Undergraduate Calendar 2002-2003

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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 235/245, or CS 114 and MATH 126, or CS 124 and MATH 126

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 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 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 135/145 or MATH 126

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 1
Cardinality, introduction to topological spaces, metric spaces, sequence spaces, completeness, Banach Fixed Point theorem, compactness, C[a,b], Stone-Weierstrass theorem, Arzela-Ascoli theorem. [Offered: F,S]
Prereq: MATH 247 or AMATH 331/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: W]
Prereq: MATH 237/247 or AMATH/PMATH 331; Not open to General Mathematics students

PMATH 354 LEC 0.50Course ID: 007674
Real Analysis 2
Zorn's lemma, separable Hilbert spaces, construction of the Lebesgue measure, sets of measure zero, definition of the Lebesgue integral, introduction to Lp-spaces, Fourier analysis on the circle: convolution, Riemann-Lebesgue lemma, Fejer's theorem and the convergence of Fourier series, functions of bounded variations, applications. [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
Curves in Euclidean 3-Space and the Serret-Frenet formulae; surfaces in 3-space and their intrinsic geometry. Gaussian curvature and the Gauss-Bonnet theorem. Coordinate transformations and tensors in n dimensions; n-dimensional Riemannian spaces; covariant differentiation; geodesics; the curvature, Ricci and Einstein tensors. Applications of tensors in Relativity and Continuum Mechanics. [Offered: F]
Prereq: AMATH 231; Not open to General Mathematics students
(Cross-listed with AMATH 333)

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, 128, 138/148 and one of MATH 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 Winter of odd years.]
Prereq: PMATH 345 or 346; Not open to General Mathematics students

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 Fall of even 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 odd 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: F]
Prereq: PMATH 354; Not open to General Mathematics students
(Cross-listed with AMATH 431)

PMATH 452 LEC 0.50Course ID: 007700
Topics in Complex Analysis
The Riemann mapping theorem and several topics such as analytic continuation, harmonic functions, elliptic functions, entire functions, univalent functions, special functions.
[Note: Offered in the Fall of odd years.]
Prereq: PMATH 352; Not open to General Mathematics students

PMATH 453 LEC 0.50Course ID: 003349
Functional Analysis
Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, Open Mapping theorem, Dual spaces, weak topologies, Tychonoff's theorem, Banach-Alaoglu theorem, reflexive spaces, compact operators, Spectral theorem, commutative Banach algebras.
[Note: AMATH 431/PMATH 451 is recommended. Offered: W]
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 algebraic geometry over the complex numbers, with applications to elliptic curves and computational algebraic geometry. Plane curves and their genus, the group law on the cubic, affine varieties, applications.
[Note: PMATH 336 is recommended as background for students who do not have PMATH 345. Offered: W]
Prereq: PMATH 334 or 345; Not open to General Mathematics students

PMATH 465 LEC 0.50Course ID: 003350
Differential Geometry
Some global aspects of surface theory, the Euler-Poincare characteristic, the global interpretation of Gaussian curvature via the Gauss-Bonnet formula. Submanifolds of En, induced Riemannian metrics, extrinsic and intrinsic curvatures, Gauss-Codazzi equations. Local Lie groups of transformations on Rn, infinitesimal generators, the Lie derivative. An introduction to differentiable manifolds, the tangent and cotangent bundles, affine connections and the Riemann curvature tensor. The above topics will be illustrated by applications to continuum mechanics and mathematical physics. [Offered: W]
Prereq: AMATH 333/PMATH 365; Not open to General Mathematics students
(Cross-listed with AMATH 433)

PMATH 467 LEC 0.50Course ID: 007704
Topology
Topics from algebraic, combinatorial and geometric topology.
[Note: Offered in the Winter of even years.]
Prereq: PMATH 351; 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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Revised April 2002