Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.

Set theory, algebra, functions and relations, trigonometry, logarithms and exponents.

Introduction to differential calculus including limits, continuity, definition of derivative, rules for differentiation, implicit differentiation, velocity, acceleration, related rates, maxima and minima, exponential functions, logarithmic functions, differentiation of exponential and logarithmic functions, curve sketching.

Primarily for students interested in pursuing a degree in one of the mathematical sciences. Logic, set theory, relations, functions and operations, careful study of the integers, discussion of the real and complex numbers, polynomials, and infinite sets.

Elementary techniques of integration; applications of Calculus such as area, volume, and differential equations; functions of several variables, Lagrange multipliers. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules. It may not be used as a prerequisite for any Calculus course numbered 1300 or above.

Permutations and combinations; probability theory. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules.

Matrix algebra including vectors and matrices, linear equations, determinants. This course is intended primarily for students in the Social Sciences, but may meet minimum requirements for some Science modules.

Review of differential calculus with transcendental functions; methods of integration; constrained and unconstrained multivariable optimization, with applications; mathematical modelling with differential equations, including applications in management, finance, economics, and social science.

Properties and applications of vectors; matrix algebra; solving systems of linear equations; determinants; vector spaces; orthogonality; eigenvalues and eigenvectors.

A rigorous development of lines and planes in Rn; linear transformations and abstract vector spaces. Determinants and an introduction to diagonalization and its applications including the characteristic polynomials, eigenvalues and eigenvectors.

A rigorous introduction to analysis on the real line. Sets and functions, logic and mathematical proof, the natural and real numbers, completeness and its consequences, limits of sequences, limits of real functions, continuity and uniform continuity.

Primarily for Mathematics students, but will interest other students with ability in and curiosity about mathematics in the modern world as well as in the past. Stresses development of students' abilities to solve problems and construct proofs. Topics will be selected from: counting, recurrence, induction; number theory; graph theory; parity, symmetry; geometry.

Logic, sets and functions, algorithms, mathematical reasoning, counting, relations, graphs, trees, Boolean Algebra, computation, modeling.

This course provides an introduction to logical reasoning and proofs. Topics include sets, counting (permutations and combinations), mathematical induction, relations and functions, partial order relations, equivalence relations, binary operations, elementary group theory and applications to error-correcting codes.

This course continues the development of logical reasoning and proofs begun in Mathematics 2155F/G. Topics include elementary number theory (gcd, lcm, Euclidean algorithm, congruences, Chinese remainder theorem) and graph theory (connectedness, complete, regular and bipartite graphs; trees and spanning trees, Eulerian and Hamiltonian graphs, planar graphs; vertex, face and edge colouring; chromatic polynomials).

Linear transformations, matrix representation, rank, change of basis, eigenvalues and eigenvectors, inner product spaces, quadratic forms and conic sections. Emphasis on problem-solving rather than theoretical development. Cannot be taken for credit by students in honors Mathematics programs.

A survey of some important basic concepts of mathematics in a historical setting, and in relation to the broader history of ideas. Topics may include: the evolution of the number concept, the development of geometry, Zeno's paradoxes.

Properties of integers, rational, real and complex numbers: commutativity, associativity, distributivity. Polynomials, prime and irreducible elements. Rings, ideals, integral and Euclidean domains, fields, and unique factorization. First isomorphism theorem, quotient rings and finite fields. Introduction to groups.

An introduction to the theory of groups: cyclic, dihedral, symmetric, alternating; subgroups, quotient groups, homomorphisms, cosets, Lagrange's theorem, isomorphism theorems; group actions, class equation, p-groups, Sylow theorems; direct and semidirect products, wreath products, finite abelian groups; Jordan-HÃ¶lder theorem, commutator subgroup, solvable and nilpotent groups; free groups, generators and relations.

A continuation of the material of Mathematics 2120A/B including properties of complex numbers and the principal axis theorem; singular value decomposition; linear groups; similarity; Jordan canonical form; Cayley-Hamilton theorem; bilinear forms; Sylvester's theorem.

Differentiation, the Mean Value Theorem, and integration. Metric spaces, including topology, convergence, compactness, completeness, and connectedness. Uniform convergence of functions. Selected additional topics.

Rigorous introduction to ordinary differential equations. Existence, uniqueness, and continuation of solutions. Linear systems with constant coefficients. Flows and dynamical systems. Series solutions.

The Cauchy-Riemann equations, elementary functions, branches of the logarithm and argument, Cauchy's integral theorem and formula, winding number, Liouville's theorem and the fundamental theorem of algebra, the identity theorem, the maximum modulus theorem, Taylor and Laurent expansions, isolated singularities, the residue theorem and applications, the argument principle and applications.

Divisibility, primes, congruences, theorems of Fermat and Wilson, Chinese remainder theorem, quadratic reciprocity, some functions of number theory, diophantine equations, simple continued fractions.

Arithmetic functions, perfect numbers, the MÃ¶bius inversion formula, introduction to Dirichlet series and the Riemann zeta function, some methods of combinatorial number theory, primitive roots and their relationship with quadratic reciprocity, the Gaussian integers, sums of squares and Minkowski's theorem, square and triangular numbers, Pell's equation, introduction to elliptic curves.

Enumeration, recursion and generating functions, linear programming, Latin squares, block designs, binary codes, groups of symmetries, orbits, and counting.

Network problems: shortest path, spanning trees, flow problems, matching, routing. Complexity. Integer programming.

Geometry of algebraic curves over the rational, real and complex fields. Classification of affine conics, singularities, intersection numbers, tangents, projective algebraic curves, multiplicity of points, flexes. Some discussion of cubic curves.

A first course in the mathematical theory of games. Topics begin with the modelling of games: extensive and strategic forms; perfect information; chance. Sprague-Grundy theory of impartial combinatorial games. Modelling preferences with utility functions. Nash equilibria, analysis of two-player games.

Modern cryptological algorithms will be discussed with an emphasis placed on their mathematical foundation. Main topics will include: basic number theory, complexity of algorithms, symmetric-key cryptosystems, public-key cryptosystems, RSA encryption, primality and factoring, discrete logarithms, elliptic curves and information theory.

Extra Information: 3 lecture hours.

Extra Information: 3 lecture hours.

Automorphisms of fields, separable and normal extensions, splitting fields, fundamental theorem of Galois theory, primitive elements, Lagrange's theorem. Finite fields and their Galois groups, cyclotomic extensions and polynomials, applications of Galois theory to geometric constructions and solution of algebraic equations.

Topological spaces, neighbourhoods, bases, subspaces, product and quotient spaces, connectedness, compactness, separation axioms.

Lebesgue measure, measurable sets and functions, Littlewood principles; the Lebesgue integral, basic convergence theorems, approximation theorems; measure spaces, signed measures, Radon-Nikodym Theorem.

Rings: fractions and localization, Chinese Remainder Theorem, factorization in commutative rings, Euclidean algorithm, PIDs, algebraic integers, polynomials and formal power series, factorization in polynomial rings; Modules: generation, direct products and sums, freeness, presentations, tensor algebras, exact sequences, projectivity, injectivity, Hom and duality, Zorn's Lemma, chain conditions, modules over PIDs.

Algebraic numbers, cyclotomic fields, low dimensional Galois cohomology, Brauer groups, quadratic forms, local and global class fields, class field theory, Galois group representations, modular forms and elliptic curves, zeta function and L-series.

Homotopy, fundamental group, Van Kampen's theorem, covering spaces, simplicial and singular homology, homotopy invariance, long exact sequence of a pair, excision, Mayer-Vietoris sequence, degree, Euler characteristic, cell complexes, projective spaces. Applications include the fundamental theorem of algebra, the Brouwer fixed point theorem, division algebras, and invariance of domain.

Affine and projective varieties, coordinate rings and function fields, birational correspondences, sheaves, dimension theory, regularity.

Hilbert spaces: L^2 spaces, orthogonal complements, dual spaces, Riesz representation theorem, the Fredholm alternative, spectral resolution of compact normal operators. Banach spaces: Hahn-Banach theorem, bounded linear operators, adjoints, closed graph and Banach Steinhaus theorems.

Manifolds (definition, examples, constructions), orientation, functions, partitions of unity, tangent bundle, cotangent bundle, vector fields, integral curves, differential forms, integration, manifolds with boundary, Stokes' theorem, submersions, immersions, embeddings, submanifolds, Sard's theorem, Whitney embedding theorem.

Linear-fractional transformations, Schwarz's lemma, Reflection Principle, the Argument principle, the Riemann mapping theorem, Runge's theorem, the Mittag-Lefler and Weierstrass theorems.

Entire and meromorphic functions, infinite products, canonical products, the Weierstrass factorization and Mittag-Leffler theorems, the Hadamard factorization theorem; simply periodic and doubly periodic functions, elliptic functions; the Picard theorems (with Schottky's, Montel's, and Landau's theorems); the prime number theorem (with the Gamma and Riemann Zeta functions).

Set theory: axioms, ordinal numbers, transfinite induction, cardinality, the axiom of choice. Foundations of mathematics: construction of the real numbers from the natural numbers by one of the standard methods. First-order logic: propositional calculus, quantifiers, truth and satisfaction, models of first-order theories, consistency, completeness and compactness.

Extra Information: 3 lecture hours

Extra Information: 3 lecture hours.