This book is concerned with one of the most fundamental questions of mathematics: the relationship between algebraic formulas and geometric images. At one of the first international mathematical congresses (in Paris in 1900), Hilbert stated a special case of this question in the form of his 16th problem (from his list of 23 problems left over from the nineteenth century as a legacy for the twentieth century). In spite of the simplicity and importance of this problem (including its numerous applications), it remains unsolved to this day (although, as you will now see, many remarkable results have been discovered).

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.

Vladimir Arnold is one of the greatest mathematical scientists of our time, as well as one of the finest, most prolific mathematical authors. This first volume of his Collected Works focuses on representations of functions, celestial mechanics and KAM theory.