25 graduate courses in Waterloo’s course catalog for Fall 2023, from UW Open Data. uwloo can’t read that term’s Graduate Calendar, so this stands in for it; the calendar itself is in Waterloo’s archive.
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.
Antireq: PMATH 432
LEC · Course ID 002339Graduate Calendar, Fall 2023PMATH 632 today
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
Antireq: PMATH 441
LEC · Course ID 002341Graduate Calendar, Fall 2023PMATH 641 today
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.
Antireq: PMATH 446
LEC · Course ID 014670Graduate Calendar, Fall 2023PMATH 646 today
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.
Antireq: PMATH 450
LEC · Course ID 013667Graduate Calendar, Fall 2023PMATH 650 today
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.
Antireq: PMATH 451
LEC · Course ID 002346Graduate Calendar, Fall 2023PMATH 651 today
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.
Prereq: PMATH 352 or consent of department.
LEC · Course ID 002347Graduate Calendar, Fall 2023PMATH 652 today
Point-set topology; smooth manifolds, smooth maps, and tangent vectors; the tangent and cotangent bundles; vector fields, tensor fields, and differential forms; Stokes's theorem; integral curves, Lie derivatives, the Frobenius theorem; de Rham cohomology.
Antireq: PMATH 465
LEC · Course ID 002349Graduate Calendar, Fall 2023PMATH 665 today
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.
Antireq: PMATH 467
LEC · Course ID 002350Graduate Calendar, Fall 2023PMATH 667 today
Reading Course
RDG · Course ID 002351Graduate Calendar, Fall 2023PMATH 690 today
Model theory: the semantics of first order logic including the compactness theorem and its consequences, elementary embedding and equivalence, the theory of definable sets and types, quantifier elimination, and w-stability. Set theory: well-orderings, ordinals, cardinals, Zermelo-Fraenkel axioms, axiom of choice, informal discussion of classes and independence results.
LEC · Course ID 013668Graduate Calendar, Fall 2023PMATH 733 today
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.
Antireq: PMATH 440
LEC · Course ID 013669Graduate Calendar, Fall 2023PMATH 740 today
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.
Antireq: PMATH 445
LEC · Course ID 014343Graduate Calendar, Fall 2023PMATH 745 today
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.
Antireq: PMATH 453
LEC · Course ID 013670Graduate Calendar, Fall 2023PMATH 753 today
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.
Antireq: PMATH 464
LEC · Course ID 002392Graduate Calendar, Fall 2023PMATH 764 today
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.
Prereq: PMATH 753
LEC · Course ID 011875Graduate Calendar, Fall 2023PMATH 810 today
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.
Antireq: PMATH 950 - Topic 9
LEC · Course ID 015700Graduate Calendar, Fall 2023PMATH 833 today
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.
LEC · Course ID 002391Graduate Calendar, Fall 2023PMATH 863 today
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.
Prereq: PMATH 665
LEC · Course ID 016231Graduate Calendar, Fall 2023PMATH 868 today
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.
LEC · Course ID 011589Graduate Calendar, Fall 2023PMATH 871 today
LEC · Course ID 014769Graduate Calendar, Fall 2023PMATH 930 today
LEC · Course ID 014770Graduate Calendar, Fall 2023PMATH 940 today
LEC · Course ID 014771Graduate Calendar, Fall 2023PMATH 945 today
LEC · Course ID 014772Graduate Calendar, Fall 2023PMATH 950 today
LEC · Course ID 014773Graduate Calendar, Fall 2023PMATH 965 today
LEC · Course ID 014774Graduate Calendar, Fall 2023PMATH 990 today