Mathematics
MAT102H5 Introduction to Mathematical Proofs (SCI)
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. [36L, 12T]MAT133Y5 Calculus and Linear Algebra for Commerce (SSc)
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. [72L, 24T]MAT134Y5 Calculus for Life Sciences (SCI)
Trigonometric functions. Limits, continuity. Techniques of differentiation and integration with applications to the life sciences. Extreme values and optimization. Graphing. The fundamental theorem of calculus. Introduction to sequences and series, power series. Introduction to discrete and continuous time modeling. [72L, 24T]MAT135Y5 Calculus (SCI)
Trigonometric functions. Limits, continuity. Review of differential calculus; applications. Graphing, extreme values and optimization. Integration and fundamental theorem; applications. Sequences and series. Power Series. Introduction to differential equations. [72L, 24T]MAT137Y5 Calculus (SCI)
A conceptual approach for students with a serious interest in mathematics. Geometric and physical intuition are emphasized but some attention is also given to the theoretical foundations of calculus. Material covers first a review of trigonometric functions followed by discussion of trigonometric identities. The basic concepts of calculus: limits and continuity, the mean value and inverse function theorem, the integral, the fundamental theorem, elementary transcendental functions, Taylor's theorem, sequences and series, power series. [72L, 24T]MAT202H5 Introduction to Discrete Mathematics (SCI)
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. [36L, 12T]MAT212H5 Modeling with Differential Equations in Life Sciences and Medicine (SCI)
Modeling with differential equations, applied to examples from Life Sciences and Medicine. Exponential and logistic growth of population, stability in first-order equations, higher order linear equations, forced oscillations, systems of first order equations, phase plane analysis, predator-prey models, modeling chemical reactions, modeling heart beat. [36L, 12T]MAT223H5 Linear Algebra I (SCI)
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. [36L, 12T]MAT224H5 Linear Algebra II (SCI)
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. [36L,12T]MAT232H5 Calculus of Several Variables (SCI)
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. [36L,12T]MAT233H5 Calculus of Several Variables (SCI)
"Bridging Course"; accepted as prerequisite for upper level courses in replacement of MAT232H5. Limited Enrolment.MAT242H5 Differential Equations I (SCI)
Solution of first order differential equations. Applications. Linear equations, especially of second order. Systems of linear equations. Nonlinear phenomena, linearization of nonlinear systems. (MAT242H5 and 252H5 replace MAT258Y5.) [36L, 12T]MAT252H5 Differential Equations II (SCI)
Power series solutions, boundary value problems, Fourier series solutions. Laplace transform, numerical methods. [36L, 12T]MAT299Y5 Research Opportunity Program (SCI)
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. SeeMAT301H5 Groups and Symmetries (SCI)
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. [36L, 12T]MAT302H5 Finite Fields and Applications (SCI)
This course will consist of an introduction to the theory of finite fields. We will also discuss some of the many practical applications of finite fields, including algebraic coding theory for the error-free transmission of information and cryptology for the secure transmission of information. [36L, 12T]MAT309H5 Introduction to Mathematical Logic (SCI)
The nature of axioms, proofs and consistency. Introduction to the theory of recursive functions. Gödel's incompleteness theorems and related results. This course emphasizes rigour. [36L, 12T]MAT311H5 Partial Differential Equations (SCI)
Partial differential equations of applied mathematics, mathematical models of physical phenomena, basic methodology. [36L,12T]MAT315H5 Introduction to Number Theory (SCI)
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). [36L,12T]MAT332H5 Introduction to Nonlinear Dynamics and Chaos (SCI)
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. [36L, 12P]MAT334H5 Complex Variables (SCI)
Theory of functions of one complex variable: analytic and meromorphic functions; Cauchy's theorem, residue calculus. Topics from: conformal mappings, analytic continuation, harmonic functions. [36L,12T]MAT344H5 Introduction to Combinatorics (SCI)
Basic counting principles, generating functions, permutations with restrictions. Fundamentals of graph theory with algorithms; applications (including network flows). [36L,12T]MAT368H5 Vector Calculus (SCI)
The implicit function theorem, vector fields. Transformations. Parametrized integrals. Line, surface and volume integrals. Theorems of Gauss and Stokes with applications. [36L,12T]MAT378H5 Introduction to Analysis (SCI)
Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension. [36L, 12T]MAT382H5 Mathematics for Teachers (SCI)
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.MAT388H5 Topics in Mathematics (SCI)
Introduction to a topic of current interest in mathematics. Content will vary from year to year. Enrolment by permission of instructors only.MAT392H5 Ideas of Mathematics (SCI)
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. [48L]MAT401H5 Polynomial Equations and Fields (SCI)
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. [36L, 12T]MAT402H5 Classical Geometries (SCI)
(Formerly MAT365H5.) Euclidean and non-Euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces. [36L, 12T]MAT405H5 Introduction to Topology (SCI)
Fundamentals of set theory. Point set topology in Rn. Metric spaces. Topological spaces and continuous mappings. Connectedness, compactness. Countability, separability. Topology of function spaces. Fundamental group and covering spaces. Brouwer fixed-point theorem. [36L, 12T]MAT406H5 Mathematical Introduction to Game Theory (SCI)
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. [36L, 12T]MAT438H5 Analysis (SCI)
Continuity, existence theorems, integration, pointwise and uniform convergence.MAT442H5 Algebraic Aspects of Cryptography (SCI)
This course will explain how number theory, group theory, and finite fields are used to build algorithms for cryptography and data integrity. We will discuss symmetric algorithms (such as AES) and asymmetric algorithms (such as RSA and discrete log based public-key schemes such as ECC). [36L]MAT478H5 Topics in Mathematics (SCI)
Introduction to a topic of current interest in mathematics. Content will vary from year to year. Enrolment by permission of instructor only. [36S]MAT488H5 Topics in Mathematics (SCI)
Introduction to a topic of current interest in mathematics. Content will vary from year to year. Enrolment by permission of instructor only. [36S]MAT492H5 Senior Thesis (SCI)
An exposition on a topic in mathematics written under the supervision of a faculty member. Open to students in Mathematical Sciences Specialist program.MAT498H5 Topics in Mathematics (SCI)
Introduction to a topic of current interest in mathematics. Content will vary from year to year. Enrolment by permission of instructor only. [36S]