Euclid's axioms, triangle centres, conic sections, compass-and-straightedge constructions, isometries of the Euclidean plane and of Euclidean space, regular and star-shaped polygons, tessellations of the Euclidean plane, regular and quasi-regular polyhedra, symmetries of polygons and polyhedra, four-dimensional polytopes, sphere packings, and the kissing problem. Applications.
An introduction to three types of non-Euclidean geometry: spherical, projective, and hyperbolic geometry. Lines, distances, circles, triangles, and areas in these non-Euclidean spaces. Conic sections in the projective plane. Inversions and orthogonal circles. Models of the hyperbolic plane (such as the Poincaré disc model or the upper-half plane model). Tilings of the hyperbolic plane.
PMATH 330Introduction to Mathematical Logic0.50 units
A broad introduction to mathematical logic. The notions of logical consequence and derivation are introduced in the settings of propositional and first order logic, with discussions of the completeness theorem and satisfiability
Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.
Prerequisites
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications.
Prerequisites
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
This course is designed to enable students who did not take the MATH 147/MATH 148/MATH 247 sequence of advanced calculus courses to fill in the missing gaps from analysis needed to prepare them for studying further topics in analysis in PMATH 351/PMATH 352. Topics discussed include: completeness of the real numbers, supremum and infimum, sequences and the Bolzano-Weierstrass theorem; the topology of Euclidean space: open, closed, connected, and compact sets, the intermediate value theorem, the Heine-Borel theorem, and the extreme value theorem; continuity and uniform continuity, sequences of functions and uniform convergence; the Riemann integral over rectangles and the integrability of continuous functions.
Prerequisites
Complete 1 of the following
Earned a minimum grade of 70% in each of the following:
PMATH 334Introduction to Rings and Fields with Applications0.50 units
Rings, ideals, factor rings, homomorphisms, finite and infinite fields, polynomials and roots, field extensions, algebraic numbers, and applications, for example, to Latin squares, finite geometries, geometrical constructions, error-correcting codes.
Prerequisites
Must have completed at least 1 of the following:
MATH235 - Linear Algebra 2 for Honours Mathematics (0.50)
MATH245 - Linear Algebra 2 (Advanced Level) (0.50)
Antirequisites
Not completed nor concurrently enrolled in any of the following:
An elementary approach to the theory of numbers; the Euclidean algorithm, congruence equations, multiplicative functions, solutions to Diophantine equations, continued fractions, and rational approximations to real numbers.
PMATH 343Introduction to the Mathematics of Quantum Information0.50 units
Finite dimensional Hilbert spaces. Unitary operators and evolution of closed systems. Hermitian and positive operators, observables, measurements, and the uncertainty principle. Tensor products, entanglement, and Bell inequalities. Superdense coding, teleportation, and the no-cloning theorem. Ensembles of states, density matrices, and the partial trace. Quantum channels, the Choi-Jamiolkowski isomorphism, and the Choi-Krauss representation. Purification and distance between states.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
MATH235 - Linear Algebra 2 for Honours Mathematics (0.50)
MATH245 - Linear Algebra 2 (Advanced Level) (0.50)
Groups, subgroups, homomorphisms and quotient groups, isomorphism theorems, group actions, Cayley and Lagrange theorems, permutation groups, Sylow theorems, and the fundamental theorem of finite abelian groups. Elementary properties of rings, subrings, ideals, homomorphisms and quotients, isomorphism theorems, and polynomial rings.
Prerequisites
Must have completed at least 1 of the following:
MATH235 - Linear Algebra 2 for Honours Mathematics (0.50)
MATH245 - Linear Algebra 2 (Advanced Level) (0.50)
Unique factorization domains, Gauss's lemma, and Eisenstein's criterion. Fields, algebraic and transcendental extensions, minimal polynomials, splitting fields, and the structure of finite fields. Galois theory. Solvable groups and the insolvability of the quintic.
Cardinality, countable and uncountable sets, normed spaces and metric spaces, open sets, continuous mappings, finite-dimensional normed spaces, sequence and function spaces, completeness, contraction mappings and the Banach fixed-point theorem, the Baire category theorem, compactness of metric spaces, the Arzela-Ascoli theorem, and the Stone-Weierstrass theorem.
Curves and hypersurfaces in Euclidean space; integration and Stokes' Theorem; Gaussian and mean curvatures; Frenet-Serret frames and geodesics; Gauss-Bonnet Theorem.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following:
MATH235 - Linear Algebra 2 for Honours Mathematics (0.50)
MATH245 - Linear Algebra 2 (Advanced Level) (0.50)
Must have completed at least 1 of the following:
MATH237 - Calculus 3 for Honours Mathematics (0.50)
Topological spaces, bases, and Hausdorff spaces. Subspace, product, and quotient topologies. Continuous maps, compactness, connectedness, and path-connectedness. Topological manifolds. The fundamental group, the Seifert-Van Kampen Theorem, and covering spaces.
Prerequisites
Must have completed at least 1 of the following:
PMATH336 - Introduction to Group Theory with Applications (0.50)
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 Gödel's incompleteness theorem.
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.
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.
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
PMATH 445Representations 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, Mackey's irreducibility criterion.
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.
PMATH 464Introduction 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.
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.
We study algebraic structures which can be associated to topological spaces, and which are invariant under homotopy equivalence. Such topological invariants include homotopy groups, homology groups, and cohomology groups or rings. Possible applications include the classification of Platonic solids, the Brouwer and Lefschetz fixed-point theorems, the hairy ball theorem, the Borsuk-Ulam theorem, and the ham sandwich theorem.
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.
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.
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.
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.
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.
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.
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations.
PMATH 745Representations 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.
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.
PMATH 764Introduction 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.
PMATH 810Banach 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.
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.
PMATH 863Introduction 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.
PMATH 868Connections 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.
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.