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. Course times can be found on the Graduate schedule of classes. See our future course offerings for information on upcoming nontopics courses.
Fall 2017 courses
 PMATH 651/451  Measure and integration  Instructor: C. Schafhauser
 PMATH 665/465  Geometry of manifolds  Instructor: D. Park
 PMATH 740/440  Analytic number theory  Instructor: Y.R. Liu
 PMATH 745/445  Representations of finite groups  Instructor: B. Webster
 PMATH 753/453  Functional analysis  Instructor: M. Kennedy
 PMATH 930  Topics in logic: introduction to universal algebra  Instructor: R. Willard
 PMATH 940  Topics in number theory: class field theory  Instructor: D. McKinnon
 PMATH 950  Topics in analysis: Hardy spaces  Instructor: K. Davidson
 PMATH 990  Topics in pure mathematics: free analogues for fundamental probabilistic structures  Instructor: A. Nica
Spring 2017 courses
 PMATH 650/450  Fourier analysis  Instructor: K.E. Hare
 PMATH 764  Introduction to algebraic geometry  Instructor: M. Satriano
Here is an archive of past course offerings.
Joint graduate/undergraduate courses
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.
 Pure Mathematics (PMATH) 632 held with PMATH 432  First order logic and computability
 PMATH 641 held with PMATH 441  Algebraic number theory
 PMATH 646 held with PMATH 446  Introduction to commutative algebra
 PMATH 650 held with PMATH 450  Lebesgue integration and Fourier analysis
 PMATH 651 held with PMATH 451  Measure and integration
 PMATH 665 held with PMATH 465  Geometry of manifolds
 PMATH 667 held with PMATH 467  Algebraic topology
Reading course
 PMATH 690  Literature and research studies
Intermediate level courses

PMATH 445 held with PMATH 745  Representations of finite groups

PMATH 463 held with PMATH 763  Introduction to Lie groups and Lie algebras

PMATH 464 held with PMATH 764  Introduction to algebraic geometry
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.
Advanced level courses
Topics courses
Topics courses are more specialized. Typically 67 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
Seminars and colloquia
There are regular, ongoing 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 course descriptions
Graduate students enrolled in the 600level "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.
PMATH 632 held with PMATH 432  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. Gödel's incompleteness theorem is discussed; making use of the halting problem of computability theory. Relative computability and the Turing degrees are further studied.
PMATH 641 held with PMATH 441  Algebraic number theory
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
PMATH 646 held with PMATH 446  Introduction to commutative algebra
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.
PMATH 650 held with PMATH 450  Lebesgue integration and Fourier analysis
Lebesgue measure on the line, the Lebesgue integral, monotone and dominated convergence theorems, Lpspaces: completeness and dense subspaces. Separable Hilbert spaces, orthonormal bases. Fourier analysis on the circle, Dirichlet kernel, RiemannLebesgue lemma, Fejer's theorem and convergence of Fourier series.
PMATH 651 held with PMATH 451  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, RadonNikodym theorem, product measures, Fubini's theorem, signed measures, Urysohn's lemma, Riesz Representation theorems for classical Banach spaces.
PMATH 665 held with PMATH 465  Geometry of Manifolds
Pointset 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.
PMATH 667 held with PMATH 467  Algebraic topology
Topological spaces and topological manifolds; quotient spaces; cut and paste constructions; classification of twodimensional manifolds; fundamental group; homology groups. Additional topics may include: covering spaces; homotopy theory; selected applications to knots and combinatorial group theory.
PMATH 433 held with PMATH 733  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 omegastability. Set theory: wellorderings, ordinals, cardinals, ZermeloFraenkel axioms, axiom of choice, informal discussion of classes and independence results.
PMATH 440 held with PMATH 740  Analytic number theory
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.
PMATH 445 held with PMATH 745  Representations of finite groups
Basic definitions and examples: subrepresentations and irreducible representations, tensor products of representations. Character theory. Representations as modules over the group ring, ArtinWedderburn structure theorem for semisimple rings. Induced representations, Frobenius reciprocity, Mackey's irreducibility criterion.
PMATH 453 held with PMATH 753  Functional analysis
Banach and Hilbert spaces, bounded linear maps, HahnBanach theorem, open mapping theorem, closed graph theorem, topologies, nets, Hausdorff spaces, Tietze extension theorem, dual spaces, weak topologies, Tychonoff's theorem, BanachAlaoglu theorem, reflexive spaces.
PMATH 463 held with PMATH 763  Introduction to Lie groups and Lie algebras
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.
PMATH 464 held with PMATH 764  Introduction to algebraic geometry
An introduction to algebraic geometry through the theory of algebraic curves. General Algebraic Geometry: affine and projective algebraic sets, Hilbert's Nullstellensatz, coordinate 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.
PMATH 810  Banach algebras and operator theory
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.