PMATH 632First Order Logic and Computability0.50 units
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.
Prerequisite: PMATH 345 or 346 or consent of department.
An introduction to algebraic number theory; unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations, Fermat's last theorem. Students without the required prerequisite may seek consent of the department.
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.
Prerequisites: PMATH 345 and 346 or consent of department.
PMATH 644Rings, Modules and Representations0.50 units
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.
Department Consent Required
Prerequisites: PMATH 345 and 346 or consent of department.
PMATH 650Lebesgue Integration and Fourier Analysis0.50 units
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.
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. Students without the required prerequisite may seek consent of the department.
Prerequisite: PMATH 354/450 or consent of department.
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.
Some global aspects of surface theory, the Euler-Poincar characteristic, the global interpretation of Gaussian curvature via the Gauss-Bonnet formula. Submanifolds of n-space, induced Riemannian metrics, extrinsic and intrinsic curvatures, Gauss-Codazzi equations. Local Lie groups of transformations on n-space, infinitesimal generators, the Lie derivative. An introduction to differentiable manifolds, the tangent and cotangent bundels, affine connections and the Riemann curavture tensor. The above topics will be illustrated by applications to continuum mechanics and mathematical physics. Students without the required prerequisite may seek consent of the department.
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 indpendence results.
An introduction to elementary and analytic number theory; primitive roots, law of quadratic reciporcity, Gaussian sums, Riemann zeta-function, distribution of prime numbers. Students without the required prerequisite may seek consent of the department.
Prerequisite: PMATH 332 or 352 or consent of department.
Techniques for classification of finite groups: p-groups, Sylow Theorems, direct and semidirect products. Representations of finite groups: character theory, restrictions and tensor products of representations, representations as modules over group algebras, induced representations, Frobenius reciprocity, Mackey's irreducibility criterion.
PMATH 763Introduction to Lie Groups and Lie Algebras0.50 units
An introduction to matrix Lie groups and their associated Lie algebra's: geometry of matrix Lie groups; relations between a matrix Lie group and its Lie algebra; representation theory of matrix Lie groups.
An introduction to the geometry of algebraic curves with applications to elliptic curves and computational algebraic geometry. Plane curves, affine varieties, the group law on the cubic, and applications.
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.