# PMATH Courses

# PMATH 320 – Euclidean Geometry

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.

# PMATH 321 – Non-Euclidean Geometry

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 330 – Introduction to Mathematical Logic

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.

# PMATH 331 – Applied Real Analysis

Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.

# PMATH 332 – Applied Complex Analysis

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.

# PMATH 333 – Introduction to Real Analysis

The purpose of the course is to present the familiar concepts of calculus at a rigorous level and to provide students who took the MATH 137/MATH 138/MATH 237 sequence with the background needed to be successful in PMATH 351 and PMATH 352. Topics discussed include the completeness properties of the reals; the density of the rationals; the topology of real n-dimensional space: open and closed sets, connectedness, compactness (by open covers), the Heine-Borel theorem, completeness; sequences in real n-dimensional space: convergence, Cauchy sequences, subsequences, the Bolzano-Weierstrass theorem; multivariable functions: limits, point-wise and uniform continuity, the extreme value theorem, uniform convergence of sequences of functions, Taylor's theorem, term-by-term differentiation of power series; integration in real n-dimensional space: Riemann integrability, Fubini's theorem for continuous functions on rectangles, term-by-term integration of power series.

# PMATH 334 – Introduction to Rings and Fields with Applications

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.

# PMATH 336 – Introduction to Group Theory with Applications

Groups, permutation groups, subgroups, homomorphisms, symmetry groups in 2 and 3 dimensions, direct products, Polya-Burnside enumeration.

# PMATH 340 – Elementary Number Theory

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 345 – Polynomials, Rings and Finite Fields

Elementary properties of rings, polynomial rings, Gaussian integers, integral domains and fields of fractions, homomorphisms and ideals, maximal ideals and fields, Euclidean rings, principal ideals, Hilbert Basis theorem, Gauss' lemma, Eisenstein's criterion, unique factorization, computational aspects of polynomials, construction of finite fields with applications, primitive roots and polynomials, additional topics. [Offered: F,S]

# PMATH 346 – Group Theory

Elementary properties of groups, cyclic groups, permutation groups, Lagrange's theorem, normal subgroups, homomorphisms, isomorphism theorems and automorphisms, Cayley's theorem and generalizations, class equation, combinatorial applications, p-groups, Sylow theorems, groups of small order, simplicity of the alternating groups, direct product, fundamental structure theorem for finitely generated Abelian groups.

# PMATH 347 – Groups and Rings

Groups, subgroups, homomorphisms and quotient groups, isomorphism theorems, group actions, Cayley and Lagrange theorems, permutation groups and the fundamental theorem of finite abelian groups. Elementary properties of rings, subrings, ideals, homomorphisms and quotients, isomorphism theorems, polynomial rings, and unique factorization domains.

# PMATH 348 – Fields and Galois Theory

Fields, algebraic and transcendental extensions, minimal polynomials, Eisenstein's criterion, splitting fields, and the structure of finite fields. Sylow theorems and solvable groups. Galois theory. The insolvability of the quintic.

# PMATH 351 – Real Analysis

Normed and metric spaces, open sets, continuous mappings, sequence and function spaces, completeness, contraction mappings, compactness of metric spaces, finite-dimensional normed spaces, Arzela-Ascoli theorem, existence of solutions of differential equations, Stone-Weierstrass theorem.

# PMATH 352 – Complex Analysis

Analytic functions, Cauchy-Riemann equations, Goursat's theorem, Cauchy's theorems, Morera's theorem, Liouville's theorem, maximum modulus principle, harmonic functions, Schwarz's lemma, isolated singularities, Laurent series, residue theorem.

# PMATH 354 – Measure Theory and Fourier Analysis

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. [Offered: W]

# PMATH 360 – Geometry

An introduction to affine, projective and non-Euclidean forms of geometry. Conic sections in the projective plane. Inversion in circles. Theorems of Desargues, Pappus, and Pascal.

# PMATH 365 – Differential Geometry

Submanifolds of Euclidean n-space; vector fields and differential forms; integration on submanifolds and Stokes's Theorem; metrics and geodesics; Gauss-Bonnet Theorem.

# PMATH 367 – Set Theory & General Topology

Relations, functions, well-orderings, Schroder-Bernstein theorem, recursion, axiom of choice and equivalents, ordinals, cardinals, continuum hypothesis, singular and inaccessible cardinals. Topological spaces, bases and sub-bases, closure and interior, product spaces, quotient spaces, nets and filters. Hausdorff spaces, completely regular and normal spaces, Urysohn's lemma, Tietze extension theorum. Compactness, Tychonoff's theorum, Stone-Cech compactification. Connectedness, path connectedness, Function spaces.

# PMATH 370 – Chaos and Fractals

The mathematics of iterated functions, properties of discrete dynamical systems, Mandelbrot and Julia sets.

# PMATH 399 – Readings in Pure Mathematics

Reading course as announced by the department.

# PMATH 432 – First Order Logic and Computability

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.

# PMATH 433 – Model Theory and Set Theory

Model theory: the semantics of first order logic including the compactness theorem and its consequences, elementary embeddings and equivalence, the theory of definable sets and types, quantifier elimination, and omega-stability. Set theory: well-orderings, ordinals, cardinals, Zermelo-Fraenkel axioms, axiom of choice, informal discussion of classes and independence results.

# PMATH 434 – Techniques in Computational Number Theory

An introduction to: integer factorization, elliptic curves methods, primality testing, fast integer arithmetic, fast Fourier transforms and quantum computing. This course is taught with a philosophy that encourages experimentation.

# PMATH 440 – Analytic Number Theory

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, and assorted topics.

# PMATH 441 – Algebraic Number Theory

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

# PMATH 442 – Fields and Galois Theory

Normal series, elementary properties of solvable groups and simple groups, algebraic and transcendental extensions of fields, adjoining roots, splitting fields, geometric constructions, separability, normal extensions, Galois groups, fundamental theorem of Galois theory, solvability by radicals, Galois groups of equations, cyclotomic and Kummer extensions.

# PMATH 444 – Rings, Modules, and Representations

Jacobson structure theory, density theorem, Jacobson radical, Maschke's theorem. Artinian rings, Artin-Wedderburn theorem, modules over semi-simple Artinian rings. Division rings. Representations of finite groups.

# PMATH 445 – Representations of Finite Groups

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.

# PMATH 446 – Introduction to Commutative Algebra

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.

# PMATH 450 – Lebesgue Integration and Fourier Analysis

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.

# PMATH 451 – Measure and Integration

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.

# PMATH 453 – Functional Analysis

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.

# PMATH 464 – Introduction to Algebraic Geometry

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 465 – Geometry of Manifolds

Point-set topology; smooth manifolds, smooth maps and tangent vectors; the tangent bundle; vector fields, tensor fields and differential forms. Other topics may include: de Rham cohomology; Frobenius Theorem; Riemannian metrics, connections and curvature.

# PMATH 467 – Algebraic Topology

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.

# PMATH 499 – Readings in Pure Mathematics

Reading course as announced by the department.

# PMATH 632 – First Order Logic and Computability

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.

# PMATH 641 – Algebraic Number Theory

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

# PMATH 642 – Fields and Galois Theory

Normal series, elementary properties of solvable groups and simple groups, algebraic and transcendental extensions of fields, adjoining roots, splitting fields, geometric constructions, separability, normal extensions, Galois groups, fundamental theorem of Galois theory, solvability by radicals, Galois groups of equations, cyclotomic and Kummer extensions. Students without the required prerequisite may seek consent of the department.

# PMATH 644 – Rings, Modules and Representations

Jacobson structure theory, density theorem, Jacobson radical, Maschke's theorem. Artinian rings, Artin-Wedderburn theorem, modules over semi-simple Artinian rings. Division rings. Representations of finite groups. Finitely generated modules over principal ideal domains. Students without the required prerequisite may seek consent of the department.

# PMATH 646 – Introduction to Commutative Algebra

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.

# PMATH 650 – Lebesgue Integration and Fourier Analysis

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.

# PMATH 651 – Measure and Integration

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.

# PMATH 652 – Topics in Complex Analysis

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.

# PMATH 665 – Geometry of Manifolds

Point-set topology; smooth manifolds, smooth maps, and tangent vectors; the tangent bundle, vector fields, tensor fields, and differential forms. Other topics may include: de Rham cohomology; Frobenius Theorem; Riemannian metrics, connections and curvature.

# PMATH 667 – Algebraic Topology

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.

# PMATH 690 – Literature and Research Studies

Reading Course

# PMATH 701 – Graduate Algebra

Isomorphism theorems, classical structures theorems for finite groups, nilpotent and solvable groups, free groups, presentation modules over Principal Ideal Domains (PIDs), Hilbert Basis Theorem, Groebner bases, Artin-Wedderbury Theorem, field extensions, decompositions, claculation of Galois groups.

# PMATH 702 – Graduate Analysis

Zorn's Lemma and the Axiom of Choice, cardinality, introduction to topological spaces, bases, nets, continuous functions and weak topologies, compactness, connectedness, Banach spaces, Contraction Mapping Principal, finite-dimensional spaces C(X) and C_O(X), Stone-Weierstrass Theorem, Arzela-Ascoli Theorem, Urysohn's Lemma, idelas in C_O(X).

# PMATH 733 – Model Theory and Set Theory

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.

# PMATH 740 – Analytic Number Theory

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.

# PMATH 745 – Representations of Finite Groups

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.

# PMATH 753 – Functional Analysis

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.

# PMATH 763 – Introduction to Lie Groups and Lie Algebras

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 764 – Introduction to Algebraic Geometry

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 800 – Topics in Real and Complex Analysis

# PMATH 810 – Banach Algebras and Operator Theory

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.