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## Pure Mathematics

*Undergraduate Officer *

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 **L*_{p}-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** ** **

**Readings in Pure Mathematics **

*PMATH400S*

**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, **L*_{p}-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** ** **

**Readings in Pure Mathematics **

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University of Waterloo

*Last modified * Feb. 1997