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.
Antireq: PMATH 432

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.
Antireq: PMATH 441

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.
Antireq: PMATH 446

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.
Antireq: PMATH 450

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.
Antireq: PMATH 451

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.
Prereq: PMATH 352 or consent of department.

PMATH 665 Geometry of Manifolds (0.50) LECCourse ID: 002349
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.
Antireq: PMATH 465

PMATH 667 Algebraic Topology (0.50) LECCourse ID: 002350
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.
Antireq: PMATH 467

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
10 Rational Homotopy Theory
11 Intro to Univer Algebra

PMATH 700s


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
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.
Antireq: PMATH 440

PMATH 745 Representations of Finite Groups (0.50) LECCourse ID: 014343
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.
Antireq: PMATH 445

PMATH 753 Functional Analysis (0.50) LECCourse ID: 013670
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.
Antireq: PMATH 453

PMATH 763 Introduction to Lie Groups and Lie Algebras (0.50) LECCourse ID: 002391
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 (0.50) LECCourse ID: 002392
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.
Antireq: PMATH 464

PMATH 800s


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.
Prereq: PMATH 753

PMATH 900s


PMATH 930 Topics in Logic (0.50) LECCourse ID: 014769
1 Computability Theory
2 Computable Structure Theory
3 Model Theory

PMATH 940 Topics in Number Theory (0.50) LECCourse ID: 014770
1 Comp Algebraic Number Theory
2 Heights and arithmetic
3 Random matrix theory
4 P-adic Analysis

PMATH 945 Topics in Algebra (0.50) LECCourse ID: 014771
1 Arithmetic Dynamics
2 Ctgry thry & homological algbr
3 Ring theory

PMATH 950 Topics in Analysis (0.50) LECCourse ID: 014772
1 Operator Algebras
2 Reps of Locally Compact Groups
3 Intro to Stochastic Integratio
4 Harmonic anlys & repres thry
5 Ergodic theory on discrete grp

PMATH 965 Topics in Gemetry and Topology (0.50) LECCourse ID: 014773
1 Kahler-Einstein Metrics
2 Tools of gauge theory

PMATH 990 Topics in Pure Mathematics (0.50) LECCourse ID: 014774
1 Matrix Analysis
2 Entanglement and nonlocality