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## Pure Mathematics

L. Cummings, MC 5047, ext. 4083

#### Note

More detailed course descriptions and availability information may be obtained upon request from the Pure Mathematics Department.

### GENERAL INTEREST COURSES

PMATH300S

PMATH 330 F,W,S 3C 0.5
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.
Prereq: MATH 235, or CS 212 and MATH 126
PMATH 432 may be substituted for PMATH 330 whenever the latter is a requirement in an Honours program.

PMATH 331 F,W 3C 0.5
Real Analysis
Topology of n-space, continuity, norms, metrics, completeness, Fourier series, and applications, for example, to ordinary differential equations, the heat problem, optimal approximation, the isoperimetric inequality.
Prereq: MATH 237
Cross-listed as AM 331
PMATH 351 may be substituted for PMATH 331 whenever the latter is a requirement in an Honours program.

PMATH 332 W,S 3C 0.5
Elementary Complex Analysis
Complex numbers, analytic functions, Cauchy-Riemann equations, conformal maps by elementary functions and applications, contour integrals, Cauchy's theorem, Taylor and Laurent expansions, residue calculus and applications.
Prereq: MATH 237
Cross-listed as AM 332
PMATH 352 may be substituted for PMATH 332 whenever the latter is a requirement in an Honours program.

PMATH 334 F,S 3C 0.5
Introduction to Rings and Fields
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.
Prereq: MATH 235
PMATH 345 may be substituted for PMATH 334 whenever the latter is a requirement in an Honours program.

PMATH 336 W,S 3C 0.5
Introduction to Group Theory
Groups, subgroups, normal subgroups, quotient groups, morphisms. Products of groups. Permutation groups. Symmetry groups.
Prereq: MATH 235
PMATH 346 may be substituted for PMATH 336 whenever the latter is a requirement in an Honours program.

PMATH 340 W 3C 0.5
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.
Prereq: MATH 126 or MATH 235
PMATH 440 may be substituted for PMATH 340 whenever the latter is a requirement in an Honours program.

### MAJOR COURSES

Note
These courses are designed to fulfill the degree requirements in the various Pure Math degree programmes. However they are open to all students.
PMATH300S

PMATH 345 F,S 3C 0.50
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.
Prereq: MATH 235, or consent of instructor

PMATH 346 W 3C 0.50
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 group.
Prereq: PMATH 235, or consent of instructor

PMATH 351 F,S 3C 0.5
Real Analysis
Cardinality, introduction to topological spaces, metric spaces, sequence spaces, completeness, Banach Fixed Point theorem, compactness, C[a,b], Stone-Weierstrass theorem, Arzela-Ascoli theorem.
Prereq: MATH 237, or consent of instructor

PMATH 352 W 3C 0.5
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.
Prereq: MATH 237, or consent of instructor

PMATH 354 W 3C .50
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.
Prereq: PMATH 351, or consent of instructor

PMATH 360 S 3C 0.5
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.
Prereq: MATH 126 or MATH 235, or consent of instructor
This course will be of interest to all math students.

PMATH 365 F 3C 0.5
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.
Prereq: AM 231, or consent of instructor
Cross-listed as AM 333

PMATH 399

PMATH 432 W 3C 0.5
Mathematical Logic
First order languages and theories.
Prereq: PMATH 345 or PMATH 346, or consent of instructor
Next offered Winter 1999, and each alternate Winter thereafter.

PMATH 440 F 3C 0.5
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.
Prereq: PMATH 352, or AM/PMATH 332
Next offered Fall 1998, and each alternate Fall thereafter.

PMATH 441 W 3C 0.5
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".
Prereq: PMATH 345
Next offered Winter 1998, and each alternate Winter thereafter.

PMATH 442 F 3C .50
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.
Prereq: PMATH 345 and PMATH 346

PMATH 444 W 3C 0.5
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.
Prereq: PMATH 345 and PMATH 346.
Next offered Winter 1999, and each alternate Winter thereafter.

PMATH 451 F 3C 0.5
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.
Prereq: PMATH 354, or consent of instructor.
Cross-listed as AM 431

PMATH 452 F 3C 0.5
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.
Prereq: PMATH 352 (Complex Analysis).
Next offered in Fall 1997, and each alternate Fall thereafter.

PMATH 453 W 3C 0.5
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.
Prereq: PMATH 354 or consent of instructor; PMATH 451/AM 431 is recommended.
Cross-listed as AM 432

PMATH 465 W 3C 0.5
Differential Geometry
Some global aspects of surface theory, the Euler-Poincar characteristic, the global interpretation of Gaussian curvature via the Gauss-Bonnet formula. Submanifolds of n-space, induced Riemannian metrics, extrinsic and intrinsic curvatures, Gauss-Codazzi equations. Local Lie groups of transformations on n-space, 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.
Prereq: PMATH 365/AM 333, or consent of instructor
Cross-listed as AM 433

PMATH 467 W 3C 0.5
Topology
Topics from algebraic, combinatorial and geometric topology.
Prereq: PMATH 346
Next offered in Winter 1998, and each alternate Winter thereafter.

PMATH 499