By Vladimir Igorevich Arnol'd

Vladimir Arnold is likely one of the most eminent mathematicians of our time a lot of those difficulties are on the entrance line of present research.

From the reports: "For a operating mathematician, it really is even more very important to understand what questions should not replied to date and did not be solved by way of the equipment already to be had, than all lists of numbers already increased, and than the erudition within the ocean of literature that has been created by way of earlier generations of researchers over twenty thousand years", V.I.Arnold states within the preface of this striking compilation. And certainly, the record of difficulties accumulated during this quantity, posed by way of him over a interval of greater than forty years in his seminar at the thought of singularities of differentiable mappings, maintains to supply worthy impetus for numerous mathematical fields (e.g., symplectic topology, dynamical structures, Kähler constructions, and lots of others). […] " M.Kunzinger, Monatshefte für Mathematik 147, factor 1, 2006 "[…] The publication is split into components – the 1st containing the issues posed in chronological order and the second one half with reviews at the difficulties. during this latter half ideas are given the place those were discovered in addition to an in depth historic bibliography of labor at the specific challenge. in a different way one could achieve a right away and succinct evaluate of present prestige of a specific open challenge from the reviews half. One novel and interesting point of the e-book is that Arnold has edited the paintings many that have contributed on to the certainty of or suggestions to the issues. in addition to Arnold there are a few fifty eight different participants, often Arnold's former scholars yet there are others open air of the Moscow tuition. […] each operating mathematician will locate whatever of direct price to their very own pursuits and locate it a useful source to dip into every now and then. One hopes that this is often an on-going undertaking and that updates will make their visual appeal usually. […]." Nicholas Witte, Australian Mathematical Society Gazette, quantity 33 quantity four 2006 "[…] those difficulties formulated through Arnold have an enourmous impact to the mathematical neighborhood to accomplish very important and gorgeous results." E.Miersemann, Zeitschrift für research und Ihre Anwendungen, quantity 24, factor four, 2005 "[…] the issues pertain to particularly varied branches of arithmetic (not in simple terms to singularity thought) and are remarkably heterogeneous of their nature. They fluctuate greatly of their scope and trouble. a few difficulties are basic for the realm into consideration; others are dedicated to minor information. […] the second one a part of the publication below evaluate […] is a set of reviews to the issues. the most aim of those reviews is to explain what development has been accomplished by way of now within the resolution of 1 challenge or one other and, on a broader scale, within the study that has arisen from the matter in query. The reviews are written by means of fifty nine individuals (including Arnold himself); they're in most cases Arnold's former scholars and/or members in his seminar. on the finish of the booklet, there's an writer index for reviews (featuring the matter numbers). a few difficulties are given a number of reviews via varied authors. […] even though the mathematical examine originating from Arnold's difficulties is way from being recorded in complete degree by means of the reviews within the moment a part of the e-book, even these reviews express what a tremendous position those difficulties have performed within the improvement of many varied components of arithmetic because the Nineteen Sixties. within the preface to the current variation of the ebook, Arnold writes approximately his difficulties: ``The saw half-life of the matter (of its roughly whole answer) is set seven years on usual. therefore, many difficulties are nonetheless open, or even those who are mostly solved continue stimulating new study showing each year in journals of assorted nations of the area. ``The invariable peculiarity of those difficulties was once that arithmetic was once thought of there now not as a video game with deductive reasonings and logos, yet as part of common technological know-how (especially of Physics), that's, as an experimental technological know-how (which is exclusive between different experimental sciences essentially through the low charges of its experiments).'' Many difficulties gathered within the first a part of the e-book less than evaluate have ended in the production of enormous new mathematical theories and retain attracting the eye of a number of actively operating mathematicians. […] interpreting the booklet less than assessment, specially its first half `The Problems', is a gripping hobby. […] The booklet permits one to plunge right into a attention-grabbing kaleidoscope of principles and effects which represent, taken jointly, a slightly immense a part of arithmetic of the second one 1/2 the final century. And, final yet no longer least, the layout of the booklet is basically appealing. To summarize, the booklet lower than overview is an excellent present PHASIS and Springer-Verlag have offered to the mathematical group. […]" Mikhail B. Sevryuk, Bulletin (New sequence) of the yankee Mathematical Society, June 2005 "Comprises lots of difficulties of assorted levels of significance … .Many difficulties accumulated … continue attracting the eye of a good number of actively operating mathematicians. … its first half ‘The Problems’ is a gripping hobby. … The ebook permits one to plunge right into a interesting kaleidoscope of principles and effects … . the layout of the e-book is admittedly attractive. … an excellent reward PHASIS and Springer-Verlag have provided to the mathematical community." Mikhail B. Sevryuk, Bulletin of the yank Mathematical Society, June 1, 2005 "This ebook incorporates a relatively whole number of difficulties … on singularities and differentiable mappings. … the issues take care of a large number of mathematical thoughts … . the entire difficulties of this e-book are on the topic of deep topics in glossy mathematical study, with purposes to varied different fields. The booklet is written via essentially the most influential modern mathematician, with huge medical horizons and enormous effect in smooth mathematics." Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1051, 2005 "The booklet below overview comprises elements: the 1st 3rd is occupied by way of formulations of the issues and the remainder contain reviews to the issues. … Arnold’s difficulties stay this present day as inspiring and stimulating as ever, and the e-book belongs to each mathematical library and the bookshelf of each study mathematician. The authors, editors and publishers of the booklet did a wonderful and intensely tricky job." (Sergei Tabachnikov, Mathematical Intelligencer, Vol. 29 (1), 2007)