[How to read the course descriptions]

*
Undergraduate Officer
*

B. Forrest, MC 5174, ext. 5560

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

**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.
*

**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**

**Readings in Pure Mathematics
**

**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**

**Readings in Pure Mathematics
**

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