Course subject: Pure Mathematics (PMATH)

For more detailed course information, click on a course title below.

Pure Mathematics (PMATH) 632 First Order Logic and Computability (0.50) LEC

Course ID: 002339
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.

Pure Mathematics (PMATH) 641 Algebraic Number Theory (0.50) LEC

Course ID: 002341
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.

Pure Mathematics (PMATH) 646 Introduction to Commutative Algebra (0.50) LEC

Course ID: 014670
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.

Pure Mathematics (PMATH) 650 Lebesgue Integration and Fourier Analysis (0.50) LEC

Course ID: 013667
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.

Pure Mathematics (PMATH) 651 Measure and Integration (0.50) LEC

Course ID: 002346
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.

Pure Mathematics (PMATH) 652 Topics in Complex Analysis (0.50) LEC

Course ID: 002347
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.

Pure Mathematics (PMATH) 665 Smooth Manifolds (0.50) LEC

Course ID: 002349
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.

Pure Mathematics (PMATH) 667 Algebraic Topology (0.50) LEC

Course ID: 002350
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.

Pure Mathematics (PMATH) 690 Literature and Research Studies (0.50) RDG

Course ID: 002351
Reading Course

Pure Mathematics (PMATH) 733 Model Theory and Set Theory (0.50) LEC

Course ID: 013668
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.

Pure Mathematics (PMATH) 740 Analytic Number Theory (0.50) LEC

Course ID: 013669
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.

Pure Mathematics (PMATH) 745 Representations of Finite Groups (0.50) LEC

Course ID: 014343
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.

Pure Mathematics (PMATH) 753 Functional Analysis (0.50) LEC

Course ID: 013670
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.

Pure Mathematics (PMATH) 764 Introduction to Algebraic Geometry (0.50) LEC

Course ID: 002392
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.

Pure Mathematics (PMATH) 810 Banach Algebras and Operator Theory (0.50) LEC

Course ID: 011875
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.

Pure Mathematics (PMATH) 833 Harmonic Analysis (0.50) LEC

Course ID: 015700
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, invrsion 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.

Pure Mathematics (PMATH) 863 Introduction to Lie Groups and Lie Algebras (0.50) LEC

Course ID: 002391
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.

Pure Mathematics (PMATH) 868 Connections and Riemannian Geometry (0.50) LEC

Course ID: 016231
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.

Pure Mathematics (PMATH) 871 Quantum Information Processing (0.50) LEC

Course ID: 011589
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.

Pure Mathematics (PMATH) 930 Topics in Logic (0.50) LEC

Course ID: 014769