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.
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.
Systems of linear equations, matrices, Gaussian elimination; basis, dimension; dot products; geometry to Rn; linear transformations; determinants, Cramer's rule; eigenvalues and eigenvectors, diagonalization.
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.
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.
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.
This is a calculus course designed primarily for students in management. The main concepts of calculus of one and several variables are studied with interpretations and applications to business and economics. Systems of linear equations and matrices are covered with applications in business.
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.
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.
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.
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
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.
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.
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.
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.
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.
Linear programming, simplex algorithm, duality theory, interior point method; quadratic and convex optimization, stochastic programming; applications to portfolio optimization and operations research.
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.
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.
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.
Fundamentals of set theory, topological spaces and continuous functions, connectedness, compactness, countability, separatability, metric spaces and normed spaces, function spaces, completeness, homotopy.
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.
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.
Topics in measure theory: the Lebesgue integral, Riemann-Stieltjes integral, Lp spaces, Hilbert and Banach spaces, Fourier series.
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.
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.
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.
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.
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.
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.
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.
An introduction to geometry with a selection of topics from the following: symmetry and symmetry groups, finite geometries and applications, non-Euclidean geometry.
This course is an introduction to axiomatic set theory and its methods. Set theory is a foundation for practically every other area of mathematics and is a deep, rich subject in its own right. The course will begin with the Zermelo-Fraenkel axioms and general set constructions. Then the natural numbers and their arithmetic are developed axiomatically. The central concepts of cardinality, cardinal numbers, and the Cantor-Bernstein theorem are studied, as are ordinal numbers and transfinite induction. The Axiom of Choice and its equivalents are presented along with applications.
A variety of topics from geometry, analysis, combinatorics, number theory and algebra, to be chosen by the instructor.
A variety of topics from geometry, analysis, combinatorics, number theory and algebra, to be chosen by the instructor.
A variety of topics from geometry, analysis, combinatorics, number theory and algebra, to be chosen by the instructor.
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.
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).
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.
This course provides an introduction and exposure to dynamical systems, with particular emphasis on low-dimensional systems such as interval maps and maps of the plane. Through these simple models, students will become acquainted with the mathematical theory of chaos and will explore strange attractors, fractal geometry and the different notions of entropy. The course will focus mainly on examples rather than proofs; students will be encouraged to explore dynamical systems by programming their simulations in Mathematica.
This course will focus on combinatorics. Topics will be selected by the instructor and will vary from year to year.
This course provides an introduction to partial differential equations as they arise in physics, engineering, finance, optimization and geometry. It requires only a basic background in multivariable calculus and ODEs, and is therefore designed to be accessible to most students. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' bag of tools.
This course introduces students to combinatorial games, two-player (matrix) games, Nash equilibrium, cooperative games, and multi-player games. Possible additional topics include: repeated (stochastic) games, auctions, voting schemes and Arrow's paradox. Numerous examples will be analyzed in depth, to offer insight into the mathematical theory and its relation to real-life situations.
Manifolds, vector fields, tangent spaces, vector bundles, differential forms, integration on manifolds.
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.
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.
Independent study under direction of a faculty member.
Independent study under direction of a faculty member.