Graduate Course Descriptions
Spring 2009

Theory of Functions of a Real Variable II

This course is a continuation of 640:501 from Fall 2008. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know.

The course will cover material from Chapters 4-8 of Folland's book:

Topological Spaces Basic properties, compact spaces, Stone-Weierstrass theorem Introduction to Functional Analysis, Normed vector spaces, Hahn-Banach theorem, bounded linear transformations, Closed graph and Open mapping theorem, applications of Baire category theorem, Hilbert spaces, topological vector spaces, weak and weak* convergence Lp Spaces, Integral inequalities, duality, bounded integral operators Introduction to Fourier analysis, Schwartz space, convolutions, Fourier transform and Fourier series, Plancherel theorem, Poisson summation formula, Lp and pointwise convergence of Fourier series, Integration on Locally Compact Spaces Continuous functions and Radon measures on locally compact spaces, dual of C(X), vague convergence of measures

Theory of Functions of a Complex Variable II

This will be a continuation of Math 503. We will emphasis on the relationship between classical complex analysis and other related fields (algebraic geometry, geometry, and analysis) through Riemann surfaces.

The theory of Riemann surface is a pillar in 20th century mathematics. It appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory. We would like to concentrate on the interaction between the complex analytic, classical geometric, and algebraic geometry points of view.

The following two parts will be covered: (1) Classical Complex Analysis and (2) Riemann surfaces.

Part 1. Analytic continuation, the monodromy theorem, normal families and Riemann mapping theorem, Picard theorems, harmonic functions and elliptic functions.

Part 2. Introduction to Riemann surfaces and algebraic curves. Hyperbolic geometry and uniformization theorem. Riemann-Roch theorem, Abel and Jacobi theorems.

The reference for the first part:

[1] Green and Krantz: Function Theory of One Complex Variable, AMS.

The reference for the second part:

[3] Farkas, H & Kra, I.: Riemann Surfaces (2nd ed.), Springer-Verlag

[4] Forster, O.: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991

[5] Narasimhan, R.: Compact Riemann surfaces. Birkh

Functional Analysis I

Selected Topics in Analysis

During the first one quarter of the course, I will use simple settiing to illustrate a few basic, and widely applicable, methods used in the study of nonlinear equations and systems: min-max methods in the calculus of variations (mountain pass lemma, Ljusternik-Schnirelman theorem); applications of the implicit function theorem (gluing of approximate solutions into genuine solutions, some bifurcation theorems); appications of the maximum principle (existence of solutions by super and sub solutions methods, a theorem of A.D. Alexandrov and the method of moving planes); A theorem of Morrey and ``small energy implies regularity''.

During the remaining three quarters of the course, I will present some results in nonlinear elliptic partial differential equations, with only outlines of the proof. Some open problems will be presented. The emphasis will not be on details of the proofs. We plan to present the following topics: Monge-Ampere equations (Bernstein type arguments and Pogorelov's estimate, C(1,alpha) and W(2,p) theory of Caffarelli); The positive mass theorem and the Penrose inequality; Chern's conjecture on affine Bernstein problem (solution by Trudinger and Wang); viscosity solutions (existence via Perron's method, Jensen approximations, and an extension to a classical Liouville theorem of the instructor as well some more recent joint work of the instructor with Caffarelli and Nirenberg). Conformally invariant fully nonlinear elliptic equations (some research activities of recent seven years or so and some ongoing ones, and some open problems).

Partial Differential Equations II

This will be a self-contained course on hyperbolic partial differential equations. We'll cover the basic theory of hyperbolic PDE's, namely, various definitions of hyperbolicity, the geometry of characteristics, energy estimates, the domain of dependence theorem, local well-posedness, and formation of singualrities. Emphasis will be on particular equations that arise in physics, i.e. Maxwell's equations of electromagnetics, Euler's equations of fluid dynamics, and Einstein's equations of General Relativity. The necessary functional-analytics tools will be developed along the way.

Harmonic Analysis on Euclidean Spaces

This course will develop basic Harmonic Analysis on Euclidean Spaces. There is no textbook but course notes from an earlier time I taught this course. Some of the topics we will cover are:

1. Interpolation theory of operators, The Riesz-Thorin and Marcinkiewicz interpolation theorem and Stein's theorem on complex interpolation.

2. Singular integral theory. Cotlar's lemma.

3. Hardy-Littlewood-Sobolev fractional integration theorem.

4. Restriction theorem of the Fourier transform.

5. Bochner-Riesz operators, Strichartz estimates for wave and Schrodinger equations.

6. The multiplier theorem of the ball of C. Fefferman.

7. BMO functions and the John-Nirenberg inequality.

8. C. Fefferman's theorem on the duality between Hardy spaces and BMO.

Algebraic Geometry II

This will be an introduction to the scheme-theoretic side of Algebraic Geometry. The first part of the course will study schemes and sheaves, especially line bundles and sheaves of differentials. This is taken from chapter 2 of Hartshorne.

The second part will be sheaf cohomology, with an emphasis on computing, including Duality and higher direct image maps. This is taken from chapter 2 of Hartshorne. In the remaining time, we will cover topics requested by the class.

Selected Topics in Geometry

Abstract Algebra II

Note: These volumes are out of print. Students may be able to obtain used copies online (be sure it is the second edition) through addall.com or other websites. In the fall, photocopies will be available for purchase.

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. Representative topics will be:

• Galois Theory
Finite algebraic extensions, resolutions of equations by radicals (and without radicals)
• Noetherian Rings
Rings of polynomials, Hilbert basis theorem, Dedekind domains, Finitely generated algebras over fields, Noether normalization, Nullstellensatz
• Basic Module Theory
Projective and injective modules, resolutions, baby homo- logical algebra, Hilbert syzygy theorem

Selected Topics in Algebra

Representation theory of vertex operator algebras is equivalent to two-dimensional conformal field theory in physics in the sense that any result or conjecture in two-dimensional conformal field theory can be reformulated precisely as a result or conjecture in the representation theory of vertex operator algebras.

In this course I will present this representation theory. The topics covered will include: Weak modules, generalized modules, N-gradable weak modules and modules for a vertex operator algebra, Zhu's algebra for a vertex operator algebra, the correspondence between modules for Zhu's algebra and N-gradable weak modules for the vertex operator algebra, reductivity of N-gradable weak modules, cofiniteness conditions, intertwining operators, differential equations of regular singular points, tensor products, modular invariance, Verlinde conjecture and Verlinde formula.

Theory of Algebras

Let g be a Kac-Moody Lie algebra of finite, affine, or hyperbolic type, over K, a field, and let G be a Kac-Moody group associated to g. If g is of finite type, then g is a finite dimensional semisimple Lie algebra, and G is a semisimple Lie group. Almost all of these occur in `space-time symmetries' and the development of the Standard Model of particle physics.

The class of affine Kac-Moody algebras have wide applications in physical theories such as elementary particle theory, quantum field theory, gauge theory, conformal field theory, gravity and string theory. Affine Kac-Moody algebras (and their generalizations by Borcherds) give rise to a rich mathematical theory, are relevant to number theory and modular forms, and they occur in the relation between the sporadic simple Monster group and symmetries of codes, lattices and conformal fields theories.

Hyperbolic Kac-Moody theory naturally generalizes the theory of finite dimensional and affine Kac-Moody Lie algebras, though many fundamental questions regarding the structure of hyperbolic groups and algebras remain open. Recently hyperbolic Kac-Moody groups and algebras have been discovered as symmetries in high-energy physics, and they have been shown to serve as duality symmetries of a theory, known as M-theory, which unifies all superstring theories. In particular hyperbolic Kac-Moody groups have been discovered to play a role in the dimensional reduction of supergravity, a theory incorporating general relativity and supersymmetry, and they describe the symmetries of a cosmological phenomenon known as "billiards".

In this course we study the mathematics suggested by the development of M-theory and its symmetries, focussing on the occurence of hyperbolic Kac-Moody groups and algebras and their properties.

Axiomatic Set Theory

This is an introductory course on proving independence results in set theory. Here a statement S is said to be independent of set theory iff S can neither be proved nor disproved from the classical ZFC axioms of set theory. For example, it is well-known that the Continuum Hypothesis CH is independent of set theory. Initially we shall follow the lazy man's approach to obtaining independence results; namely, we shall study the consequences of the following two extra set-theoretic axioms.

• The Diamond Axiom(♢): a combinatorial principle which says intuitively that there exists a fortune-teller who correctly predicts the future often enough to be useful.
• Martin's Axiom(MA +