35 courses in the 1994–95 Undergraduate Calendar, read from the scanned original. The text comes from OCR, so check the scanned page before relying on a detail.
HONOURS SEMINARS
Consult departmental listings for topics and prerequisites for 1994-95, Open to third and fourth year Honours Psychology or Make-Up Psychology students, or by consent of instructor.
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.
Topology of R°, 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 fatter is a requirement in an Honours program.
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.
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.
Course Descriptions Pure Mathematics
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.
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.
GROUP 2 COURSES
The following courses may be used to fulfill the various requirements of any of the Honours programs leading to a degree with a Pure Mathematics designation. Group 2 courses are open to students in any Honours program.
Groups: examples of groups, permutation groups, groups of low order, hornomorphisms, 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
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
Metric spaces, compactness, completeness, continuity, convergence, integration, function spaces.
Prereq: MATH 237, or consent of instructor
Analytic functions, Cauchy’s theorem, Laurent series, the residue theorem, integral evaluation, Mébius and other conformal maps.
Prereq: MATH 237, or consent of instructor ’
Antireq: AM/PMATH 332
Fourier analysis: A descriptive introduction to L, spaces, inner products and Hilbert spaces, Fourier series on the circle, convergence theorems, the Fourier transform. Other topics: The heat equation, the Dirichlet probiem on the disk, approximation theory and orthogonal polynomials. Prereq: PMATH 351 or AM/PMATH 331 with consent of instructor
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.
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
Intuitive set theory, metric spaces, point
set topology. Prereq: MATH 237. PMATH 351 is
strongly recommended
Variable length coding. The Shannon entropy as a measure of uncertainty and expected information. Minima! 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
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
First order languages and theories. Next offered Fall 1995, and each
alternate Fall thereafter
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
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet’s unit theorem, solutions of Diophantine equations, Fermat's ‘ast theorem’. Prereq: PMATH 334 or 344 Next offered in Fall 1994, and each alternate Fall thereafter
Course Descriptions Pure Mathematics
Continuation of linear algebra. Main topics.
representations of endomorphisms, structure of bilinear forms, multilinear products. Prereg: MATH 235 Not offered every year
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 Fall 1994
nilpotent and solvable groups, direct and semidirect products, free groups. Prereq: PMATH 334 or 344 Next offered in Fall 1995
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 Winter 1995
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
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
16:127
Banach spaces, linear operators, geometry of Hilbert spaces, Hahn-Banach theorem, open mapping theorem, compact operators, applications.
Topics chosen from: Charts and atlases, Manifolds and Diffeomorphisms, Tangent Spaces, Submanifalds, 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
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 R®, 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
Topics from aigebraic, combinatorial and geometric topology. Prereq: PMATH 336, 367
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
Not offered in 1994–95.
Not offered in 1994–95.