PMATH Course Descriptions

Pure Mathematics (2009-2010)

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.

Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.

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.

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.

Groups, permutation groups, subgroups, homomorphisms, symmetry groups in 2 and 3 dimensions, direct products, Polya-Burnside enumeration.

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.

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]

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. [Offered: W]

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. [Offered: F,S]

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. [Offered: F]

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. [Offered: W]

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.

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]

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.

The mathematics of iterated functions, properties of discrete dynamical systems, Mandelbrot and Julia sets.

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.

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.

An introduction to: integer factorization, elliptic curves methods, primality testing, fast integer arithmetic, fast Fourier transforms and quantum computing. This course is taught with a philosophy that encourages experimentation. [Offered: F]

An introduction to elementary and analytic number theory; primitive roots, law of quadratic reciprocity, Gaussian sums, Riemann zeta-function, distribution of prime numbers.

An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations, Fermat's "last theorem".

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. [Offered: F]

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.

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. [Offered: W]

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. [Offered: F]

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.

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.

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.