This course examines cross-cultural language use by second language learners from both a theoretical and pedagogical perspective. Topics addressed include the role of pragmatic transfer between native and target languages, individual differences, learning context, and instruction in the development of second language pragmatic competence. Written work to be completed in French/Italian for credit towards a Specialist (French or Italian) or Major (French/Italian).

This course offers a comprehensive survey and analysis of fundamental concepts and issues related to second, bilingual, and foreign language instruction by developing students' knowledge of second language acquisition, approaches to language teaching, computer-assisted teaching, and pedagogical design and implementation in the language classroom. Written work to be completed in French/Italian for credit towards a Specialist (French/Italian) or Major (French/Italian).

This course will conduct a critical appraisal of online course materials, and formulate appropriate pedagogical strategies for their exploitation. This course is taught in English and is open to students from other disciplines. Written work to be completed in French/Italian for credit towards a Specialist (French/Italian) or Major (French/Italian).

A research or reading project undertaken by the student under the supervision of a faculty member. Written work to be completed in French/Italian for credit towards an LTL program in French or Italian.

A research or reading project undertaken by the student under the supervision of a faculty member. Written work to be completed in French/Italian for credit towards an LTL program in French or Italian.

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.

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.

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.

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.

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.

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.

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.

Continuation of MAT137H5. 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.

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.

Continuation of MAT157H5. 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.

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.

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.

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.

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.

"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.

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

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.

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.

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.

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.  

Most applications of Mathematics involve the use of a computer. Numerical analysis studies how formulas can be transformed into computations. The topics covered may include: numerical methods in Calculus, such as series expansions and rates of convergence, numerical integration and differentiation, finite interpolation methods, splines; and numerical methods for ordinary differential equations, such as root-finding methods, Fourier series and Fourier transform, least-squares approximation, regression, and principal component analysis.

This course provides a richly rewarding opportunity for students in their second year to work in the research project of a professor in return for 299H 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 Research Opportunity Program (ROP) for more details.

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.

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.

(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.