The international symposium on number theory and analysis in memory of the late famous Chinese mathematician Prof. Hua Loo Keng was co-sponsored by the Institute of Mathematics, Academia Sinica and the University of Science and Technology of China. lt took place between August Ist and 7th of 1988 on the campus of Tsing Hua University, and some 150 mathematicians were pres- ent. The symposium was carried out in two separate sections: number theory and analysis. This is retlected in the publication ofa set oftwo volumes, the first one on Number Theory edited by Professor Wang Yuan and the second on Analysis by Professors Gong Sheng, Lu Qi-keng and Yang Lo. The distinguished list of main speakers and the contents of these two vol- umes reflect the high level of the mathematical activity throughout the seven days. W e pay special tribute to our main speakers professors Chuang, Conn, Ding, Drasin, Fitzgerald, Gaier, Gong, Grauert, Gu, Hejhal, Iyanaga, Karatsuba, Koranyi, Liao, Lu, Pan, Richert, Satake, Schmidt, Siu, Tatuzawa, Tsang, Vladimirov, Y. Wang, G. Y. Wang, Wustholz and Yang, who gave the excellent one hour lectures, and also to the participants who gave contributed talks on their own research work. The discussions among the mathematicians were always in a warm atmosphere. Our thanks go to professors Chern, Subbarao and Yau for their contributions to these proceedings.

This interesting book deals with the theory of convex and starlike biholomorphic mappings in several complex variables. The underly- ing theme is the extension to several complex variables of geometric aspects of the classical theory of univalent functions. Because the author's introduction provides an excellent overview of the content of the book, I will not duplicate the effort here. Rather, I will place the book into historical context. The theory of univalent functions long has been an important part of the study of holomorphic functions of one complex variable. The roots of the subject go back to the famous Riemann Mapping Theorem which asserts that a simply connected region n which is a proper subset of the complex plane C is biholomorphically equivalent to the open unit disk ~. That is, there is a univalent function (holo- morphic bijection) I : ~ -+ n. In the early part of this century work began to focus on the class S of normalized (f (0) = 0 and I' (0) = 1) univalent functions defined on the unit disk. The restriction to uni- valent functions defined on the unit disk is justified by the Riemann Mapping Theorem. The subject contains many beautiful results that were obtained by fundamental techniques developed by many mathe- maticians, including Koebe, Bieberbach, Loewner, Goluzin, Grunsky, and Schiffer. The best-known aspect of univalent function theory is the so-called Bieberbach conjecture which was proved by de Branges in 1984.

H.Weyl studied harmonic analysis on compact groups of finite di mension. He proved that an orthonormal system exists and that any continuous function on these groups can be approximated by some tinite linear combination of functions in this system. His research, however, seems to be too abstract to yield an explicit expression for the orthonormal system. Thus, we cannot talk about the form of the approximation, nor about its convergence. iO The simplest example of compact groups is {e }, on which there exists an orthonormal system inO { e }, n = 0, +/- 1, +/- 2 , ... , namely 1 J2" ." ." {I, for n = m; - e,n"e-1m"dO = 2n 0 0, for n =;6 m. The harmonic analysis on this compact group refers to the whole Fourier analysis. So far, extensive literature has been available on this topic. Its remarkable progress is evidenced by the great monograph of seven-hundred pages in two volumes written by A. Zygmund in 1959. iO An immediate extension for {e } is group U", which consists of all n X n square matrices U satisfying ufj' = I, where fj' denotes the conjugate transpose matrix of U. As for construction, there is a close relation between the group U and the group S03. Besides, 2 the application of U" has been found more and more important in physics.

Harmonic Analysis in China is a collection of surveys and research papers written by distinguished Chinese mathematicians from within the People's Republic of China and expatriates. The book covers topics in analytic function spaces of several complex variables, integral transforms, harmonic analysis on classical Lie groups and manifolds, LP- estimates of the Cauchy-Riemann equations and wavelet transforms. The reader will also be able to trace the great influence of the late Professor Loo-keng Hua's ideas and methods on research into harmonic analysis on classical domains and the theory of functions of several complex variables. Western scientists will thus become acquainted with the unique features and future trends of harmonic analysis in China. Audience: Analysts, as well as engineers and physicists who use harmonic analysis.

This interesting book deals with the theory of convex and starlike biholomorphic mappings in several complex variables. The underly- ing theme is the extension to several complex variables of geometric aspects of the classical theory of univalent functions. Because the author's introduction provides an excellent overview of the content of the book, I will not duplicate the effort here. Rather, I will place the book into historical context. The theory of univalent functions long has been an important part of the study of holomorphic functions of one complex variable. The roots of the subject go back to the famous Riemann Mapping Theorem which asserts that a simply connected region n which is a proper subset of the complex plane C is biholomorphically equivalent to the open unit disk ~. That is, there is a univalent function (holo- morphic bijection) I : ~ -+ n. In the early part of this century work began to focus on the class S of normalized (f (0) = 0 and I' (0) = 1) univalent functions defined on the unit disk. The restriction to uni- valent functions defined on the unit disk is justified by the Riemann Mapping Theorem. The subject contains many beautiful results that were obtained by fundamental techniques developed by many mathe- maticians, including Koebe, Bieberbach, Loewner, Goluzin, Grunsky, and Schiffer. The best-known aspect of univalent function theory is the so-called Bieberbach conjecture which was proved by de Branges in 1984.

On the basis of Hua Loo-Kengs results on harmonic analysis on classical groups, the author Gong Sheng develops his subject further, drawing togetherresults of his own research as well as works from other Chinese mathematicians. The book is divided into three parts studying harmonic analysis of various groups. Starting with the discussion on unitary groups in part one, the author moves on to rotation groups and unitary symplectic groups in parts 2 and 3. Thus the book provides a survey of harmonic analysis on characteristic manifold of classical domain of first type for real fields, complex fields and quaternion fields. This study will appeal to a wide range of readers from senior mathematics students up to graduate students and to teachers in this field of mathematics.

A concise textbook on complex analysis for undergraduate and graduate students, this book is written from the viewpoint of modern mathematics: the Bar {Partial}-equation, differential geometry, Lie groups, all the traditional material on complex analysis is included. Setting it apart from others, the book makes many statements and proofs of classical theorems in complex analysis simpler, shorter and more elegant: for example, the Mittag-Leffer theorem is proved using the Bar {Partial}-equation, and the Picard theorem is proved using the methods of differential geometry.

Mathematics is the fundamental knowledge for every scientist. As an academic at the University of Science and Technology of China, Professor Sheng Gong takes his passion for mathematics teaching even further. Besides imparting knowledge to students from the Department of Mathematics, he has created and developed his method of teaching Calculus to help students from physics, engineering and other sciences disciplines understand Calculus faster and deeper in order to meet the needs of applications in their own fields.

This book is based on Professor Sheng Gong's 42 years of teaching experience along with a touch of applications of Calculus in other fields such as computer science, engineering. Science students will benefit from the unique way of illustrating theorems in Calculus and also perceive Calculus as a whole instead of a combination of separate topics. The practical examples provided in the book bring motivation to students to learn Calculus.

Mathematics is the fundamental knowledge for every scientist. As an academic at the University of Science and Technology of China, Professor Sheng Gong takes his passion for mathematics teaching even further. Besides imparting knowledge to students from the Department of Mathematics, he has created and developed his method of teaching Calculus to help students from physics, engineering and other sciences disciplines understand Calculus faster and deeper in order to meet the needs of applications in their own fields.

This book is based on Professor Sheng Gong's 42 years of teaching experience along with a touch of applications of Calculus in other fields such as computer science, engineering. Science students will benefit from the unique way of illustrating theorems in Calculus and also perceive Calculus as a whole instead of a combination of separate topics. The practical examples provided in the book bring motivation to students to learn Calculus.