Pure Mathematics (PMATH)

28 graduate courses in the Spring 2026 Graduate Calendar.

Older calendars, Fall 2006 to Winter 2024

Year	Fall	Winter	Spring
2023–24	Fall	Winter	Spring
2022–23	Fall	Winter	Spring
2021–22	Fall	Winter	Spring
2020–21	Fall	Winter	Spring
2019–20	Fall	Winter	Spring
2018–19	Fall	Winter	Spring
2017–18	Fall	Winter	Spring
2016–17	Fall	Winter	Spring
2015–16	Fall	Winter	Spring
2014–15	Fall	Winter	Spring
2013–14	Fall	Winter	Spring
2012–13	Fall	Winter	Spring
2011–12	Fall	Winter	Spring
2010–11	Fall	Winter	Spring
2009–10	Fall	Winter	Spring
2008–09	Fall	Winter	Spring
2007–08	Fall	Winter	Spring
2006–07	Fall	Winter	Spring

Terms on uwloo come from Waterloo’s archived copies of the calendar. Where uwloo can’t read a term’s calendar, its course list is Waterloo’s course catalog for that term (UW Open Data). Terms marked ↗ aren’t on uwloo yet: they go to Waterloo’s archive.

PMATH 632 Mathematical Logic0.50 units

First-order languages and structures, syntactic provability and semantic truth, the soundness and completeness theorems, compactness and its consequences, elementary equivalence, the ZFC axioms of set theory, ordinals, cardinals, computability and Goedel’s incompleteness theorem.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH432

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 646 Introduction to Commutative Algebra0.50 units

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.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH446

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 650 Measure Theory0.50 units

Review of Riemann integration. Measure spaces: sigma-algebras, Borel sets, outer measure, Caratheodory condition, Lebesgue-Steiltjes measures, translation invariance and regularity of Lebesgue measure. Measurable functions: simple approximation, Egorov's and Lusin's theorems. Integration: monotone convergence theorem, Fatou's lemma, Lebesgue dominated convergence theorem. Signed measures: Hahn and Jordan decompositions, absolute continuity of measures, Radon-Nikodym theorem. Lp spaces: completeness, separability, Riesz representation theorem.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH450

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 652 Topics in Complex Analysis0.50 units

The Riemann mapping theorem and several topics such as analytic continuation, harmonic functions, elliptic functions, entire functions, univalent functions, special functions. Students without the required prerequisite may seek consent of the department.

Prerequisites

Must have completed the following:
- PMATH352

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 655 Convex Analysis and Geometry0.50 units

Basic convexity, Caratheodory theorem, Radon and Helly theorems, geometry of polytopes, isoperimetric inequality, symmetrizations, rearrangement inequalities, duality.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH455

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 665 Smooth Manifolds0.50 units

Smooth manifolds, smooth maps, and tangent vectors; the tangent and cotangent bundles; vector fields, tensor fields, and differential forms; Stokes' theorem; integral curves, Lie derivatives, the Frobenius theorem; de Rham cohomology.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH465

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 667 Algebraic Topology0.50 units

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.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH467

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 690 Literature and Research Studies0.50 units

Reading Course

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 733 Model Theory0.50 units

Definable sets, quantifier elimination, algebraically closed fields, real closed fields, omitting types and prime models, interpretation and imaginaries, types, saturation, strongly minimal sets and the Zilber trichotomy, forking and independence.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH433

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 734 Set Theory0.50 units

ZFC axioms, well-orders, ordinals, regular and singular cardinals, the cumulative hierarchy, Boolean algebras, ultrafilters, club and stationary sets, measurable cardinals, the reflection principle, Godel’s constructible universe, ordinal definable sets.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH434

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 740 Analytic Number Theory0.50 units

Summation methods, analytic theory of the Riemann zeta function, Prime Number Theorem, primitive roots. Dirichlet characters and infinitude of primes in arithmetic progressions, and assorted topics.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH440

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 741 Algebraic Number Theory0.50 units

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

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH441
- PMATH641

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 745 Representations of Finite Groups0.50 units

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, Mackeys irreducibility criterion.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH445

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 753 Functional Analysis0.50 units

Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, open mapping theorem, closed graph theorem, topologies, nets, Hausdorff spaces, dual spaces, weak topologies, Tychonoff's theorem, Banach-Alaoglu theorem, reflexive spaces.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH453

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 754 Fourier Analysis0.50 units

Fourier analysis on the circle: Fourier series; Hilbertian analysis; convolutions and summability kernels; pointwise, uniform and Lp convergence of Fourier series. Fourier analysis on the real line: test and Schwarz class functions, distributions and tempered distributions, Fourier transforms, inversion theorem, L1 and L2 analysis. Selected applications.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH457

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 757 Topological Dynamics and Ergodic Theory0.50 units

Compact topological dynamical systems, the greatest ambit, spaces of ultrafilters, Van der Waerden and Hales Jewett theorems, minimal flows, boundary actions, invariant measures, probability measure preserving actions, Rokhlin’s lemma, ergodic theorems, orbit equivalence, entropy, Ornstein’s theorem.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH457

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 764 Introduction to Algebraic Geometry0.50 units

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.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH464

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 810 Banach Algebras and Operator Theory0.50 units

Banach algebras, functional calculus, Gelfan transform, Jacobson radical, Banach space and Hilbert space operators, Fredholm alternative, spectral therorem for compact normal operators, ideals in C^*-algebras, linear functionals and states, Gelfand-Naimark-Segar (GNS) construction, von Neumann algebras, strong/weak operator topologies, Double Commutant theorem, Kaplansky's density theorem, spectral theorem for normal operators. Students are expected to have background in functional analysis covering the topics of PMATH 753.

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 833 Harmonic Analysis0.50 units

Basic topics in Fourier analysis on locally compact groups, in particular abelian groups: Haar measure, convolution, characters, the dual group, Fourier transform, Parseval's theorem, Plancherel theorem, Pontryagin duality theorem, inversion theorem, Bochener's theorem. Other topics such as harmonic analysis on compact or non-albian locally compact groups, Peter-Weyl theorem, amenability, probabilistic methods in harmonic analysis.

Antirequisites

Not completed nor concurrently enrolled in:
- PMATH950

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 863 Introduction to Lie Groups and Lie Algebras0.50 units

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.

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 868 Connections and Riemannian Geometry0.50 units

Review of smooth manifolds. Vector bundles. Connections and curvature, holonomy, characteristic classes. Connections on tangent bundle: torsion, geodesics, exponential map. Riemannian geometry: Levi-Civita connection, Riemannian geodesics, Hopf-Rinow Theorem. Additional topics if time permits. Students are expected to have background in smooth manifolds covering the topics of PMATH 665.

Archived calendar entry Requirements, Spring 2026 Course outlines

PMATH 871 Quantum Information Processing0.50 units

Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.