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Complex analysis is a cornerstone of mathematics, making it an
essential element of any area of study in graduate mathematics.
Schlag's treatment of the subject emphasizes the intuitive
geometric underpinnings of elementary complex analysis that
naturally lead to the theory of Riemann surfaces. The book begins
with an exposition of the basic theory of holomorphic functions of
one complex variable. The first two chapters constitute a fairly
rapid, but comprehensive course in complex analysis. The third
chapter is devoted to the study of harmonic functions on the disk
and the half-plane, with an emphasis on the Dirichlet problem.
Starting with the fourth chapter, the theory of Riemann surfaces is
developed in some detail and with complete rigor. From the
beginning, the geometric aspects are emphasized and classical
topics such as elliptic functions and elliptic integrals are
presented as illustrations of the abstract theory. The special role
of compact Riemann surfaces is explained, and their connection with
algebraic equations is established. The book concludes with three
chapters devoted to three major results: the Hodge decomposition
theorem, the Riemann-Roch theorem, and the uniformization theorem.
These chapters present the core technical apparatus of Riemann
surface theory at this level. This text is intended as a fairly
detailed, yet fast-paced intermediate introduction to those parts
of the theory of one complex variable that seem most useful in
other areas of mathematics, including geometric group theory,
dynamics, algebraic geometry, number theory, and functional
analysis. More than seventy figures serve to illustrate concepts
and ideas, and the many problems at the end of each chapter give
the reader ample opportunity for practice and independent study.
This two-volume text in harmonic analysis introduces a wealth of
analytical results and techniques. It is largely self-contained and
useful to graduates and researchers in pure and applied analysis.
Numerous exercises and problems make the text suitable for
self-study and the classroom alike. The first volume starts with
classical one-dimensional topics: Fourier series; harmonic
functions; Hilbert transform. Then the higher-dimensional
Calderon-Zygmund and Littlewood-Paley theories are developed.
Probabilistic methods and their applications are discussed, as are
applications of harmonic analysis to partial differential
equations. The volume concludes with an introduction to the Weyl
calculus. The second volume goes beyond the classical to the highly
contemporary and focuses on multilinear aspects of harmonic
analysis: the bilinear Hilbert transform; Coifman-Meyer theory;
Carleson's resolution of the Lusin conjecture; Calderon's
commutators and the Cauchy integral on Lipschitz curves. The
material in this volume has not previously appeared together in
book form.
This two-volume text in harmonic analysis introduces a wealth of
analytical results and techniques. It is largely self-contained and
will be useful to graduate students and researchers in both pure
and applied analysis. Numerous exercises and problems make the text
suitable for self-study and the classroom alike. This first volume
starts with classical one-dimensional topics: Fourier series;
harmonic functions; Hilbert transform. Then the higher-dimensional
Calderon Zygmund and Littlewood Paley theories are developed.
Probabilistic methods and their applications are discussed, as are
applications of harmonic analysis to partial differential
equations. The volume concludes with an introduction to the Weyl
calculus. The second volume goes beyond the classical to the highly
contemporary and focuses on multilinear aspects of harmonic
analysis: the bilinear Hilbert transform; Coifman Meyer theory;
Carleson's resolution of the Lusin conjecture; Calderon's
commutators and the Cauchy integral on Lipschitz curves. The
material in this volume has not previously appeared together in
book form.
This book contains five theses in analysis, by A C Gilbert, N
Saito, W Schlag, T Tao and C M Thiele. It covers a broad spectrum
of modern harmonic analysis, from Littlewood-Paley theory
(wavelets) to subtle interactions of geometry and Fourier
oscillations. The common theme of the theses involves intricate
local Fourier (or multiscale) decompositions of functions and
operators to account for cumulative properties involving size or
structure.
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