Arnold's Problems contains mathematical problems brought up by Vladimir Arnold in his famous seminar at Moscow State University over several decades. In addition, there are problems published in his numerous papers and books.
The invariable peculiarity of these problems was that Arnold did not consider mathematics a game with deductive reasoning and symbols, but a part of natural science (especially of physics), i.e. an experimental science. Many of these problems are still at the frontier of research today and are still open, and even those that are mainly solved keep stimulating new research, appearing every year in journals all over the world.
The second part of the book is a collection of commentaries, mostly by Arnold's former students, on the current progress in the problems' solutions (featuring a bibliography inspired by them).
This book will be of great interest to researchers and graduate students in mathematics and mathematical physics.
From the reviews:
"For a working mathematician, it is much more important to know what questions are not answered so far and failed to be solved by the methods already available, than all lists of numbers already multiplied, and than the erudition in the ocean of literature that has been created by previous generations of researchers over twenty thousand years," V.I.Arnold states in the preface of this impressive compilation. And indeed, the list of problems collected in this volume, posed by him over a period of more than 40 years in his seminar on the theory of singularities of differentiable mappings, continues to provide invaluable impetus for a variety of mathematical fields (e.g., symplectic topology, dynamical systems, Kahler structures, and many others). [...] "
M.Kunzinger, Monatshefte fur Mathematik 147, Issue 1, 2006
"[...] The book is divided into two parts - the first containing the problems posed in chronological order and the second part with comments on the problems. In this latter part solutions are given where these have been found along with an extensive historical bibliography of work on the particular problem. Otherwise one can gain an immediate and succinct overview of current status of a particular open problem from the comments part. One novel and fascinating aspect of the book is that Arnold has edited the work many who have contributed directly to the understanding of or solutions to the problems.
Besides Arnold there are some 58 other contributors, mostly Arnold's former students but there are others outside of the Moscow school. [...]
Every working mathematician will find something of direct value to their own interests and find it an invaluable resource to dip into from time to time. One hopes that this is an on-going project and that updates will make their appearance regularly. [...]."
Nicholas Witte, Australian Mathematical Society Gazette, Volume 33 Number 4 2006
"[...] These problems formulated by Arnold have an enourmous influence to the mathematical community to achieve important and beautiful results."
E.Miersemann, Zeitschrift fur Analysis und Ihre Anwendungen, Volume 24, Issue 4, 2005
"[...] The problems pertain to quite diverse branches of mathematics (not only to singularity theory) and are remarkably heterogeneous in their nature. They differ very much in their scope and difficulty. Some problems are fundamental for the area under consideration; others are devoted to minor details. [...]
The second part of the book under review [...] is a collection of comments to the problems. The main goal of these comments is to describe what progress has been achieved by now in the solution of one problem or another and, on a broader scale, in the research that has arisen from the problem in question. The comments are written by 59 persons (including Arnold himself); they are mostly Arnold's former students and/or participants in his seminar. At the end of the book, there is an author index for comments (featuring the problem numbers). Some problems are given several comments by different authors. [...]
Although the mathematical research originating from Arnold's problems is far from being recorded in full measure by the comments in the second part of the book, even those comments show what an enormous role these problems have played in the development of many diverse areas of mathematics since the 1960s. In the preface to the present edition of the book, Arnold writes about his problems: ``The observed half-life of the problem (of its more or less complete solution) is about seven years on average. Thus, many problems are still open, and even those that are mainly solved keep stimulating new research appearing every year in journals of various countries of the World.
``The invariable peculiarity of these problems was that Mathematics was considered there not as a game with deductive reasonings and symbols, but as a part of natural science (especially of Physics), that is, as an experimental science (which is distinguished among other experimental sciences primarily by the low costs of its experiments).''
Many problems collected in the first part of the book under review have led to the creation of vast new mathematical theories and keep attracting the attention of a great number of actively working mathematicians. [...]
Reading the book under review, especially its first part `The Problems', is a gripping pastime. [...] The book enables one to plunge into a fascinating kaleidoscope of ideas and results which constitute, taken together, a rather sizeable part of mathematics of the second half of the last century. And, last but not least, the design of the book is really beautiful. To summarize, the book under review is a wonderful gift PHASIS and Springer-Verlag have presented to the mathematical community. [...]"
Mikhail B. Sevryuk, Bulletin (New Series) of the American Mathematical Society, June 2005
"Comprises plenty of problems of various degrees of importance ... .Many problems collected ... keep attracting the attention of a great number of actively working mathematicians. ... its first part 'The Problems' is a gripping pastime. ... The book enables one to plunge into a fascinating kaleidoscope of ideas and results ... . the design of the book is really beautiful. ... a wonderful gift PHASIS and Springer-Verlag have presented to the mathematical community."
Mikhail B. Sevryuk, Bulletin of the American Mathematical Society, June 1, 2005
"This book contains a fairly complete selection of problems ... on singularities and differentiable mappings. ... The problems deal with a multitude of mathematical concepts ... . All the problems of this book are related to deep subjects in modern mathematical research, with applications to various other fields. The book is written by one of the most influential contemporary mathematician, with large scientific horizons and huge impact in modern mathematics."
Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1051, 2005
"The book under review consists of two parts: the first third is occupied by formulations of the problems and the rest comprise comments to the problems. ... Arnold's problems remain today as inspiring and stimulating as ever, and the book belongs to every mathematical library and the bookshelf of every research mathematician. The authors, editors and publishers of the book did a fantastic and very difficult job." (Sergei Tabachnikov, Mathematical Intelligencer, Vol. 29 (1), 2007)