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In this book on smooth and non-smooth harmonic analysis, the notion of dual variables is adapted to fractals. In addition to harmonic analysis via Fourier duality, the author also covers multiresolution wavelet approaches as well as a third tool, namely, $L^2$ spaces derived from appropriate Gaussian processes.

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The emphasis is on the global methods and the use of Fourier integral operator methods to analyse norms and nodal sets of eigenfunctions.

Provides introductory material to give the reader an accessible entry point to this vast subject matter. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.

Describes a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations. The author shows how these equations are derived, and how to obtain the concentration of measure estimates required to study these equations asymptotically.

Zeta and $L$-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and $L$-functions as a central theme.

"Support from the National Science Foundation."

Examines some recent developments in the theory of $C^*$-algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries.

Based on CBMS lectures given at Texas Christian University, this book describes some of the interplay between string dualities and topology and operator algebras. It shows how several seemingly disparate subjects are closely linked with one another. It gives readers an overview of some areas of research.

Includes an analytic solution to the Busemann-Petty problem, which asks whether bodies with smaller areas of central hyperplane sections necessarily have smaller volume, characterizations of intersection bodies, extremal sections of certain classes of bodies, and a Fourier analytic solution to Shephard's problem on projections of convex bodies.

Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications.

These lectures recount an application of stable homotopy theory to a concrete problem in low energy physics: the classification of special phases of matter. While the joint work of the author and Michael Hopkins is a focal point, a general geometric frame of reference on quantum field theory is emphasized.

Based on 10 lectures given at the CBMS workshop on spectral graph theory in June 1994 at Fresno State University, this exposition can be likened to a conversation with a good teacher - one who not only gives you the facts, but tells you what is really going on, why it is worth doing, and how it is related to familiar ideas in other areas.

Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. Numerous open problems appropriate for graduate students and post-docs are included throughout.

Contains lectures from the CBMS Regional Conference held at Harvey Mudd College, June 1977. This monograph consists of applications to nonlinear differential equations of the author's coincidental degree. It includes an bibliography covering many aspects of the modern theory of nonlinear differential equations and the theory of nonlinear analysis.

The Uncertainty Principle in Harmonic Analysis (UP) is a classical, yet rapidly developing, area of modern mathematics. These notes are devoted to the so-called Toeplitz approach to UP which recently brought solutions to some of the long-standing problems posed by the classics.

Introduces a systematic approach to the construction and analysis of semi simple $p$-adic groups. This book presents an overview of the representation theory of GL$_n$ over finite groups.

Based on lectures presented by Popa at the NSF-CBMS Regional Conference held in Eugene, Oregon, in August 1993, this title offers a unified and self-contained presentation of the results presented in Popa's earlier papers.

Surveys some of the remarkable developments that have taken place in operator theory over the years. This monograph is largely expository and should be accessible to those who have had a course in functional analysis and operator theory.

Brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available.

The study of group actions on manifolds is the meeting ground of a variety of mathematical areas. In particular, interesting geometric insights can be obtained by applying measure-theoretic techniques. This book provides an introduction to some of the important methods, major developments, and open problems on the subject.

Offers an overview of a number of significant ideas and results developed in the geometrical study of differential equations. This book covers topics that include symmetries of differential equations and variational problems, the variational bi-complex and conservation laws, as well as geometric integrability for hyperbolic equations.

Introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. Suitable for graduate students in mathematics, this book describes the construction and computation of intersection products by means of the geometry of normal cones.

Reflects the themes of student learning and calculus. This volume includes overviews of calculus reform in France and in the US and large-scale and small-scale longitudinal comparisons of students enrolled in first-year reform courses and in traditional courses. It concludes with a study of a concept that overlaps the areas of focus, quantifiers.

Among nonlinear PDEs, dispersive and wave equations form an important class of equations, including the nonlinear Schrodinger equation, nonlinear wave equation, Korteweg de Vries equation, and the wave maps equation. This book offers an introduction to the methods and results used in the modern analysis of the Cauchy problem for such equations.