematical Sciences]

[MATA36H3 Calculus II for Physical Sciences OR 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]

(*) MATA31H3 is required for MATA37H3

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

MATB43H3 Introduction to Analysis

MATC27H3 Introduction to Topology

MATC34H3 Complex Variables

MATC37H3 Introduction to Real Analysis

MATC46H3 Differential Equations II

MATD34H3 Complex Variables II

**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

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

MATB61H3 Linear Programming and Optimization

STAB57H3 Introduction to Statistics

Any C- or D-level MAT, STA, or CSC course, excluding 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@utsc.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, [MATA30H3 or MATA31H3], and [MATA36H3 or MATA37H3]). Students with a CGPA of at least 2.5 across the core A-level courses, as well as a CGPA of at least 2.5 across all attempted courses, are guaranteed admission.

*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, [MATA30H3 or MATA31H3], and [MATA36H3 or MATA37H3]). Only students with a CGPA of at least 2.5 across the core A-level courses, as well as a CGPA of at least 2.5 across all attempted courses, will be considered for admission to the Co-op Program.

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 (http://www.utsc.utoronto.ca/askcoop/future-co-op-students). 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 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. 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 of their first year at UTSC

- Current UTSC students entering Co-op in April/May will complete this course in the summer term

- Current UTSC students entering Co-op in July/August will complete this course in the fall term

2. COPD03H3 – Preparing to Compete for your Co-op Work Term

- Prerequisite: COPD01H3

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

3. COPD11H3 – Managing your Work Term Search & Transition to Work

- Prerequisite: COPD03H3

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

4. COPD12H3 – Integrating Your Work Term Experience Part I

- Prerequisite: COPD11H3 and one Co-op work term

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

5. COPD13H3 – Integrating Your Work Term Experience Part II

- Prerequisite: COPD12H3 and two Co-op work terms

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

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

For information on fees, status in Co-op programs, and certification of completion of Co-op programs, see Section 6B.5 of 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

### 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), 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), 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 or MATA30H3

**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 MAT223H

**Exclusion:**MAT224H

**Breadth Requirements:**Quantitative Reasoning

### 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:**MATB24H3 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:**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 and [an additional 1.0 credit at the A-level in MAT courses [excluding MATA02H3]]

**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:**MATC34H3

**Exclusion:**MAT354H, ( MATC65H3)

**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:**