Tony Gardiner

Alexandre Borovik

The Essence of Mathematics Through Elementary Problems

The author's goal is to start a dialogue between mathematicians and cognitive scientists. He discusses, from a working mathematician's point of view, the mystery of mathematical intuition: why are certain mathematical concepts more intuitive than others? To what extent does the ``small scale'' structure of mathematical concepts and algorithms reflect the workings of the human brain? What are the ``elementary particles'' of mathematics that build up the mathematical universe? The book is saturated with amusing examples from a wide range of disciplines--from turbulence to error-correcting codes to logic--as well as with just puzzles and brainteasers. Despite the very serious subject matter, the author's approach is lighthearted and entertaining. This is an unusual and unusually fascinating book. Readers who never thought about mathematics after their school years will be amazed to discover how many habits of mind, ideas, and even material objects that are inherently mathematical serve as building blocks of our civilization and everyday life. A professional mathematician, reluctantly breaking the daily routine, or pondering on some resisting problem, will open this book and enjoy a sudden return to his or her young days when mathematics was fresh, exciting, and holding all promises. And do not take the word ``microscope'' in the title too literally: in fact, the author looks around, in time and space, focusing in turn on a tremendous variety of motives, from mathematical ``memes'' (genes of culture) to an unusual life of a Hollywood star. --Yuri I. Manin, Max-Planck Institute of Mathematics, Bonn, and Northwestern University

Notes on Cognitive Aspects of Mathematical Practice

Mathematics Under the Microscope

Alexei G. Myasnikov

Since its origin in the early 20th century, combinatorial group theory has been primarily concerned with algorithms for solving particular problems on groups given by generators and relations: word problems, conjugacy problems, isomorphism problems, etc. Recent years have seen the focus of algorithmic group theory shift from the decidability/undecidability type of result to the complexity of algorithms. Papers in this volume reflect that paradigm shift. Articles are based on the AMS/ASL Joint Special Session, Interactions Between Logic, Group Theory and Computer Science. The volume is suitable for graduate students and research mathematicians interested in computational problems of group theory.

AMS-ASL Joint Special Session, Interactions Between Logic, Group Theory, and Computer Science, January 15-16, 2003, Baltimore, Maryland

Computational and Experimental Group Theory

Ali Nesin

Lecturer in Mathematics Alexandre Borovik

Associate Professor of Mathematics Ali Nesin

The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed fields. The book contains almost all the known results in the subject. Trying to attract pure group theorists in thesubject and to prepare the graduate student to start the research in the area, the authors adopted an algebraic and self evident point of view rather than a model theoretic one, and developed the theory from scratch. All the necessary model theoretical and group theoretical notions are explained inlength. The book is full of exercises and examples and one of its chapters contains a discussion of open problems and a program for further research.

Groups of Finite Morley Rank

Alexandre Borovik

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