Alexandre V. Borovik – författare

"Mirrors and Reflections" presents a systematic and elementary introduction to the properties of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups. The exposition is directed at advanced undergraduates and first-year graduate students, and features a large number of exercises at various levels of difficulty.

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.Key topics and features:* Systematic, clearly written exposition with ample references to current research* Matroids are examined in terms of symmetric and finite reflection groups* Finite reflection groups and Coxeter groups are developed from scratch* The Gelfand-Serganova theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties* Matroid representations in buildings and combinatorial flag varieties are studied in the final chapter* Many exercises throughout* Excellent bibliography and indexAccessible to graduate students and research mathematicians alike, "Coxeter Matroids" can be used as an introductory survey, a graduate course text, or a reference volume.

The book gives a detailed presentation of the classification of the simple groups of finite Morley rank which contain a nontrivial unipotent 2-subgroup. They are linear algebraic groups over algebraically closed fields of characteristic 2. Although the story told in the book is inspired by the classification of the finite simple groups, it goes well beyond this source of inspiration. Not only do the techniques adapted from finite group theory cover, in a peculiar way, various portions of the three generations of approaches to finite simple groups but model theoretic methods also play an unexpected role. The book contains a complete account of all this material, part of which has not been published. In addition, almost every general result about groups of finite Morley rank is exposed in detail and the book ends with a chapter where the authors provide a list of open problems in the relevant fields of mathematics. As a result, the book provides food for thought to finite group theorists, model theorists, and algebraic geometers who are interested in group theoretic 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.