Undergraduate Calendar 2006-2007

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Notes

  1. In some areas, the Department of Pure Mathematics offers two distinct streams of courses, one for students in a Pure Mathematics major plan, and another for students in other majors. PMATH courses numbered from 345 to 354 are designed for Pure Mathematics majors. However they are open to all students. The PMATH courses numbered from 331 to 336 cover similar topics at a less intensive level.
  2. More detailed course descriptions and availability information can be obtained from the Pure Mathematics departmental web pages.

PMATH 300s


PMATH 330 LEC 0.50Course ID: 007659
Introduction to Mathematical Logic
A broad introduction to Mathematical Logic. The logic of sentences: truth-functions and axiomatic approaches (eg. Natural Deduction and Gentzen sequences). A brief introduction to the logic of predicates and to the foundations of mathematics.
[Note: PMATH 432 may be substituted for PMATH 330 whenever the latter is a requirement in an Honours plan. Offered: F,W,S]
Prereq: (MATH 126 and CS 126/124/114) or MATH 235/245; Not open to Computer Science students.
Antireq: CS 245

PMATH 331 LEC 0.50Course ID: 003323
Applied Real Analysis
Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.
[Note: PMATH 351 may be substituted for AMATH/PMATH 331 whenever the latter is a requirement in an Honours plan. Offered: F,W]
Prereq: MATH 237/247; Not open to General Mathematics students
(Cross-listed with AMATH 331)

PMATH 332 LEC 0.50Course ID: 003324
Applied Complex Analysis
Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications.
[Note: PMATH 352 may be substituted for AMATH/PMATH 332 whenever the latter is a requirement in an Honours plan. Offered: W,S]
Prereq: MATH 237/247; Not open to General Mathematics students
(Cross-listed with AMATH 332)

PMATH 334 LEC 0.50Course ID: 007662
Introduction to Rings and Fields with Applications
Rings, ideals, factor rings, homomorphisms, finite and infinite fields, polynomials and roots, field extensions, algebraic numbers, and applications, for example, to Latin squares, finite geometries, geometrical constructions, error-correcting codes.
[Note: PMATH 345 may be substituted for PMATH 334 whenever the latter is a requirement in an Honours plan. Offered: F,S]
Prereq: MATH 235/245; Not open to General Mathematics students

PMATH 336 LEC 0.50Course ID: 007663
Introduction to Group Theory with Applications
Groups, permutation groups, subgroups, homomorphisms, symmetry groups in 2 and 3 dimensions, direct products, Polya-Burnside enumeration.
[Note: PMATH 346 may be substituted for PMATH 336 whenever the latter is a requirement in an Honours plan. Offered: W,S]
Prereq: MATH 235/245; Not open to General Mathematics students

PMATH 340 LEC 0.50Course ID: 007664
Elementary Number Theory
An elementary approach to the theory of numbers; the Euclidean algorithm, congruence equations, multiplicative functions, solutions to Diophantine equations, continued fractions, and rational approximations to real numbers.
[Note: PMATH 440 may be substituted for PMATH 340 whenever the latter is a requirement in an Honours plan. Offered: W]
Prereq: MATH 126 or 135/145

PMATH 345 LEC 0.50Course ID: 007667
Polynomials, Rings and Finite Fields
Elementary properties of rings, polynomial rings, Gaussian integers, integral domains and fields of fractions, homomorphisms and ideals, maximal ideals and fields, Euclidean rings, principal ideals, Hilbert Basis theorem, Gauss' lemma, Eisenstein's criterion, unique factorization, computational aspects of polynomials, construction of finite fields with applications, primitive roots and polynomials, additional topics. [Offered: F,S]
Prereq: MATH 235/245; Not open to General Mathematics students

PMATH 346 LEC 0.50Course ID: 007668
Group Theory
Elementary properties of groups, cyclic groups, permutation groups, Lagrange's theorem, normal subgroups, homomorphisms, isomorphism theorems and automorphisms, Cayley's theorem and generalizations, class equation, combinatorial applications, p-groups, Sylow theorems, groups of small order, simplicity of the alternating groups, direct product, fundamental structure theorem for finitely generated Abelian groups. [Offered: W]
Prereq: MATH 235/245; Not open to General Mathematics students

PMATH 351 LEC 0.50Course ID: 007669
Real Analysis
Normed and metric spaces, open sets, continuous mappings, sequence and function spaces, completeness, contraction mappings, compactness of metric spaces, finite-dimensional normed spaces, Arzela-Ascoli theorem, existence of solutions of differential equations, Stone-Weierstrass theorem. [Offered: F,S]
Prereq: MATH 247 or AMATH/PMATH 331; Not open to General Mathematics students

PMATH 352 LEC 0.50Course ID: 007672
Complex Analysis
Analytic functions, Cauchy-Riemann equations, Goursat's theorem, Cauchy's theorems, Morera's theorem, Liouville's theorem, maximum modulus principle, harmonic functions, Schwarz's lemma, isolated singularities, Laurent series, residue theorem. [Offered: F]
Prereq: MATH 237/247 or AMATH/PMATH 331; Not open to General Mathematics students

PMATH 354 LEC 0.50Course ID: 007674
Measure Theory and Fourier Analysis
Lebesgue measure on the line, the Lebesgue integral, monotone and dominated convergence theorems, Lp-spaces: completeness and dense subspaces. Separable Hilbert space, orthonormal bases. Fourier analysis on the circle, Dirichlet kernel, Riemann-Lebesgue lemma, Fejer's theorem and convergence of Fourier series. [Offered: W]
Prereq: PMATH 351; Not open to General Mathematics students

PMATH 360 LAB,LEC 0.50Course ID: 007675
Geometry
An introduction to affine, projective and non-Euclidean forms of geometry. Conic sections in the projective plane. Inversion in circles. Theorems of Desargues, Pappus, and Pascal.
[Note: This course will be of interest to all math students. Offered: S]
Prereq: MATH 126 or MATH 235/245

PMATH 365 LEC 0.50Course ID: 003325
Elementary Differential Geometry and Tensor Analysis
An introduction to local differential geometry, laying the groundwork for both global differential geometry and general relativity. Embedded curves and the intrinsic geometry of surfaces in Euclidean 3-space. Differential forms, vector fields, and the Stokes Theorem in n dimensions. Tensors, n-dimensional Riemannian spaces, covariant differentiation, geodesics, and curvature. Gaussian curvature and the Gauss-Bonnet theorem.
[Note: Offered in the Winter and the Spring of odd years.]
Prereq: AMATH 231 or MATH 247; Not open to General Mathematics students
(Cross-listed with AMATH 333)

PMATH 367 LEC 0.50Course ID: 007677
Set Theory & General Topology
Relations, functions, well-orderings, Schroder-Bernstein theorem, recursion, axiom of choice and equivalents, ordinals, cardinals, continuum hypothesis, singular and inaccessible cardinals. Topological spaces, bases and sub-bases, closure and interior, product spaces, quotient spaces, nets and filters. Hausdorff spaces, completely regular and normal spaces, Urysohn's lemma, Tietze extension theorum. Compactness, Tychonoff's theorum, Stone-Cech compactification. Connectedness, path connectedness, Function spaces.
[Note: Offered in the Fall of even years.]
Prereq: AMATH/PMATH 331 or PMATH 351; Not open to General Mathematics students

PMATH 370 LEC 0.50Course ID: 009496
Chaos and Fractals
The mathematics of iterated functions, properties of discrete dynamical systems, Mandelbrot and Julia sets.
[Note: Programming experience on one computer language with graphical output is recommended. Offered in the Fall of even years.]
Prereq: One of MATH 118, 119, 128, 138/148 and one of MATH 114, 115, 126, 235/245; Not open to General Mathematics students

PMATH 399 RDG 0.50Course ID: 007680
Readings in Pure Mathematics
Prereq: Not open to General Mathematics students

PMATH 400s


PMATH 432 LEC 0.50Course ID: 007687
Mathematical Logic
First order languages and theories.
[Note: Offered in the Fall of odd years.]
Prereq: PMATH 345 or 346; Not open to General Mathematics students

PMATH 434 LEC 0.50Course ID: 012236
Techniques in Computational Number Theory
An introduction to: integer factorization, elliptic curves methods, primality testing, fast integer arithmetic, fast Fourier transforms and quantum computing. This course is taught with a philosophy that encourages experimentation. [Offered: F]
Prereq: CM 339/CS 341 or PMATH 334 or 336 or 345 or 346; Not open to General Mathematics students
(Cross-listed with CM 434)

PMATH 440 LEC 0.50Course ID: 007690
Analytic Number Theory
An introduction to elementary and analytic number theory; primitive roots, law of quadratic reciprocity, Gaussian sums, Riemann zeta-function, distribution of prime numbers.
[Note: Offered in the Winter of odd years.]
Prereq: PMATH 352 or AMATH/PMATH 332; Not open to General Mathematics students

PMATH 441 LEC 0.50Course ID: 007691
Algebraic Number Theory
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations, Fermat's "last theorem".
[Note: Offered in the Winter of even years.]
Prereq: PMATH 345; Not open to General Mathematics students

PMATH 442 LEC 0.50Course ID: 007692
Fields and Galois Theory
Normal series, elementary properties of solvable groups and simple groups, algebraic and transcendental extensions of fields, adjoining roots, splitting fields, geometric constructions, separability, normal extensions, Galois groups, fundamental theorem of Galois theory, solvability by radicals, Galois groups of equations, cyclotomic and Kummer extensions. [Offered: F]
Prereq: PMATH 345, 346; Not open to General Mathematics students

PMATH 444 LEC 0.50Course ID: 007694
Non-Commutative Algebra
Jacobson structure theory, density theorem, Jacobson radical, Maschke's theorem. Artinian rings, Artin-Wedderburn theorem, modules over semi-simple Artinian rings. Division rings. Representations of finite groups.
[Note: Offered in the Winter of even years.]
Prereq: PMATH 345; Not open to General Mathematics students.
Coreq: PMATH 346

PMATH 451 LEC 0.50Course ID: 003348
Measure and Integration
General measures, measurability, Caratheodory Extension theorem and construction of measures, integration theory, convergence theorems, Lp-spaces, absolute continuity, differentiation of monotone functions, Radon-Nikodym theorem, product measures, Fubini's theorem, signed measures, Urysohn's lemma, Riesz Representation theorems for classical Banach spaces. [Offered: W]
Prereq: PMATH 354; Not open to General Mathematics students
(Cross-listed with AMATH 431)

PMATH 453 LEC 0.50Course ID: 003349
Functional Analysis
Banach and Hilbert spaces, Hahn-Banach theorems, weak topologies, dual spaces, Banach-Alaoglu theorem, open mapping theorem, closed graph theorem, uniform boundedness principle. Selected topics from topological vector spaces and commutative algebras of operators.[Offered: F]
Prereq: PMATH 354; Not open to General Mathematics students
(Cross-listed with AMATH 432)

PMATH 464 LEC 0.50Course ID: 010733
Algebraic Curves
An introduction to the geometry of algebraic curves with applications to elliptic curves and computational algebraic geometry. Plane curves, affine varieties, the group law on the cubic, and applications.
[Note: Offered in the Winter of odd years.]
Prereq: PMATH 345; Not open to General Mathematics students

PMATH 465 LEC 0.50Course ID: 003350
Differential Geometry
Curves and surfaces in Euclidean space, an introduction to differentiable manifolds, the tangent and cotangent bundles. Vector fields and differential forms. The Lie bracket and Lie derivative of vector fields. Exterior differentiation and integration of differential forms. Riemannian manifolds, affine connections and the Riemann curvature tensor. Other topics such as Gauss-Bonnet Theorem and applications to mathematical physics.
[Note: Offered in the Winter of even years.]
Prereq: (AMATH/PMATH 332 or PMATH 352) and AMATH 333/PMATH 365 and MATH 235/245; Not open to General Mathematics students
(Cross-listed with AMATH 433)

PMATH 467 LEC 0.50Course ID: 007704
Topology
Review of general topology, quotient spaces, scissors and glue constructions. Basics on homotopy and topological manifolds. The fundamental group. Compact surfaces. Introduction to homology. Selected applications to covering spaces, homotopy theory, general manifolds, knots, differential equation, combinatorial group theory.
[Note: Offered in the Winter of odd years.]
Prereq: PMATH 351 or 367; Not open to General Mathematics students.
Coreq: PMATH 346

PMATH 499 RDG 0.50Course ID: 007706
Readings in Pure Mathematics
Prereq: Not open to General Mathematics students

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