Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

Let n ∈ℕ and let A and B be rings. An additive map h : A → B is called an n -Jordan homomorphism if h ( a n )=( h ( a )) n for all a ∈ A . Every Jordan homomorphism is an n -Jordan homomorphism, for all n ≥2, but the converse is false in general. In this paper we investigate the n -Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.

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n -JORDAN HOMOMORPHISMS

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

An Introduction to K -Theory for C *-Algebras: Preface

We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifoldO via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly’s work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds. CONTENTS

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Asymptotic expansion of the heat kernel for orbifolds

We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. Introduction. Let M be a compact oriented manifold and let be a transver- sally oriented foliation on M. A foliation is a Riemannian foliation if there exists a Riemannian metric on M with the property that the distance from one leaf of to another is locally constant; such a metric is called a bundle-like metric for . Associated to are the space of basic forms: Ω B(M )= Ω B ( M , )= Ω ( M ): i(X) =0 , i ( X ) d = 0 for all X Γ(T ) , where i(X) is the interior product with the vector field X and Γ(T ) denotes the sections of the distribution T associated to . The exterior derivative d maps basic forms to basic forms; let dB denote d restricted to Ω B (M). The basic Laplacian is the operator ΔB = dB B + BdB on basic forms, where B is the adjoint of dB on Ω B (M). The analytic and geometric properties of this operator have been studied by several researchers. In (5), the basic Laplacian was studied as an operator on basic functions (i.e., functions that are constant on leaves of ), and the author proved the existence of the heat kernel in this case. In (13), the existence of the heat kernel on basic forms was proved for the case where the mean curvature form of the foliation is basic. There are also "basic" Hodge theorems, for example (6) and (10). However, the proof of the Hodge theorem in (6) does not yield various estimates that are important in applications, while the theorem proved in (10) has the same restriction as the results in (13), namely that the authors require the mean curvature form to be basic. In this paper, we study the basic Laplacian on forms, without any restriction on the mean curvature. We prove the existence and uniqueness of the heat kernel for ΔB on forms for any Riemannian foliation, and we write down an explicit formula for the heat kernel. We also present a proof of the Hodge theorem for

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The basic Laplacian of a Riemannian foliation

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.

Smooth distributions are finitely generated

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1. Preliminaries 2. K-Theory 3. Additional structure 4. Characteristic classes Bibliography Symbol index Subject index.

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Complex topological K-theory

We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method.

http://faculty.tcu.edu/epark/papers/Hopf_DGA.pdf

Efton Park

Papers

The basic Laplacian of a Riemannian foliation

Smooth distributions are finitely generated

Smooth distributions are finitely generated

Complex topological K-theory

A Hopf index theorem for foliations