Five recent book reviews
The book consists of eleven chapters which cover basic concepts, methods and techniques of Complex Analysis through theory and problem-solving. Emphasis is given to both examples and exercises which enlighten the theory.
The book provides a brief presentation of theory of the fundamentals of complex valued functions of a complex variable, together with several worked in detail examples and exercises. Emphasis is given to the understanding of properties of complex numbers, convergence of sequences and series, general properties of complex functions, conformal mappings, computational techniques of integrals, the power series representation of analytic functions, isolated singularities, Laurent series, residues, analytic continuation and integral transforms.
The book will be particularly useful to undergraduate students and beginning graduate students of both pure and applied Mathematics departments and engineering schools. The author has succeeded to provide a large collection of examples and problems with step-by-step solutions, computations and proofs.
Overall, the book is recommended for class use, as well as a supplement of standard textbooks.
This is a soft cover edition of the autobiography of Benoit Mandelbrot. He exposes himself as a rebel against established mathematics, which turned him into a fiercely independent scientist, but it also brought him to develop the theory of fractals that properly model many natural phenomena. The recognition of the richness of the Mandelbrot set has eventually given him, relatively late in life, the recognition he has been looking for since he was a child.
This is the autobiography of Benoit Mandelbrot (1929-2010) who died shortly before the manuscript went to the publisher. So there was some editing by friends and colleagues before it was finally published with Pantheon Books in 2012. This edition is a soft cover reprint of the same book.
Benoit Mandelbrot was born in Warsaw in a Lithuanian Jewish family. His uncle Szolem Mandelbrojt, an influential mathematician, student of Hadamard and member of Bourbaki, helped them escape to Paris in 1936. The family was scattered during WW II in different parts of France, but they all survived. Benoit passed entrance exams for the prestigious École Normale Supérieure but left after two days to choose for the military regime of the École Polytechnique instead, much to the dislike of his uncle. This illustrates the rebellious character of Benoit, always avoiding the well established, predictable and paved road. Among his teachers at the Polytechnique were Paul Lévy and Gaston Julia.
Already as a child, Benoit was fascinated by Kepler because he could resolve anomalies in the solar system by attributing elliptical trajectories to the planets. Benoit always wanted to realize a similar achievement in his life, which, in his conviction, this was only possible by leaving the predefined paths of science. Order in chaos can not be obtained by adding modifications and exceptions to existing models. In the period 1947-49 he studied at Caltech where he witnessed the birth of molecular biology which later has culminated in the discovery of the DNA double helix. This enforced his belief in his Keplerian dream.
After an hilarious intermezzo of a year he had to spent in the French Air Force for bureaucratic reasons, Benoit registered at the Université de Paris. He presented a hastily written thesis in which he elaborated what is now called the Zipf-Mandelbrot law of word frequencies. This is a distribution of sharp decrease but with a long tail. After that he left again for the US to spend postdoc positions at MIT (with Norbert Wiener), Princeton (with John von Neumann) and returned to Paris in 1954 to rejoin the CNRS. He met his wife in that period. They married and spent a two year honeymoon in Genève where he worked at the university. Back in France he got a teaching position in Lille. However, restless as he was, instead of choosing for a tenure position in France, he chose in 1958 to work for IBM in Yorktown Heights.
This move to IBM starts the third part of this autobiography where he develops into the fractalist that we know. First by modelling stock prices, later he promoted fractal behaviour of the Nile Basin and the fractal distribution of galaxies. None of his work was well received by the economic and scientific establishment, not being embedded in pure traditional mathematics or statistics. When in 1979-80 the Mandelbrot set occurred in his lectures and after his first book Les object fractales (1975) but even more so after publishing in 1982 his book The fractal geometry of nature, his Kepler dream finally came true.
This story is written down with the skills of a great story-teller. Mandelbrot exposes himself as a scientific loner, rebel without a cause, and outlier par excellence, There was an obvious mismatch between him and the academia. Either he himself or the institute had to decline a university position at several occasions and he got tenure at Yale only at the very end of his career. He often questions himself whether he has taken the right decisions, but he never regrets, although he thinks that some good advise might have given a different turn to his life. So he never had a formal teacher, nor had he formal students. His low opinion of exam ratings, of persons, events or institutions, are not hidden, and his own achievements are told without the inhibition of false modesty. A very well documented story of a scientific maverick indeed featuring many of the `big shots' he has met, from J.R. Oppenheimer to Victoria de los Angeles to Valéry Giscard d'Estaing. Whether this is an example to be followed is left to the reader, but Mandelbrot seems to be convinced that he has made the proper choices all along the way.
This book is a rigorous introduction to ergodic theory and dynamical systems, paying special attention to the understanding of certains applications of ergodic theory to problems in number theory. Among other applications we can find Weyl's theorem on uniform distribution of polynomials, the ergodic proof of Szemeredi's theorem on arithmetic progressions or the proof of the equidistribution of horocycle orbits.
The book is an introduction to ergodic theory and dynamical systems. Some topics have been selected with the applications to number theory in mind, but contains some others to aid motivation and to give a complete picture of ergodic theory.
The book is organized into 11 chapters. The first chapter contains some motivating examples and results from ergodic theory. Chapter 2 introduces measure-preserving transformations and some basic facts on the notions of ergodicity, weak-mixing and strong-mixing. Continued fractions decompositions of real numbers, continued fraction map and Gauus measure are introduced in chapter 3, as well as some classical results on Diophantine approximations.
Chapter 4 has to do with the existence of invariant measures for continuous maps on compact metric spaces, ergodic decomposition and unique ergodicity. The proof of Weyl’s theorem on uniform distribution of polynomials is included in this chapter.
Chapter 5 provides more advanced background in measure theory which is used in chapter 6, where factors and joinings of measure-preserving systems are discussed and a proof of the ergodic decomposition result is given.
The ergodic proof of Szemeredi's theorem on arithmetic progressions, due to Furstenberg, is included in chapter 7. This is the first example of how the ergodic methods lead to results in combinatorics.
The ergodic theory for group actions is studied in chapter 8. The analysis of actions on locally homogeneous spaces is introduced in chapter 9 by studying the geodesic flow on hyperbolic surfaces. Chapter 10 is devoted to rotations on quotients of nilpotents groups and to the study of the continuous Heisenberg group.
The last chapter deals with the the ergodicity of the horocycle flow and its properties. lattices in SL(2,R), the Mautner phenomenon, the Howe-Moore theorem, rigidity of invariant measures for the horocyclic flow and the equidistribution of horocyclic orbits.
The book is intended for graduate students and researchers with some background in measure theory and functional analysis. Definitely, it is a book of great interest for researchers in ergodic theory, homogeneous dynamics or number theory.
"Circles disturbed" is a collection of sixteen independent essays/chapters which takes us along the frontier between the realms of mathematics and narrative.
The book offers a proof of a complete Gross-Zagier formula
on quaternionic Shimura curves over totally real fields.