26 graduate courses in the Spring 2024 Graduate Calendar.
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
Reading Course
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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 entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
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.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines
Consult the schedule of classes for specific topics being offered.
Archived calendar entryRequirements, Spring 2024Course outlines