PMATH 300s P U R E M A T H E M A T I C S. Notes

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1 P U R E M A T H E M A T I C S Notes 1. In some areas, the Department of Pure Mathematics offers two distinct streams of courses, one for students in a Pure Mathematics major plan, and another for students in other majors. PMATH courses numbered from 345 to 352 are designed for Pure Mathematics majors. However they are open to all students. The PMATH courses numbered from 331 to 336 cover similar topics at a less intensive level. 2. More detailed course descriptions and availability information can be obtained from the Pure Mathematics departmental web pages. PMATH 300s PMATH 330 LEC 0.50 Course ID: 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. [Note: PMATH 432 may be substituted for PMATH 330 whenever the latter is a requirement in an Honours plan.] Prereq: (MATH 225/126 and CS 126/124/114) or MATH 235 or 245; Not open to Computer Science students. Antireq: CS 245 PMATH 331 LEC 0.50 Course ID: Applied Real Analysis Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation. [Note: PMATH 351 may be substituted for AMATH/PMATH 331 whenever the latter is a requirement in an Honours plan] Prereq: MATH 237 or 247; Not open to General Mathematics students (Cross-listed with AMATH 331) PMATH 332 LEC 0.50 Course ID: Applied Complex Analysis Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications. [Note: PMATH 352 may be substituted for AMATH/PMATH 332 whenever the latter is a requirement in an Honours plan.] Prereq: MATH 237 or 247; Not open to General Mathematics students. Antireq: PHYS 365 (Cross-listed with AMATH 332) PMATH 334 LEC 0.50 Course ID: Introduction to Rings and Fields with Applications 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. [Note: PMATH 345 may be substituted for PMATH 334 whenever the latter is a requirement in an Honours plan.] PMATH 336 LEC 0.50 Course ID: Introduction to Group Theory with Applications

2 Groups, permutation groups, subgroups, homomorphisms, symmetry groups in 2 and 3 dimensions, direct products, Polya-Burnside enumeration. [Note: PMATH 346 may be substituted for PMATH 336 whenever the latter is a requirement in an Honours plan.] PMATH 340 LEC 0.50 Course ID: 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. [Note: PMATH 440 may be substituted for PMATH 340 whenever the latter is a requirement in an Honours plan.] Prereq: MATH 225/126 or 135 or 145 PMATH 345 LEC 0.50 Course ID: 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. [Offered: F,S] PMATH 346 LEC 0.50 Course ID: 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 groups. PMATH 351 LEC 0.50 Course ID: Real Analysis Normed and metric spaces, open sets, continuous mappings, sequence and function spaces, completeness, contraction mappings, compactness of metric spaces, finite-dimensional normed spaces, Arzela-Ascoli theorem, existence of solutions of differential equations, Stone-Weierstrass theorem. Prereq: MATH 247 or PMATH 352; Not open to General Mathematics students PMATH 352 LEC 0.50 Course ID: 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 247 or AMATH/PMATH 331; Not open to General Mathematics students PMATH 360 LAB,LEC 0.50 Course ID: Geometry

3 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. [Note: This course will be of interest to all math students.] Prereq: MATH 225/126 or MATH 235 or 245 PMATH 365 LEC 0.50 Course ID: Elementary Differential Geometry An introduction to local differential geometry, laying the groundwork for both global differential geometry and general relativity. Submanifolds of n-dimensional Euclidean space. Embedded curves and the intrinsic geometry of surfaces in Euclidean 3-space. Metrics, geodesics, and curvature. Gaussian curvature and the Gauss-Bonnet theorem. [Offered: W] Prereq: (AMATH 231 or MATH 247) and MATH 235 or 245; Not open to General Mathematics students PMATH 367 LEC 0.50 Course ID: Set Theory & General Topology Relations, functions, well-orderings, Schroder-Bernstein theorem, recursion, axiom of choice and equivalents, ordinals, cardinals, continuum hypothesis, singular and inaccessible cardinals. Topological spaces, bases and sub-bases, closure and interior, product spaces, quotient spaces, nets and filters. Hausdorff spaces, completely regular and normal spaces, Urysohn's lemma, Tietze extension theorum. Compactness, Tychonoff's theorum, Stone-Cech compactification. Connectedness, path connectedness, Function spaces. Prereq: AMATH/PMATH 331 or PMATH 351; Not open to General Mathematics students PMATH 370 LEC 0.50 Course ID: Chaos and Fractals The mathematics of iterated functions, properties of discrete dynamical systems, Mandelbrot and Julia sets. [Note: Programming experience on one computer language with graphical output is recommended.] Prereq: (One of MATH 118, 119, 128, 138, 148) and (One of MATH 114, 115, 225/126, 235, 245); Not open to General Mathematics students (Cross-listed with CM 370) PMATH 399 RDG 0.50 Course ID: Readings in Pure Mathematics Prereq: Not open to General Mathematics students PMATH 400s PMATH 432 LEC 0.50 Course ID: First Order Logic and Computability The concepts of formal provability and logical consequence in first order logic are introduced, and their equivalence is proved in the soundness and completeness theorems. Goedel's incompleteness theorem is discussed, making use of the halting problem of computability theory. Relative computability and the Turing degrees are further studied. Prereq: PMATH 345 or 346; Not open to General Mathematics students

4 PMATH 433 LEC 0.50 Course ID: Model Theory and Set Theory Model theory: the semantics of first order logic including the compactness theorem and its consequences, elementary embeddings and equivalence, the theory of definable sets and types, quantifier elimination, and omega-stability. Set theory: well-orderings, ordinals, cardinals, Zermelo-Fraenkel axioms, axiom of choice, informal discussion of classes and independence results. Prereq: PMATH 345 or 346; Not open to General Mathematics students PMATH 440 LEC 0.50 Course ID: 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 AMATH/PMATH 332; Not open to General Mathematics students PMATH 441 LEC 0.50 Course ID: 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; Not open to General Mathematics students PMATH 442 LEC 0.50 Course ID: 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, 346; Not open to General Mathematics students PMATH 444 LEC 0.50 Course ID: Rings, Modules, and Representations 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; Not open to General Mathematics students. Coreq: PMATH 346 PMATH 450 LEC 0.50 Course ID: Lebesgue Integration and Fourier Analysis Lebesgue measure on the line, the Lebesgue integral, monotone and dominated convergence theorems, Lp-spaces: completeness and dense subspaces. Separable Hilbert space, orthonormal bases. Fourier analysis on the circle, Dirichlet kernel, Riemann-Lebesgue lemma, Fejer's theorem and convergence of Fourier series. Prereq: PMATH 351; Not open to General Mathematics students PMATH 451 LEC 0.50 Course ID: Measure and Integration

5 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 354/450; Not open to General Mathematics students PMATH 453 LEC 0.50 Course ID: Functional Analysis Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, open mapping theorem, closed graph theorem, topologies, nets, Hausdorff spaces, Tietze extension theorem, dual spaces, weak topologies, Tychonoff's theorem, Banach-Alaoglu theorem, reflexive spaces. Prereq: PMATH 354/450; Not open to General Mathematics students PMATH 464 LEC 0.50 Course ID: Algebraic Curves An introduction to the geometry of algebraic curves with applications to elliptic curves and computational algebraic geometry. Plane curves, affine varieties, the group law on the cubic, and applications. Prereq: PMATH 345; Not open to General Mathematics students PMATH 465 LEC 0.50 Course ID: Differential Geometry An introduction to differentiable manifolds. The tangent and cotangent bundles. Vector fields and differential forms. The Lie bracket and Lie derivative of vector fields. Exterior differentiation, integration of differential forms, and Stokes's Theorem. Riemannian manifolds, affine connections, and the Riemann curvature tensor. [Note: Offered in the Winter of even years.] Prereq: AMATH 333/PMATH 365; Not open to General Mathematics students PMATH 467 LEC 0.50 Course ID: Topology Review of general topology, quotient spaces, scissors and glue constructions. Basics on homotopy and topological manifolds. The fundamental group. Compact surfaces. Introduction to homology. Selected applications to covering spaces, homotopy theory, general manifolds, knots, differential equation, combinatorial group theory. Prereq: PMATH 351 or 367; Not open to General Mathematics students. Coreq: PMATH 346 PMATH 499 RDG 0.50 Course ID: Readings in Pure Mathematics Prereq: Not open to General Mathematics students