Dynamics, Statistics and Projective Geometry of Galois Fields by V. I. Arnold (PDF)

Description

V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐VI Arnold was not only one of the greatest Russian mathematicians of the past few decades, but the most cited Russian SCIENTIST in all fields from 08 to 09. He died in July of 2010. He was the only living mathematician with a new planet named after him, and one of the (last) and rarest of mathematicians to not only cover nearly all 500 fields of math, but additionally amazing fields as eclectic as astronomy, chemistry, biology, medicine and physics.This short little group of essays was originally given as a 2 hour lecture to Russian HIGH SCHOOL students (not joking), which might say something about the state of STEM education in US schools. Even MIT dropped its graduate projective geometry class as a requirement, but new applications in projective geometry in fields as diverse as machine vision and game programming is gradually creating renewed interest. With the demise of generalists like Arnold, we’re on our own to make the amazing connections he was able to make BETWEEN fields of not only math, but many other sciences.Reading this little diamond, translated from the Russian, is breathtaking. Arnold just doesn’t seem to care how fast he crosses, sentence to sentence, between the major “language” classes of math, much like Gauss. Also like Gauss, his ideas, problems and suggestions will likely provide research material for decades to come in vastly different areas of mathematics and other sciences. As an example, Arnold is detailing geometric progressions when he suddenly digresses into a fascinating (p/np) type discussion on the mathematics of computing logarithms (p. 30), which in the rest of the world would fall under the topic of computational complexity, but which he handles mathematically.He also gives many examples of why projective geometry (PG) needs to be considered before the more modern “linear algebra first” strategies that seemingly cut to the chase. As Dirac used PG in his private notes to first understand many aspects of hyperspace before formally portraying his analysis in linear algebra, tensors, etc., Arnold obviously has MANY insights that can easily be forgotten in today’s specialist driven math domains.In U.S. or U.K. schools this material would be advanced undergrad and grad, so please don’t take the little book’s origin as in any way related to the complexity of the material. Even a seasoned graduate mathematician will likely find many “connections” (like dynamical systems) between “math genres” stimulating and suggestive here. It will take a lot more reading to confirm these connections, but an adventure awaits you if you invest the time. With dynamical systems paths self selecting strange attraction, chaos and disintegration, or oscillation, Arnold’s contributions (here and in thousands of articles) “bring back” many aspects of projective and even parallel geometries (affine) to help visualize and cross reference those paths.The book is tiny for the investment, but gives a very interesting taste of the amazing connections that can be made by that rare breed of generalist who saw all the fields of math as different languages, the translation of which causes both new insights and meanings, and explains and clears up age old confusion.Library Picks reviews only for the benefit of Amazon buyers and has nothing to do with Amazon, the publishers or authors, and we always buy the books we review.

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