Oral Qualifying Exam - Ben Kennedy 
	     December 1, 2003

TOPIC ONE: FUNCTIONAL ANALYSIS

	TOPOLOGICAL VECTOR SPACES

Metrizability and normability; finite-dimensional spaces (local
compactness); local convexity (sublinear functionals, seminorms); dual
spaces (the Hahn-Banach theorems, weak and weak-* topologies, Alaoglu's
theorem, weak closure and weak boundedness); the Banach-Steinhaus theorem
(principle of uniform boundedness).

	BANACH SPACES

Bounded linear maps (Hahn-Banach theorems, open mapping theorem, closed
graph theorem), Hilbert spaces (Riesz lemma, Hilbert space bases); adjoint
maps (definition and basic properties, basic properties of normal, unitary
and projection operators); complexification of real Banach spaces.

	CALCULUS

The derivative (inverse and implicit function thoerems); vector-valued
integration, holomorphism of Banach-valued functions.

	SPECTRAL THEORY

Spectra in Banach algebras (nonemptiness, the spectral radius formula, the
functional calculus, the spectral mapping theorem); different types of
spectral points in algebras of linear operators; the spectral theorem for
compact operators; normal operators in Hilbert space (spectral measures
and the spectral theorem).

	DISTRIBUTIONS

The topology of $\mathcal{D}(\Omega)$ and $\mathcal{S}$; definition of
distributions and tempered distributions, definition of convolution and
Fourier transform, an application (Malgrange's theorem).

TOPIC TWO: RIEMANNIAN GEOMETRY

	DIFFERENTIAL GEOMETRIC BACKGROUND

Manifolds and their vector bundles; integration on manifolds.

	CONNECTIONS AND COVARIANT DIFFERENTIATION

Motivation in Euclidean space; definitions.

	BASIC METRIC STRUCTURE

Geodesics; the exponential map; existence of normal neighborhoods; the
first and second variation formulas, convex neighborhoods; the Hopf-Rinow
theorem.

	CURVATURE

Definition and interpretation; Gaussian curvature; the second fundamental
form; Gauss' theorem on sectional curvature of immersed manifolds; totally
geodesic immersions; why Gaussian curvature in two dimensions is
especially important.

	GLOBAL EFFECTS OF CURVATURE

Hadamard-Cartan and Bonnet-Myers theorems; spaces of constant sectional
curvature.

Application to complex analysis (Little Picard Theorem).