Mathematical Sciences

Mathematical Sciences Programs | Mathematical Sciences Courses

Faculty and Staff List

Professors Emeriti
I.R. Graham, B.Sc., Ph.D.
F.D. Tall, A.B., Ph.D.
S. Tanny, B.Sc., Ph.D.
A. Wilk, B.Sc.
W. Weiss, B.Sc., M.Sc., Ph.D.

Professors
I. Binder, B.Sc., Ph.D.
A. Burazin, Hon.B.Sc., M.Sc, Ph.D.
D. Dauvergne, B.Sc., M.Sc, Ph.D.
J. De Simoi, B.Sc., M.Sc., Ph.D.
J. Desjardins, B.Sc., M.Sc., Ph.D.
S. Fuchs, B.Sc., M.Sc., Ph.D.
M. Groechenig, B.Sc., Ph.D.
T. Holden, B.Math., M.Math., Ph.D.
M. Karrass, Hons B.A., B.Ed., M.A., Ph.D.
Y. Karshon, B.Sc., M.Sc., Ph.D.
K. Khanin, M.Sc., Ph.D.
Y. Liokumovich, B.Sc., M.Sc., Ph.D.
M. Pawliuk, B.Sc. (Hons), M.Sc., Ph.D.
A. Rennet, B.A., Ph.D.
L. Seco, B.Sc, Ph.D
A. Shankar, B.Sc., Ph.D.
Y. Shlapentokh-Rothman, B.A., M.A., Ph.D.
J. Thind, Honours B.Sc., M.Sc., Ph.D.
M. Tvalavadze, B.Sc., Ph.D.
S. Unger, B.Sc., Ph.D.
M. Wesslén, B.Sc., M.Sc., Ph.D.
M. Yampolsky, Ph.D.
T. Yusun, B.Sc., M.Sc, Ph.D.
K. Zhang, B.Sc., Ph.D.

Chair
Ilia Binder
Room 3016, Deerfield Hall
905-828-3834
chairmcs.utm@utoronto.ca

Vice-Chair, Mathematics and Statistics
Jacopo De Simoi
Room 3040, Deerfield Hall
905-569-5698
jacopods@math.utoronto.ca

Associate Chair, Mathematics
Maria Wesslén
Room 3048, Deerfield Hall
905-828-5323
maria.wesslen@utoronto.ca

Academic Advisor and Undergraduate Program Administrator
Laura Ferlito
laura.ferlito@utoronto.ca
www.utm.utoronto.ca/math-cs-stats

 

Mathematics teaches one to think analytically and creatively. It is a foundation for advanced careers in a knowledge-based economy. The past century has been a remarkable one for discovery in mathematics. Problems in computer science, physics, biology, and economics have opened new fields of mathematical inquiry, and discoveries at the most abstract level, for example in number theory, have led to breakthroughs in applied areas.

The Mathematical Sciences Specialist Program at U of T Mississauga provides students with a solid foundation in the fundamental theoretical aspects of the mathematical sciences along with a broad range of techniques for applying this theory. The Major and Minor Programs in Mathematical Sciences consist largely of MAT courses, and may be combined with programs in other subjects.

First-year Courses

Most first-year students at U of T Mississauga take courses in calculus. This could be either a year-long Y course (MAT133Y5) or two one-term H courses (MAT132H5 and MAT134H5) or (MAT135H5 and MAT136H5) or (MAT137H5 and MAT139H5) or (MAT157H5 and MAT159H5).

MAT133Y5 serves students in Commerce, Economics or Management who do not wish to take courses in Mathematics or Statistics beyond the 100 level. It cannot be used as a prerequisite for any other MAT course, except for students who have also completed MAT233H5. MAT133Y5 counts as a Science course for distribution purposes.

Students in the above subjects who wish to do a Major in Statistics or Mathematics should take (MAT135H5 and MAT136H5) or (MAT137H5 and MAT139H5) or (MAT157H5 and MAT159H5) instead.

(MAT132H5 and MAT134H5) and (MAT135H5 and MAT136H5) are comparable in terms of mathematical content and difficulty, but differ in the nature of applications. (MAT132H5 and MAT134H5) are specifically designed for students in the Life Sciences, and are often taken at the same time as (BIO152H5 and BIO153H5). (MAT135H5 and MAT136H5) give a sense of the wide-ranging applications of calculus to the physical, biological and social sciences.

(MAT137H5 and MAT139H5) and (MAT157H5 and MAT159H5) are for students who know when they enter university that they wish to study Mathematics or Computer Science). Students in Statistics, Physics or Economics who are mathematically inclined will also enjoy these courses. (MAT157H5 and MAT159H5) offer the rigour and depth needed to prepare students for advanced studies in mathematics. Students with a very serious interest in mathematics, including those who wish to pursue graduate-level studies in mathematics, should therefore consider taking (MAT157H5 and MAT159H5).

*Note: Effective September 1, 2023, in the Fall 2023 – Winter 2024 academic session, MAT137Y5 and MAT157Y5 will be split into two one-semester H courses. MAT137Y5 will be split into MAT137H5 and MAT139H5, and MAT157Y5 will be split into MAT157H5 and MAT159H5.

MAT102H5 is a special course for beginning Mathematical Sciences students. It is intended to bridge the gap between high school mathematics, where mathematical proofs and logical arguments are often omitted, and university-level mathematics, where proofs are critical to a full understanding of the material.

A wide variety of upper-level courses is available to students who have the proper prerequisites. Students should feel free to consult the department regarding course selection.

Students should also review the Degree Requirements section prior to selecting courses

Program website: www.utm.utoronto.ca/math-cs-stats

Mathematical Sciences Programs

Mathematical Sciences - Specialist (Science)

Mathematical Sciences - Specialist (Science)

The Specialist Program in Mathematical Sciences is primarily directed toward students who hope to pursue graduate studies in, or related to mathematics.

Enrolment Requirements:

Limited Enrolment — Enrolment in the Specialist program is limited to students with a minimum of 4.0 credits, including:

  1. MAT102H5 (minimum 65%);
  2. MAT137Y5 or MAT139H5 (minimum 60%) or MAT157Y5 or MAT159H5; and
  3. A minimum cumulative grade point average (CGPA), to be determined annually.
  4. All students must complete 4.0 U of T credits before requesting this program. Courses with a grade of CR/NCR will not count as a part of the 4.0 credits required for program entry.

Completion Requirements:

13.5 credits are required.

First Year:

  1. CSC108H5 and CSC148H5
  2. MAT102H5 and MAT240H5
  3. [( MAT137H5 or MAT157H5) and ( MAT139H5 or MAT159H5)] or MAT137Y5 or MAT157Y5

Second Year:

  1. CSC236H5
  2. MAT202H5 and MAT244H5 and MAT247H5 and MAT257Y5
  3. STA256H5 and ( STA258H5 or STA260H5)

Higher Years:

  1. MAT301H5 and ( MAT334H5 or MAT354H5) and MAT392H5
  2. MAT302H5 or MAT315H5
  3. 2.0 additional credit from MAT302H5 or MAT309H5 or MAT311H5 or MAT315H5 or MAT332H5 or ( MAT337H5 or MAT378H5) or MAT344H5
  4. 1.0 additional credits in MAT at the 400 level ( MAT401H5 is recommended)
  5. 1.0 additional credits at the 300/400 level in CSC or MAT/ STA, except MAT322H5
  6. 0.5 additional credits in MAT at the 300+level, except MAT322H5

NOTES:

  1. Mathematical Science Specialists are strongly encouraged to enroll in MAT157H5, MAT159H5, MAT257Y5, and MAT354H5.
  2. Students are strongly encouraged to familiarize themselves with the 100-level calculus pre-requisites to select the correct courses.
  3. Students may replace MAT257Y5 with [( MAT232H5 or MAT233H5) and MAT236H5), but if they do then MAT337H5 AND MAT405H5 are required as part of “Higher Years”.
  4. Students who do not feel ready for MAT257Y5 in their Second Year, may wish to take MAT232H5 that year, and then take MAT257Y5 in their Third Year.

ERSPE2511

Mathematical Sciences - Major (Science)

Mathematical Sciences - Major (Science)

Enrolment Requirements:

Limited Enrolment — Enrolment in the Major program is limited to students with a minimum of 4.0 credits, including:

  1. MAT102H5 (minimum 60%);
  2. MAT134H5 or MAT136H5 or MAT139H5 or MAT134Y5 or MAT135Y5 or MAT233H5 (minimum 60%) or MAT159H5 or MAT137Y5 or MAT157Y5; and
  3. A minimum cumulative grade point average (CGPA), to be determined annually.
  4. All students must complete 4.0 U of T credits before requesting this program. Courses with a grade of CR/NCR will not count as a part of the 4.0 credits required for program entry.

Completion Requirements:

8.0 credits are required.

First Year:

  1. MAT102H5
  2. [( MAT132H5 or MAT135H5 or MAT137H5 or MAT157H5) and ( MAT134H5 or MAT136H5 or MAT139H5 or MAT159H5)] or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT157Y5
  3. MAT223H5 or MAT240H5

Second Year:

  1. MAT202H5 and MAT244H5
  2. [( MAT232H5 or MAT233H5) and MAT236H5] or MAT257Y5
  3. MAT224H5 or MAT247H5

Higher Years:

  1. MAT301H5 and ( MAT334H5 or MAT354H5)
  2. MAT337H5 or MAT378H5 or MAT392H5 or MAT405H5
  3. MAT305H5 or MAT311H5 or MAT332H5
  4. MAT302H5 or MAT315H5 or MAT344H5
  5. STA256H5 or 0.5 credit of MAT at the 300/ 400 level, except MAT322H5
  6. 0.5 additional credits in MAT at the 400 level

NOTES:

  1. MAT137H5 and MAT139H5 are recommended.
  2. Students are strongly encouraged to familiarize themselves with the 100-level calculus pre-requisites to select the correct courses.
  3. Mathematical Majors are strongly encouraged to enroll in MAT240H5 followed by MAT247H5.

ERMAJ2511

Mathematical Sciences - Minor (Science)

Mathematical Sciences - Minor (Science)

Completion Requirements:

4.0 credits in MAT are required, including 1.0 credit of MAT at the 300/400 level.

First Year:

  1. MAT102H5
  2. [( MAT132H5 or MAT135H5 or MAT137H5 or MAT157H5) and ( MAT134H5 or MAT136H5 or MAT139H5 or MAT159H5)] or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT157Y5

Second Year:

  1. MAT223H5 or MAT240H5
  2. [ MAT232H5 and ( MAT202H5 or MAT224H5 or MAT236H5 or MAT240H5 or MAT244H5 or MAT247H5 or CSC236H5)] or MAT257Y5

Higher Years:

  1. 1.0 credit of MAT at the 300+ level

NOTES:

  1. MAT223H5 or MAT240H5 may be taken in the first year.
  2. Students may replace the combination [( MAT132H5 or MAT135H5 or MAT137H5 or MAT157H5) and ( MAT134H5 or MAT136H5 or MAT139H5 or MAT159H5)] or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT157Y5 and MAT232H5 with the combination ( MAT133Y5 and MAT233H5)
  3. Students are strongly encouraged to familiarize themselves with the 100-level calculus pre-requisites to select the correct courses.

ERMIN2511

Mathematical Sciences Courses

MAT102H5 • Introduction to Mathematical Proofs

Understanding, using and developing precise expressions of mathematical ideas, including definitions and theorems. Set theory, logical statements and proofs, induction, topics chosen from combinatorics, elementary number theory, Euclidean geometry.

Prerequisites: Minimum 70% in Grade 12 Advanced Functions (MHF4U)
Exclusions: MAT138H1 or MAT246H1 or CSC165H1 or CSCA67H3
Recommended Preparation: Minimum 70% in Grade 12 Calculus and Vectors (MCV4U)

Distribution Requirement: Science
Total Instructional Hours: 41L/12T
Mode of Delivery: In Class

MAT132H5 • Differential Calculus for Life Sciences

Review of functions and their graphs, trigonometry, exponentials and logarithms. Limits and continuity of functions of a single variable. Derivatives and differentiation techniques. Applications of differentiation, including extreme values, related rates and optimization. Life science applications are emphasized.

Prerequisites: Minimum 70% in Grade 12 Advanced Functions (MHF4U) 
Exclusions: MAT133Y5 or MAT134Y5 or MAT135H5 or MAT135Y5 or MAT137H5 or MAT137Y5 or MAT157H5 or MAT157Y5 or MAT133Y1 or MAT135Y1 or MAT135H1 or MAT137Y1 or MAT157Y1 or MATA29H3 or MATA30H3 or MATA31H3 or MATA32H3
Recommended Preparation: Highly Recommended: Minimum 70% in Grade 12 Calculus and Vectors (MCV4U)
Enrolment Limits: Restricted to students in a Life Science Program.

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class

MAT133Y5 • Calculus and Linear Algebra for Commerce

Mathematics of finance, matrices and linear equations. Review of differential calculus; applications. Integration and fundamental theorem; applications. Introduction to partial differentiation; applications. NOTE: This course cannot be used as the calculus prerequisite for any 200-level MAT or STA course, except in combination with MAT233H5.

Prerequisites: Minimum 70% in Grade 12 Advanced Functions (MHF4U).
Exclusions: MAT132H5 or MAT134H5 or MAT135H5 or MAT136H5 or MAT137H5 or MAT139H5 or MAT157H5 or MAT159H5 or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT157Y5 or MAT135H1 or MAT136H1 or MAT133Y1 or MAT135Y1 or MAT137Y1 or MAY157Y1 or MATA30H3 or MATA31H3 or MATA32H3 or MATA33H3 or MATA35H3 or MATA36H3 or MATA37H3
Recommended Preparation: Highly Recommended: Minimum 70% in Grade 12 Calculus and Vectors (MCV4U).
Enrolment Limits: This course cannot be used for the specialist or major programs in Mathematics, Statistics or Computer Science, except in combination with MAT233H5. Restricted to students admitted into Management or Commerce.

Distribution Requirement: Science
Total Instructional Hours: 80L/24T
Mode of Delivery: In Class

MAT134H5 • Integral Calculus for Life Sciences

Continuation of MAT132H5. Antiderivatives and indefinite integrals in one variable, definite integrals and the fundamental theorem of calculus. Integration techniques and applications of integration. Infinite sequences, series and convergence tests. Power series, Taylor and Maclaurin series. Life science applications are emphasized.

Prerequisites: MAT132H5 or MAT135H5 or MAT137H5 or MAT157H5 or MAT135H1 or MATA29H3 or MATA30H3 or MATA31H3
Exclusions: MAT133Y5 or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT139H5 or MAT133Y1 or MAT135Y1 or MAT136H1 or MAT136H5 or MAT137Y1 or MAT157Y1 or MAT157Y5 or MAT159H5 or MATA33H3 or MATA35H3 or MATA36H3 or MATA37H3
Enrolment Limits: Restricted to students in a Life Science Program.

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class

MAT135H5 • Differential Calculus

Review of functions and their graphs, trigonometry, exponentials and logarithms. Limits and continuity of functions of a single variable. Derivatives and differentiation techniques. Applications of differentiation, including extreme values, related rates and optimization. A wide range of applications from the sciences will be discussed.

Prerequisites: Minimum 70% in Grade 12 Advanced Functions (MHF4U) 
Exclusions: MAT132H5 or MAT133Y5 or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT137H5 or MAT133Y1 or MAT135Y1 or MAT135H1 or MAT137Y1 or MAT157Y1 or MAT157Y5 or MAT157H5 or MATA29H3 or MATA30H3 or MATA31H3 or MATA32H3
Recommended Preparation: Highly Recommended: Minimum 70% in Grade 12 Calculus and Vectors (MCV4U)

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class

MAT136H5 • Integral Calculus

Continuation of MAT135H5. Antiderivatives and indefinite integrals in one variable, definite integrals and the fundamental theorem of calculus. Integration techniques and applications of integration. Infinite sequences, series and convergence tests. Power series, Taylor and Maclaurin series. A wide range of applications from the sciences will be discussed. 

Prerequisites: MAT132H5 or MAT135H5 or MAT137H5 or MAT157H5 or MAT135H1 or MATA29H3 or MATA30H3 or MATA31H3
Exclusions: MAT133Y5 or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT139H5 or MAT133Y1 or MAT135Y1 or MAT136H1 or MAT134H5 or MAT137Y1 or MAT157Y1 or MAT157Y5 or MAT159H5 or MATA33H3 or MATA35H3 or MATA36H3 or MATA37H3

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class

MAT137H5 • Differential Calculus for Mathematical Sciences

A conceptual approach to calculus. A focus on theoretical foundations and proofs as well as some emphasis on geometric and physical intuition. Limits and continuity, differentiation, the mean value, extreme value and inverse function theorems. Applications typically include related rates and optimization.

Prerequisites: Minimum 70% in Grade 12 Advanced Functions (MHF4U) and Minimum 70% in Grade 12 Calculus and Vectors (MCV4U).
Exclusions: MAT132H5 or MAT133Y5 or MAT134Y5 or MAT135H5 or MAT137Y5 or MAT135Y5 or MAT157H5 or MAT157Y5 or MAT133Y1 or MAT135Y1 or MAT135H1 or MAT137Y1 or MAT157Y1 or MATA30H3 or MATA31H3 or MATA32H3 or MATA33H3 or MATA35H3 or MATA36H3 or MATA37H3

Distribution Requirement: Science
Total Instructional Hours: 40L/24T
Mode of Delivery: In Class

MAT139H5 • Integral Calculus for Mathematical Sciences

A conceptual sequel to MAT137H5. Integration, the fundamental theorem of calculus, sequences and series, power series and Taylor’s theorem. Applications typically include approximation, integration techniques, areas and volumes.

Prerequisites: MAT137H5 or MAT157H5
Exclusions: MAT133Y5 or MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137H5 or MAT157H5 or MAT157Y5 MAT133Y1 or MAT135Y1 or MAT135H1 or MAT137Y1 or MAT157Y1 or MATA30H3 or MATA31H3 or MATA32H3 or MATA33H3 or MATA35H3 or MATA36H3 or MATA37H3

Distribution Requirement: Science
Total Instructional Hours: 40L/24T
Mode of Delivery: In Class

MAT157H5 • Analysis I

A rigorous and proof-intensive introduction to the analysis of single variable real-valued functions for students with a serious interest in mathematics. Topics typically include the construction of the real numbers, the epsilon-delta definition of the limit, continuity, and differentiation.

Prerequisites: [Minimum 70% in Grade 12 Advanced Functions (MHF4U)] and [Minimum 70% in Grade 12 Calculus and Vectors (MCV4U)]
Corequisites: MAT102H5 (strongly recommended in the Fall term for students taking MAT157H5 in their first year).
Exclusions: MAT157Y5 or MAT157Y1 or MATA37H3

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT159H5 • Analysis II

A rigorous and proof-intensive sequel to MAT157H5 for students with a serious interest in mathematics. Topics typically include sequences, series, and integration of single variable real-valued functions.

Prerequisites: MAT157H5
Exclusions: MAT157Y5 or MAT157Y1 or MATA37H3

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT202H5 • Introduction to Discrete Mathematics

Mathematics derives its great power from its ability to formulate abstract concepts and techniques. In this course, students will be introduced to abstraction and its power through a study of topics from discrete mathematics. The topics covered will include: Sets, relations and functions; Basic counting techniques: subsets, permutations, finite sequences, inclusion-exclusion; Discrete probability: random variables paradoxes and surprises; Basic number theory: properties of the integers and the primes. The course will emphasize active participation of the students in discussion and written assignments.

Prerequisites: MAT102H5 and (MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5 or MAT233H5)
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT223H5 • Linear Algebra I

Systems of linear equations, matrix algebra, determinants. Vector geometry in R2 and R3. Complex numbers. Rn: subspaces, linear independence, bases, dimension, column spaces, null spaces, rank and dimension formula. Orthogonality, orthonormal sets, Gram-Schmidt orthogonalization process, least square approximation. Linear transformations from Rn to Rm. The determinant, classical adjoint, Cramer's rule. Eigenvalues, eigenvectors, eigenspaces, diagonalization. Function spaces and applications to a system of linear differential equations. The real and complex number fields.

Prerequisites: Grade 12 Advanced Functions (MHF4U) and Grade 12 Calculus and Vectors (MCV4U or MAT102H5).
Exclusions: MAT223H1 or MATA22H3 or MATA23H3 or MAT240H1 or MAT240H5

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class, Hybrid

MAT224H5 • Linear Algebra II

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

Prerequisites: MAT102H5 and MAT223H5
Exclusions: MAT240H5 or MAT224H1 or MATB24H3

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT232H5 • Calculus of Several Variables

Differential and integral calculus of several variables: partial differentiation, chain rule, extremal problems, Lagrange multipliers, classification of critical points. Multiple integrals, Green's theorem and related topics.

Prerequisites: MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5
Exclusions: MAT233H5 or MAT235Y1 or MAT237Y1 or MAT257Y5 or MAT257Y1 or MATB41H3
Recommended Preparation: MAT223H5 or MAT240H5

Distribution Requirement: Science
Total Instructional Hours: 40L/12T
Mode of Delivery: In Class

MAT233H5 • Calculus of Several Variables

"Bridging Course"; accepted as prerequisite for upper level courses in replacement of MAT232H5. Limited Enrolment. Sequences and series, power series, Taylor series, trigonometric and inverse trigonometric functions and their use in integrations. Differential and integral calculus of several variables; partial differentiation, chain rule, extremal problems, Lagrange multipliers, classification of critical points. Multiple integrals, Green's theorem and related topics.

Prerequisites: MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5 or 65% in MAT133Y5
Exclusions: MAT232H5 or MAT235Y1 or MAT237Y1 or MAT257Y1 or MAT257Y5 or MATB41H3
Recommended Preparation: MAT223H5 or MAT240H5

Distribution Requirement: Science
Total Instructional Hours: 48L/12T
Mode of Delivery: In Class

MAT236H5 • Vector Calculus

The implicit function theorem, vector fields. Transformations. Parametrized integrals. Line, surface and volume integrals. Theorems of Gauss and Stokes with applications.

Prerequisites: MAT102H5 and (MAT223H5 or MAT240H5) and (MAT232H5 or MAT233H5)
Exclusions: MAT235Y1 or MAT237Y1 or MAT257Y1 or MAT257Y5 or MATB42H3
Enrolment Limits: Priority is given to students enrolled in Mathematics, Computer Science and Statistics Specialist or Major programs; Astronomical Sciences Specialist (ERSPE1025) and Astronomy Major (ERMAJ2204) programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT240H5 • Algebra I

A theoretical approach to Linear Algebra and its foundations, aimed at students with a serious interest in Mathematics. Topics to be covered: Vector spaces over arbitrary fields (including C and finite fields), linear equations and matrices, bases and linear independence, linear transformations, determinants, eigenvalues and eigenvectors, similarity, change of basis, diagonalization, the characteristic and minimal polynomials, the Cayley-Hamilton theorem.

Prerequisites: 65% in MAT102H5
Exclusions: MAT224H5 or MAT224H1 or MAT240H1

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT244H5 • 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; linearization of non-linear systems. Applications in life and physical sciences. Power series solutions, boundary value problems, Fourier series solutions, numerical methods.

Prerequisites: (MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5 or MAT233H5) and (MAT223H5 or MAT240H5).
Exclusions: MAT322H5 or MAT244H1 or MAT267H1 or MATB44H3
Enrolment Limits: Priority is given to students enrolled in Mathematics, Computer Science and Statistics Specialist or Major programs; Astronomical Sciences Specialist (ERSPE1025), Astronomy Major (ERMAJ2204), Biophysics Specialist (ERSPE1944), and Physics Major (ERMAJ1944).

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT247H5 • Algebra II

Continuation of MAT240H5. A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra.

Prerequisites: MAT240H5 or MAT240H1
Exclusions: MAT247H1

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT257Y5 • Analysis III

A rigorous and proof-intensive course in multivariable calculus for students with a serious interest in mathematics.  Topology of metric spaces; compactness, functions and continuity, the extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima. Integration; Fubini's theorem, partitions of unity, change of variables. Integration on manifolds; Stokes' theorem.  

Prerequisites: (MAT157Y5 or MAT159H5) and MAT240H5
Exclusions: MAT237Y1 or MAT257Y1

Distribution Requirement: Science
Total Instructional Hours: 72L/48T
Mode of Delivery: In Class

MAT299Y5 • Research Opportunity Program

This courses provides a richly rewarding opportunity for students in their second year to work in the research project of a professor in return for 299Y course credit. Students enrolled have an opportunity to become involved in original research, learn research methods and share in the excitement and discovery of acquiring new knowledge. Participating faculty members post their project descriptions for the following summer and fall/winter sessions in early February and students are invited to apply in early March. See Experiential and International Opportunities for more details.

Prerequisites: Departmental permission.

Distribution Requirement: Science
Mode of Delivery: In Class

MAT301H5 • Groups and Symmetries

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. Classification of finitely generated Abelian Groups. Emphasis on examples and calculations.

Prerequisites: MAT102H5 and (MAT202H5 or MAT224H5 or MAT240H5)
Exclusions: MAT301H1 or MAT347Y1 or MATC01H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT302H5 • Introduction to Algebraic Cryptography

(Cross list with CSC322H5) The course will take students on a journey through the methods of algebra and number theory in cryptography, from Euclid to Zero Knowledge Proofs. Topics include: block ciphers and the Advanced Encryption Standard (AES); algebraic and number-theoretic techniques and algorithms in cryptography, including methods for primality testing and factoring large numbers; encryption and digital signature systems based on RSA, factoring, elliptic curves and integer lattices; and zero-knowledge proofs.

Prerequisites: (MAT224H5 or MAT240H5) and MAT301H5
Exclusions: CSC322H5 or MATD16H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT305H5 • Elementary Lie Theory

This course is an introduction to the theory of matrix groups with a particular emphasis on applications and examples. This course will cover orthogonal transformations in two and three dimensions, quaternions, isometries of Euclidean space, Lie algebras and matrix exponentials.

Prerequisites: [(MAT224H5 or MAT240H5) and MAT236H5] or MAT257Y5
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT307H5 • Curves and Surfaces

This course is an introduction to the theory of curves and surfaces with a particular emphasis on applications and computational techniques. This course will cover curves in R2 and R3, curvature, torsion, differential of maps, First Fundamental Form, Parallel transport, Bishop Frames, Geodesics, Gauss-Bonnet Theorem, and Gaussian curvature.


Prerequisites: [( MAT224H5 or MAT240H5) and (MAT232H5 or MAT233H5)] or MAT257Y5
Exclusions: MAT363H1 or MAT367H1or MATC63H3 or MATD26H3 or MATD67H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT309H5 • Introduction to Mathematical Logic

The relationships among axioms, proofs, consistency and truth in mathematics. Soundness and Completeness. Introductions to model theory, set theory, and computability; arithmetic as a central example. Gödel's incompleteness theorems; outlines of their proofs. This course emphasizes rigour.

Prerequisites: MAT257Y5 or [MAT236H5 and (MAT202H5 or MAT224H5 or MAT240H5) and 0.5 additional credit of MAT at the 300+ level]
Exclusions: CSC438H1 or MAT309H1 or MAT409H1 or MAT357Y1 or MATC09H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics and Computer Science Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT311H5 • Partial Differential Equations

Partial differential equations of applied mathematics, mathematical models of physical phenomena, basic methodology.

Prerequisites: MAT257Y5 or (MAT236H5 and MAT244H5)
Exclusions: APM346H1 or APM351Y1 or MAT351Y1 or MATC46H3
Enrolment Limits: Priority is given to students enrolled in Mathematics, Computer Science and Statistics Specialist or Major programs; Astronomical Sciences Specialist (ERSPE1025), Astronomy Major (ERMAJ2204).

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT315H5 • Introduction to Number Theory

Elementary topics in number theory such as: prime numbers; arithmetic with residues; Gaussian integers, quadratic reciprocity law, representation of numbers as sums of squares. (This course emphasizes rigour). 

Prerequisites: MAT102H5 and [MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5 or (MAT133Y5 and MAT233H5)] and (MAT224H5 or MAT240H5) and MAT301H5
Exclusions: MAT315H1 or MATC15H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT322H5 • Mathematical Modelling in Biology

The course will serve as an introduction to mathematical modelling of biological processes. It will cover a selection of the following topics: Difference equations and applications. Linear differential equations and systems; phase plane analysis; nonlinear systems of differential equations and linearization; Poincaré-Bendixson Theorem. Applications of differential equations to biology, including a logistic population with harvesting; predator-prey model; competing species; epidemic models. Examples of partial differential equations; reaction-diffusion equation; pattern formation.

Prerequisites: MAT102H5 and (MAT134H5 or MAT136H5 or MAT139H5 or MAT159H5 or MAT134Y5 or MAT135Y5 or MAT137Y5 or MAT157Y5 or MAT233H5) and (MAT223H5 or MAT240H5)
Exclusions: MAT388H5 (Fall 2019 and Fall 2020) or MAT244H5 or MAT244H1 or MATB44H3. or MATC58H3
Enrolment Limits: Restricted at all times to students in the Math Minor program.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: Online, In Class

MAT332H5 • Introduction to Nonlinear Dynamics and Chaos

Stability in nonlinear systems of differential equations, bifurcation theory, chaos, strange attractors, iteration of nonlinear mappings and fractals. This course will be geared towards students with interest in sciences.

Prerequisites: MAT257Y5 or [MAT236H5 and (MAT223H5 or MAT240H5) and MAT244H5]
Exclusions: MAT335H1 or MATC35H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics or Statistics Specialist or Major programs and Bioinformatic Specialist.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT334H5 • Complex Variables

Theory of functions of one complex variable: analytic and meromorphic functions; Cauchy's theorem, residue calculus. Topics from: conformal mappings, analytic continuation, harmonic functions.

Prerequisites: MAT257Y5 or [(MAT232H5 or MAT233H5) and (MAT202H5 or MAT240H5 or 0.5 additional credit of MAT at the 300+ level with a mark of at least 60%)]
Exclusions: MAT334H1 or MAT354H5 or MAT354H1 or MATC34H3
Enrolment Limits: Priority is given to students enrolled in Mathematics, Computer Science and Statistics Specialist or Major programs; Astronomical Sciences Specialist (ERSPE1025), Astronomy Major (ERMAJ2204).

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT337H5 • Introduction to Real Analysis

(Formerly MAT378H5) The real numbers; Sequences and series; Functional limits; Topology in R^n; Differentiation and Integration; Power Series; Metric Spaces; Integrability and sets of measure zero. The course emphasizes rigour and theory.

Prerequisites: MAT257Y5 or [(MAT224H5 or MAT240H5) and MAT236H5 and MAT244H5]
Exclusions: MAT378H5 or MAT337H1 or MAT357H1 or MATB43H3 or MATC37H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics or Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT344H5 • Introduction to Combinatorics

Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows).

Prerequisites: MAT102H5 and (MAT223H5 or MAT240H5)
Exclusions: MAT344H1 or MATC44H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics or Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT354H5 • Complex Analysis

Complex numbers, the complex plane and Riemann sphere, Möbius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy's theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz' lemma, residue theorem and residue calculus.

Prerequisites: MAT257Y5 or [(MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5) and (MAT202H5 or MAT240H5 or MAT337H5) and (MAT232H5 or MAT233H5)]
Exclusions: MAT334H1 or MAT334H5 or MAT354H1 or MATC34H3 or MATD34H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics or Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT382H5 • 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 the proofs, applications of Mathematics, and the connection of Mathematics to other subjects. Restricted to students in the MAT major and specialist programs. 

Prerequisites: (Minimum 60% in MAT134H5 or MAT134Y5 or MAT135Y5 or MAT136H5 or MAT137Y5 or MAT139H5 or MAT157Y5 or MAT159H5 or MAT233H5) and [minimum 60% in MAT102H5 and (MAT223H5 or MAT240H5)] and 0.5 additional credit of MAT at the 200+ level.
Exclusions: MAT329Y1 or MATC82H3

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT387H5 • Topics in Mathematics

Introduction to a topic of current interest in mathematics. Content will vary from year to year. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

Prerequisites: Appropriate prerequisite requirement(s) will be available on the UTM timetable along with the topic title prior to course registration.
Enrolment Limits: Priority is given to students enrolled in the Mathematics or Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT388H5 • Topics in Mathematics

Introduction to a topic of current interest in mathematics. Content will vary from year to year. This course may include a tutorial and/or practical section in some years. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

Prerequisites: Appropriate prerequisite requirement(s) will be available on the UTM timetable along with the topic title prior to course registration.
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT392H5 • Ideas of Mathematics

This is a one-term course to give students extensive practice in the writing of mathematics. The format will be to study excellent expositions of important ideas of mathematics and then to assign short writing assignments based on them.

Prerequisites: MAT202H5 and MAT244H5 and (MAT236H5 or MAT257Y5) and (MAT224H5 or MAT247H5)
Exclusions: MATC90H3
Enrolment Limits: Limited enrolment. The course is open only to students in the MAT major/specialist programs, with priority to students in the specialist program.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT397H5 • Further Studies in Mathematics

Students explore a topic in mathematics under the supervision of a faculty member. Interested students must consult with mathematics faculty at least two months prior to registration, to determine the topic and scope.

Prerequisites: Departmental permission and a minimum CGPA of 2.5
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Course Experience: University-Based Experience
Distribution Requirement: Science
Mode of Delivery: In Class

MAT398H5 • Further Studies in Mathematics

Students explore a topic in mathematics under the supervision of a faculty member. Interested students must consult with mathematics faculty at least two months prior to registration, to determine the topic and scope.

Prerequisites: Departmental permission and a minimum CGPA of 2.5
Enrolment Limits: Priority is given to students enrolled in Mathematics Specialist or Major programs.

Course Experience: University-Based Experience
Distribution Requirement: Science
Mode of Delivery: In Class

MAT399Y5 • Research Opportunity Program

This courses provides a richly rewarding opportunity for students in their third year to work in the research project of a professor in return for 399Y course credit. Students enrolled have an opportunity to become involved in original research, learn research methods and share in the excitement and discovery of acquiring new knowledge. Participating faculty members post their project descriptions for the following summer and fall/winter sessions in early February and students are invited to apply in early March. See Experiential and International Opportunities for more details.

Prerequisites: Departmental permission.

Distribution Requirement: Science
Mode of Delivery: In Class

MAT401H5 • Polynomial Equations and Fields

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

Prerequisites: (MAT224H5 or MAT240H5) and (MAT236H5 or MAT257Y5) and MAT301H5
Exclusions: MAT347Y1 or MAT401H1 or MATD01H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT402H5 • Classical Geometries

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

Prerequisites: MAT102H5 and (MAT232H5 or MAT233H5 or MAT257Y5) and (MAT224H5 or MAT240H5) and 0.5 additional credit of MAT at the 300+ level.
Exclusions: MAT402H1 or MATD02H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT405H5 • Introduction to Topology

Sets and functions; Topology in R^n; Topological spaces; Open and closed sets; Closure and interior; Continuous functions; Quotient spaces; Connectedness and compactness; Separation axioms and related theorems.

Prerequisites: MAT257Y5 or [(MAT224H5 or MAT240H5) and MAT236H5 and at least 0.5 additional credit of MAT at the 300+ level with a mark of at least 65%]
Exclusions: MAT327H1 or MATC27H3
Recommended Preparation: MAT337H5
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT406H5 • Mathematical Introduction to Game Theory

Combinatorial games: Nim and other impartial games; Sprague-Grundy value; existence of a winning strategy in partisan games. Two-player (matrix) games: zero-sum games and Von-Neuman's minimax theorem; general sum-matrix games, prisoner's dilemma, Nash equilibrium, cooperative games, asymmetric information. Multi-player games: coalitions and the Shapley value. Possible additional topics: repeated (stochastic) games; auctions; voting schemes and Arrow's paradox. Mathematical tools that may be introduced include hyperplane separation of convex sets and Brouwer's fixed point theorem. Numerous examples will be analyzed in depth, to offer insight to the mathematical theory and its relation with real life situations.

Prerequisites: MAT102H5 and (MAT223H5 or MAT240H5) and (STA246H5 or STA256H5)
Exclusions: ECO316H1 or MATD50H3
Enrolment Limits: Priority is given to students enrolled in the Mathematics, Computer Science and Statistics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/12T
Mode of Delivery: In Class

MAT478H5 • Topics in Mathematics

Introduction to a topic of current interest in mathematics. Content will vary from year to year. This course may include a tutorial and/or practical section in some years. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

Prerequisites: Appropriate prerequisite requirement(s) will be available on the UTM timetable along with the topic title prior to course registration.
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT488H5 • Topics in Mathematics

Introduction to a topic of current interest in mathematics. Content will vary from year to year. This course may include a tutorial and/or practical section in some years. The contact hours for this course may vary in terms of contact type (L, T) from year to year, but will be between 36-60 contact hours in total. See the UTM Timetable.

Prerequisites: Appropriate prerequisite requirement(s) will be available on the UTM timetable along with the topic title prior to course registration.
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Distribution Requirement: Science
Total Instructional Hours: 36L/24T
Mode of Delivery: In Class

MAT492H5 • Senior Thesis

An exposition on a topic in mathematics written under the supervision of a faculty member. Open to students in Mathematical Sciences Specialist program.

Prerequisites: MAT392H5 and 2.0 additional credits in MAT at the 300 level and a minimum CGPA of 2.5
Enrolment Limits: Only open to students in the MAT major/specialist programs.

Course Experience: University-Based Experience
Distribution Requirement: Science
Mode of Delivery: In Class

MAT497H5 • Further Studies in Mathematics

Students explore a topic in mathematics under the supervision of a faculty member. Interested students must consult with mathematics faculty at least two months prior to registration, to determine the topic and scope.

Prerequisites: Departmental permission and a minimum CGPA of 2.5
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Course Experience: University-Based Experience
Distribution Requirement: Science
Mode of Delivery: In Class

MAT498H5 • Further Studies in Mathematics

Students explore a topic in mathematics under the supervision of a faculty member. Interested students must consult with mathematics faculty at least two months prior to registration, to determine the topic and scope.

Prerequisites: Departmental permission and a minimum CGPA of 2.5
Enrolment Limits: Priority is given to students enrolled in the Mathematics Specialist or Major programs.

Course Experience: University-Based Experience
Distribution Requirement: Science
Mode of Delivery: In Class

MAT499Y5 • Research Opportunity Program

This courses provides a richly rewarding opportunity for students in their fourth year to work in the research project of a professor in return for 499Y course credit. Students enrolled have an opportunity to become involved in original research, learn research methods and share in the excitement and discovery of acquiring new knowledge. Participating faculty members post their project descriptions for the following summer and fall/winter sessions in early February and students are invited to apply in early March. See Experiential and International Opportunities for more details.

Prerequisites: Departmental permission.

Distribution Requirement: Science
Mode of Delivery: In Class

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