Demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. This book presents an introduction to the subject as a whole and an overview of those branches of harmonic analysis that are relevant to the Kakeya conjecture.

Since the 1980s, the study of superprocesses has expanded into a major industry and can now be regarded as a central theme in modern probability theory. This book is intended as a rapid introduction to the subject, geared toward graduate students and researchers in stochastic analysis. A variety of different approaches to the superprocesses have emerged, yet no one approach has superseded any others. In this book, readers are exposed to a number of different ways of thinking about the processes, and each is used to motivate some key results. The emphasis is on why results are true rather than...

This title gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor. The idea that these topics are related first appeared in the physics literature in the study of quantum field theory....

This book contains the latest developments in a central theme of research on analysis of one complex variable. The material is based on lectures at the University of Michigan. The exposition is about understanding the geometry of interpolating and sampling sequences in classical spaces of analytic functions. The subject can be viewed as arising from three classical topics: Nevanlinna-Pick interpolation, Carleson's interpolation theorem for $Hinfty$, and the sampling theorem, also known as the Whittaker-Kotelnikov-Shannon theorem. The author clarifies how certain basic properties of the space...

Covers developments in the theory of Teichmuller spaces and offers references to the literature on Teichmuller spaces and quasiconformal mappings. This work describes how quasiconformal mappings have revitalized the subject of complex dynamics. It illustra

Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems.

An expanded edition of Kac's original introduction to algebraic aspects of conformal field theory which was published by the AMS in 1996. This revised edition is based on courses given by the author at MIT and at Rome University in Spring 1997. New material is added, including the foundations of a rapidly growing area of algebraic conformal theory. Also, in some places, the exposition is significantly simplified.

Explores genericity of approximation in various categories and presents many applications, including spectral multiplicity and properties of the maximal spectral type. This book contains a treatment of various constructions of cohomological nature with an emphasis on obtaining asymptotic behavior from approximate pictures at different time scales.

Differential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solution of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the...

Coarse geometry is the study of spaces (particularly metric spaces) from a ''large scale'' point of view, so that two spaces that look the same from a great distance are actually equivalent. This point of view is effective because it is often true that the relevant geometric properties of metric spaces are determined by their coarse geometry. Two examples of important uses of coarse geometry are Gromov's beautiful notion of a hyperbolic group and Mostow's proof of his famous rigidity theorem. The first few chapters of the book provide a general perspective on coarse structures. Even when only...