PMATH 600s


PMATH 632 First Order Logic and Computability (0.50) LECCourse 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.
Prerequisite: PMATH 345 or 346 or consent of department.

PMATH 641 Algebraic Number Theory (0.50) LECCourse ID: 002341
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.
Prerequisite: PMATH 345 or consent of department.

PMATH 642 Fields and Galois Theory (0.50) LECCourse ID: 002342
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 644 Rings, Modules and Representations (0.50) LECCourse ID: 002343
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 646 Introduction to Commutative Algebra (0.50) LECCourse 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.
Department Consent Required
Prereq: PMATH 347, 348

PMATH 650 Lebesgue Integration and Fourier Analysis (0.50) LECCourse 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.
Instructor Consent Required
Prereq: PMATH 351

PMATH 651 Measure and Integration (0.50) LECCourse 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. Students without the required prerequisite may seek consent of the department.
Prerequisite: PMATH 354/450 or consent of department.

PMATH 652 Topics in Complex Analysis (0.50) LECCourse 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.
Prerequisite: PMATH 352 or consent of department.

PMATH 665 Differential Geometry (0.50) LECCourse ID: 002349
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 adn the Riemann curvature 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. Prerequisites: PMATH 365 or AM 333 or consent of Department.
Prerequisites: PMATH 365 or AM 333 or consent of Department

PMATH 667 Topology (0.50) LECCourse ID: 002350
Topics from algebraic, combinatorial and geometric topology. Students without the required prerequisite may seek consent of the department.
Prerequisite: PMATH 351 or consent of Department

PMATH 690 Literature and Research Studies (0.50) RDGCourse ID: 002351
Reading Course
Department Consent Required
1 Lie Algebras
2 Topology
3 Representation Theory
4 C*-algebras and groups
5 Integration and Probability
6 Kaehler geometry
7 Smooth Mthds in Algbrc Tpology
8 Oprtr Algebras & q - probablty
9 Class Field Theory

PMATH 700s


PMATH 701 Graduate Algebra (0.50) LECCourse ID: 011873
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 (0.50) LECCourse ID: 011874
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 (0.50) LECCourse 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.

PMATH 740 Analytic Number Theory (0.50) LECCourse ID: 013669
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.

PMATH 745 Groups and Representations (0.50) LECCourse ID: 014343
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.
Department Consent Required
Prereq: PMATH 347

PMATH 753 Functional Analysis (0.50) LECCourse ID: 013670
Banach and Hilvert 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.
Prerequisite: PMATH 354/450 or consent of department.

PMATH 763 Introduction to Lie Groups and Lie Algebras (0.50) LECCourse ID: 002391
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.
Instructor Consent Required
Prerequisite: PMATH 346, 351 and 365

PMATH 764 Algebraic Curves (0.50) LECCourse ID: 002392
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.
Instructor Consent Required

PMATH 800s


PMATH 800 Topics in Real and Complex Analysis (0.50) LECCourse ID: 010486
Department Consent Required
1 Fractal Geometry
2 Probabilistic methods
3 Cnvrgnc - Probability Measures
4 Riemann Surfaces

PMATH 810 Banach Algebras and Operator Theory (0.50) LECCourse 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.
Prerequisite: PMATH 653

PMATH 811 Topics in Functional Analysis (0.50) LECCourse ID: 010487
Department Consent Required
1 Random Matrices & Asymptotics
2 Intro to Banach&Operator Alg
3 von Neuman II-1 Factors
4 H^p spaces
5 Intro to Banach spaces

PMATH 822 Topics in Operator Theory (0.50) LECCourse ID: 002408
Department Consent Required
1 Approximate Operator Theory
2 von Neumann Algebras
3 Simultaneous Triangularization
4 Operator Spaces
5 Von Neumann algebras & noncomm
6 Nonself-adj. Operator Algebras
7 Intro. to Operator Algebras
8 Introduction to K-theory
9 Free Probability
10 Quantum Groups

PMATH 833 Topics in Harmonic Analysis (0.50) LECCourse ID: 002412
Department Consent Required
1 Harmonic Anlys on the Circle
2 Harmonic Analysis
3 Semisimple Lie Groups SL(2,R)
4 Quantum Groups

PMATH 844 Topics in Functional Equations (0.50) LECCourse ID: 002414
Department Consent Required

PMATH 900s


PMATH 900 Topics in Algebra (0.50) LECCourse ID: 010483
Department Consent Required
1 Commutative Rings
2 Complex Semisimple Lie Algebra
3 Rep. Theory of Compact Groups
4 Topics: Infinite Group Theory
5 Foundations of Quantum Theory
6 Valuation Theory
7 Valued Fields
8 Special Algebraic Structures
9 Groebner Bases

PMATH 911 Topics in Mathematical Logic (0.50) LECCourse ID: 010484
Department Consent Required
1 Computability Theory
2 Model Theory
3 Homotopy Type Theory

PMATH 922 Topics in Universal Algebra (0.50) LECCourse ID: 002366
Department Consent Required
1 Algebra Cnstrnt Satsfctn Probs
2 Intro to Universal Algebra

PMATH 933 Topics in Group Theory (0.50) LECCourse ID: 002377
Department Consent Required
1 Intro:Combinatorial Grp Theory
2 Represent. of Compact Groups
3 Representations of Compact Lie

PMATH 944 Topics in Number Theory (0.50) LECCourse ID: 002383
Department Consent Required
1 Modular Forms
2 Diophatine Equations
3 P-Adic Analysis, Trees, Sieves
4 Goldbach Conj & Waring's Prob
5 Introduction to Circle Method
6 Computational Number Theory
7 Diophantine Inequalities
8 Diophantine Approximation
9 Number thry & randm matrx thry
10 P-adic analysis
11 Geometry of numbers
12 Riemann Surfaces

PMATH 955 Topics in Geometry (0.50) LECCourse ID: 002388
Department Consent Required
1 Model Theory of Fields
2 Riemann Surfaces
3 Gauge Theory
4 Complex and Kahler Manifolds
5 Generalized Complex Geometry
6 Holomorphic Vector Bundles
7 Advanced Algebraic Geometry
8 Atiyah-Singer Index Theorem

PMATH 966 Topics in Topology (0.50) LECCourse ID: 010485
Department Consent Required
1 Introduction to Knot Theory
2 Smooth 4-manifolds