PMATH 600s. Prerequisite: PMATH 345 or 346 or consent of department.

Transcription

1 PMATH 600s PMATH 632 First Order Logic and Computability (0.50) LEC Course ID: 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 PMATH 633 Model Theory and Set Theory (0.50) LEC Course ID: 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 w-stability. Set theory: well-orderings, ordinals, cardinals, Zermelo-Fraenkel axioms, axiom of choice, informal discussion of classes and indpendence results. PMATH 640 Analytic Number Theory (0.50) LEC Course ID: 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 Prerequisite: PMATH 332 or 352 or consent of PMATH 641 Algebraic Number Theory (0.50) LEC Course ID: 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 Prerequisite: PMATH 345 or consent of PMATH 642 Fields and Galois Theory (0.50) LEC Course ID: 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 Prerequisites: PMATH 345 and 346 or consent of PMATH 644 Rings, Modules and Representations (0.50) LEC Course ID: 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 Prerequisites: PMATH 345 and 346 or consent of PMATH 651 Measure and Integration (0.50) LEC Course ID: 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

2 spaces. Students without the required prerequisite may seek consent of the Prerequisite: PMATH 354 or consent of PMATH 652 Topics in Complex Analysis (0.50) LEC Course ID: 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 Prerequisite: PMATH 352 or consent of PMATH 653 Functional Analysis (0.50) LEC Course ID: 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 or consent of PMATH 664 Algebraic Curves (0.50) LEC Course ID: 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 665 Differential Geometry (0.50) LEC Course ID: 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 Prerequisites: PMATH 365 or AM 333 or consent of Department PMATH 667 Topology (0.50) LEC Course ID: Topics from algebraic, combinatorial and geometric topology. Students without the required prerequisite may seek consent of the Prerequisite: PMATH 351 or consent of Department PMATH 690 Literature and Research Studies (0.50) RDG Course ID: Reading Course 1 Lie Algebras 2 Topology 3 Representation Theory 4 C*-algebras and groups 5 Integration and Probability 6 Kaehler geometry

3 PMATH 700s PMATH 700 Topics in Algebra (0.50) LEC Course ID: Commutative Rings 2 Complex Semisimple Lie Algebra PMATH 701 Graduate Algebra (0.50) LEC Course ID: 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 711 Topics in Mathematical Logic (0.50) LEC Course ID: Computability Theory 2 Model Theory PMATH 722 Topics in Universal Algebra (0.50) LEC Course ID: PMATH 733 Topics in Group Theory (0.50) LEC Course ID: Intro:Combinatorial Grp Theory 2 Represent. of Compact Groups 3 Representations of Compact Lie PMATH 744 Topics in Number Theory (0.50) LEC Course ID: 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

4 PMATH 755 Topics in Geometry (0.50) LEC Course ID: Model Theory of Fields 2 Riemann surfaces PMATH 766 Topics in Topology (0.50) LEC Course ID: Introduction to Knot Theory PMATH 800s PMATH 800 Topics in Real and Complex Analysis (0.50) LEC Course ID: Fractal Geometry PMATH 801 Graduate Analysis (0.50) LEC Course ID: 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 810 Banach Algebras and Operator Theory (0.50) LEC Course ID: 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) LEC Course ID: Random Matrices & Asymptotics 2 Intro to Banach&Operator Alg 3 von Neuman II-1 Factors PMATH 822 Topics in Operator Theory (0.50) LEC Course ID: Approximate Operator Theory 2 von Neumann Algebras 3 Simultaneous Triangularization 4 Operator Spaces 5 Von Neumann algebras & noncomm

5 6 Nonself-adj. Operator Algebras 7 Intro. to Operator Algebras 8 Introduction to K-theory PMATH 833 Topics in Harmonic Analysis (0.50) LEC Course ID: Harmonic Anlys on the Circle 2 Harmonic Analysis 3 Semisimple Lie Groups SL(2,R) PMATH 844 Topics in Functional Equations (0.50) LEC Course ID: