Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
The Department of Pure Mathematics offers many courses each year. Graduate courses consist of lectures given by faculty members, usually two or three hours per week. Graduate students can take graduate courses for credit, or simply for interest. They can also attend undergraduate courses if extra background is desired.
Please take a look at our current course offerings.
In order to gain credit for a 6xx course a graduate student must do work in addition to that required for the corresponding 4xx course. Typically this involves a reading project accompanied by the presentation of a talk and/or submission of an essay.
Our 600 and 700 level courses are offered on a regular basis and are typically graded on the basis of assignments and a final examination.
Topics courses are more specialized. Typically 6-7 are offered each year, with contents varying according to timing and instructor availability.
930 - Topics in logic
940 - Topics in number theory
945 - Topics in algebra
950 - Topics in analysis
965 - Topics in geometry and topology
990 - Topics in pure mathematics
There are regular, on-going seminars in analysis, geometry and topology, number theory and logic. Visitors, professors, postdoctoral fellows and graduate students present their recent research in these seminars. As well, learning seminars are frequently held in the area of any major research strengths. In these seminars, a group of interested people study a new topic.
There is a regular Departmental Colloquium at which distinguished visitors speak. Speakers in the Colloquium are asked to aim their talk to an audience of general mathematicians, rather than the specialists in their area.
The graduate students also organize a Graduate Student Colloquium in which graduate students speak about their mathematical interests. The Graduate Student Colloquium is typically held monthly and is followed by a social event.
Graduate students also organize other specialized seminars.
Graduate students enrolled in the 600-level "held with" courses will be required to complete additional assignments, a final project and/or an oral presentation over and above the corresponding undergraduate course requirements.
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. Gödel's incompleteness theorem is discussed; making use of the halting problem of computability theory. Relative computability and the Turing degrees are further studied.
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
Module theory: classification of finitely generated modules over PIDs, exact sequences and tensor products, algebras, localisation, chain conditions. Primary decomposition, integral extensions, Noether's normalisation lemma, and Hilbert's Nullstellensatz.
Lebesgue measure on the line, the Lebesgue integral, monotone and dominated convergence theorems, Lp-spaces: completeness and dense subspaces. Separable Hilbert spaces, orthonormal bases. Fourier analysis on the circle, Dirichlet kernel, Riemann-Lebesgue lemma, Fejer's theorem and convergence of Fourier series.
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.
Point-set topology; smooth manifolds, smooth maps and tangent vectors; the tangent bundle; vector fields, tensor fields and differential forms. Other topics may include: de Rham cohomology; Frobenius Theorem; Riemannian metrics, connections and curvature.
Topological spaces and topological manifolds; quotient spaces; cut and paste constructions; classification of two-dimensional manifolds; fundamental group; homology groups. Additional topics may include: covering spaces; homotopy theory; selected applications to knots and combinatorial group 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.
Summation methods; analytic theory of the Riemann zeta function; Prime Number Theorem; primitive roots; quadratic reciprocity; Dirichlet characters and infinitude of primes in arithmetic progressions; assorted topics.
Basic definitions and examples: subrepresentations and irreducible representations, tensor products of representations. Character theory. Representations as modules over the group ring, Artin-Wedderburn structure theorem for semisimple rings. Induced representations, Frobenius reciprocity, Mackey's irreducibility criterion.
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.
An introduction to matrix Lie groups and their associated Lie algebras: geometry of matrix Lie groups; relations between a matrix Lie group and its Lie algebra; representation theory of matrix Lie groups.
An introduction to algebraic geometry through the theory of algebraic curves. General Algebraic Geometry: affine and projective algebraic sets, Hilbert's Nullstellensatz, co-ordinate rings, polynomial maps, rational functions and local rings. Algebraic Curves: affine and projective plane curves, tangency and multiplicity, intersection numbers, Bezout's theorem and divisor class groups.
Banach algebras, functional calculus, Gelfand transform, Jacobson radical, Banach space and Hilbert space operators, Fredholm alternative, spectral theorem for compact normal operators, ideals in C*-algebras, linear functionals and states, GNS construction, von Neumann algebras, strong/weak operator topologies, Double Commutant theorem, Kaplansky's Density Theorem, spectral theorem for normal operators.
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca