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## Pure Mathematics

B. Forrest, MC 5174, ext. 5560

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

#### Pure Mathematics - Group 1 Courses

PMATH 330 F,W,S 3C 0.5
Introduction to Mathematical Logic 1
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.
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 Rn, 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
Antireq: PMATH 351
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
Complex Analysis
Complex numbers; continuity, differentiability, analyticity of functions; the Cauchy-Riemann equations; solution of Laplace's equation; conformal mapping by elementary functions, and applications; contour integration, the Cauchy and allied theorems; Taylor and Laurent expansions, uniform convergence and power series; the residue calculus, and applications.
Prereq: MATH 237
Antireq: PMATH 352
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 W,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 135, 235
Antireq: PMATH 344
PMATH 344 may be substituted for PMATH 334 whenever the latter is a requirement in an Honours program.

PMATH 336 F,S 3C 0.5
Introduction to Group Theory
Groups, subgroups, normal subgroups, quotient groups, morphisms. Products of groups. Permutation groups. Symmetry groups.
Prereq: MATH 135, 235
Antireq: PMATH 343
PMATH 343 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
Antireq: PMATH 440
PMATH 440 may be substituted for PMATH 340 whenever the latter is a requirement in an Honours program.

#### Pure Mathematics - Group 2 Courses

PMATH 343 F,W 3C 0.5
Abstract Algebra 1
Groups: examples of groups, permutation groups, groups of low order, homomorphisms, subgroups and normal subgroups, factor groups, Lagrange's theorem, Cayley's theorem, Abelian groups, direct products, the structure of finitely generated Abelian groups, applications. Rings: ideals, quotient rings, homomorphisms, domains, primes, maximal ideals and fields, field of fractions, Euclidean domains, principal ideal domains, unique factorization, polynomial extensions of unique factorization domains, applications.
Prereq: MATH 235
Antireq: PMATH 336

PMATH 344 F,S 3C 0.5
Abstract Algebra 2
Field theory, examples of fields, field of fractions, algebraic extensions, construction of roots, separable extensions, splitting fields, classification of finite fields. Finite non-Abelian groups, Sylow theorems. Introduction to Galois theory.
Prereq: PMATH 343 or 336 with consent of instructor
Antireq: PMATH 334

PMATH 351 F,S 3C 0.5
Real Analysis
Metric spaces, compactness, completeness, continuity, convergence, integration, function spaces.
Prereq: MATH 237, or consent of instructor
Antireq: AM/PMATH 331

PMATH 352 W 3C 0.5
Complex Analysis
Analytic functions, Cauchy's theorem, Laurent series, the residue theorem, integral evaluation, Möbius and other conformal maps.
Prereq: MATH 237, or consent of instructor
Antireq: AM/PMATH 332

PMATH 353 W 3C 0.5
Fourier Analysis
Fourier analysis: A descriptive introduction to Lp spaces, inner products and Hilbert spaces, Fourier series on the circle, convergence theorems, the Fourier transform. Other topics: The heat equation, the Dirichlet problem on the disk, approximation theory and orthogonal polynomials.
Prereq: PMATH 351 or AM/PMATH 331 with 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 consent of instructor
This course will be of interest to all math students.

PMATH 365 F,S 3C 0.5
Elementary Differential Geometry and Tensor Analysis
Curves in Euclidean 3-space (E3) and the Serret-Frenet formulae; surfaces in E3 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 367 W 3C 0.5
Set Theory and General Topology
Intuitive set theory, metric spaces, point set topology.
Prereq: MATH 237. PMATH 351 is strongly recommended

PMATH 380A 3C 0.5
Introduction to Information Theory
Variable length coding. The Shannon entropy as a measure of uncertainty and expected information. Minimal average length coding; the Shannon entropy as lower bound. Source entropy. Channels. Transinformation. Capacity. Applications to problem solving, information transmission, logics, science, linguistics and communications (TV, music, etc.). Determination of practical measures of information.
Prereq: Consent of instructor
Not offered every year

PMATH 380B 3C 0.5
Applications of Information Theory
Measures of information: probabilistic, mixed, and nonprobabilistic. Measures of uncertainty (entropies). Entropies for continuous random variables. The maximum entropy principle and its applications to: statistical mechanics, optimal discretization, and quantization of random variables, image representation and pattern recognition. Applications to questionnaires. Optimal questionnaires.
Prereq: Consent of instructor
Not offered every year

PMATH 399

PMATH 432 F 3C 0.5
Mathematical Logic
First order languages and theories.
Next offered Fall 1995, and each alternate Fall thereafter

PMATH 440 W 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: AM/PMATH 332 or PMATH 352
Antireq: PMATH 340
Next offered in Winter 1996, and each alternate Winter thereafter

PMATH 441 F 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 334 or 344
Next offered in Fall 1996, and each alternate Fall thereafter

PMATH 443 3C 0.5
Multilinear Algebra
Continuation of linear algebra. Main topics: representations of endomorphisms, structure of bilinear forms, multilinear products.
Prereq: MATH 235
Not offered every year

PMATH 444 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 344
Next offered in Winter 1996

PMATH 446 3C 0.5
Group Theory
Permutations, Cayley Theorem, Sylow Theorem, Jordan-Hölder Theorem, nilpotent and solvable groups, direct and semidirect products, free groups.
Prereq: PMATH 336 or 344
Next offered in Fall 1995

PMATH 448 3C 0.5
Commutative Algebra
Prime ideals, Krull dimension, integral elements, localization, discrete valuations, Dedekind domains, Noetherian domains. Algebraic and transcendental field extensions, algebraic closure. Introduction to algebraic geometry.
Prereq: PMATH 344
Next offered in Fall 1996

PMATH 451 F 3C 0.5
Measure and Integration
Lebesgue measure and integral for the real line, general measure and integration theory, convergence theorems, Fubini's theorem, absolute continuity, Radon-Nikodym theorem, LP-spaces.
Prereq: PMATH 351 or PMATH 353
Cross-listed as AM 431

PMATH 452 W 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
Next offered in Winter 1996, and each alternate Winter thereafter

PMATH 453 W 3C 0.5
Functional Analysis
Banach spaces, linear operators, geometry of Hilbert spaces, Hahn-Banach theorem, open mapping theorem, compact operators, applications.
Prereq: PMATH 353 or AM 431/
PMATH 451
Cross-listed as AM 432

PMATH 463 3C 0.5
Differentiable Manifolds
Topics chosen from: Charts and atlases, Manifolds and Diffeomorphisms, Tangent Spaces, Submanifolds, Vector Bundles, Tensor and Exterior Algebras, Differential Forms, Oriented Manifolds and Geometry, Homogeneous Spaces and Lie Groups.
Prereq: PMATH 365 or consent of instructor
Not offered every year

PMATH 465 3C 0.5
Differential Geometry
Some global aspects of surface theory, the Euler-Poincaré 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.
Prereq: AM 333/PMATH 365 or consent of instructor
Cross-listed as AM 433

PMATH 467 3C 0.5
Topology
Topics from algebraic, combinatorial and geometric topology.
Prereq: PMATH 336, 367

PMATH 470 3C 0.5
Functional Equations
Cauchy's, Pexider's, and similar equations. Equations for polynomials and trigonometric functions. Reduction to different equations. Applications.
Prereq: Consent of instructor
Not offered every year

PMATH 499