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 truly is even more vital to understand what questions aren't replied to this point and didn't be solved by means of the tools 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 past generations of researchers over twenty thousand years", V.I.Arnold states within the preface of this striking compilation. And certainly, the checklist of difficulties gathered 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, keeps to supply beneficial impetus for various mathematical fields (e.g., symplectic topology, dynamical platforms, Kähler buildings, and lots of others). […] " M.Kunzinger, Monatshefte für Mathematik 147, factor 1, 2006 "[…] The ebook 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 chanced on besides an in depth historic bibliography of labor at the specific challenge. in a different way possible achieve an instantaneous and succinct evaluation of present prestige of a selected open challenge from the reviews half. One novel and interesting element 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, generally Arnold's former scholars yet there are others outdoor of the Moscow college. […] 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 on a regular basis. […]." Nicholas Witte, Australian Mathematical Society Gazette, quantity 33 quantity four 2006 "[…] those difficulties formulated by means of Arnold have an enourmous effect to the mathematical group to accomplish vital and lovely results." E.Miersemann, Zeitschrift für research und Ihre Anwendungen, quantity 24, factor four, 2005 "[…] the issues pertain to really varied branches of arithmetic (not merely to singularity thought) and are remarkably heterogeneous of their nature. They fluctuate a great deal 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 booklet below overview […] is a suite of reviews to the issues. the most objective of those reviews is to explain what growth has been accomplished through now within the answer of 1 challenge or one other and, on a broader scale, within the learn that has arisen from the matter in query. The reviews are written by means of fifty nine individuals (including Arnold himself); they're quite often Arnold's former scholars and/or members in his seminar. on the finish of the ebook, there's an writer index for reviews (featuring the matter numbers). a few difficulties are given a number of reviews through varied authors. […] even if the mathematical learn originating from Arnold's difficulties is much from being recorded in complete degree via the reviews within the moment a part of the publication, even these reviews exhibit what an important 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 booklet, Arnold writes approximately his difficulties: ``The saw half-life of the matter (of its roughly whole answer) is ready seven years on common. therefore, many difficulties are nonetheless open, or even those who are ordinarily solved maintain stimulating new examine showing each year in journals of varied international locations of the realm. ``The invariable peculiarity of those difficulties used to be that arithmetic used to be thought of there no longer as a online game with deductive reasonings and logos, yet as part of usual technological know-how (especially of Physics), that's, as an experimental technological know-how (which is wonderful between different experimental sciences essentially through the low expenditures of its experiments).'' Many difficulties gathered within the first a part of the e-book less than evaluate have ended in the production of huge new mathematical theories and preserve attracting the eye of lots of actively operating mathematicians. […] studying the e-book less than assessment, specifically its first half `The Problems', is a gripping hobby. […] The publication permits one to plunge right into a interesting kaleidoscope of rules and effects which represent, taken jointly, a slightly significant a part of arithmetic of the second one half the final century. And, final yet now not least, the layout of the e-book is admittedly attractive. To summarize, the e-book below evaluate is an excellent present PHASIS and Springer-Verlag have awarded to the mathematical neighborhood. […]" Mikhail B. Sevryuk, Bulletin (New sequence) of the yankee Mathematical Society, June 2005 "Comprises lots of difficulties of assorted levels of significance … .Many difficulties amassed … maintain attracting the eye of plenty 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 booklet is admittedly attractive. … an excellent reward PHASIS and Springer-Verlag have awarded to the mathematical community." Mikhail B. Sevryuk, Bulletin of the yankee Mathematical Society, June 1, 2005 "This e-book features a particularly entire choice of difficulties … on singularities and differentiable mappings. … the issues take care of a mess of mathematical suggestions … . all of the difficulties of this e-book are with regards to deep topics in smooth mathematical examine, with functions to varied different fields. The ebook is written by way of probably the most influential modern mathematician, with huge clinical horizons and large effect in smooth mathematics." Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1051, 2005 "The e-book below evaluation includes components: the 1st 3rd is occupied by means 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 booklet belongs to each mathematical library and the bookshelf of each examine mathematician. The authors, editors and publishers of the ebook did a phenomenal and intensely tricky job." (Sergei Tabachnikov, Mathematical Intelligencer, Vol. 29 (1), 2007)

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Is it true that in the complex case the complement of the bifurcation diagram of a function is always a K(%, 1) space? Are the components contractible in the real case? Conjecturally no, although R. Thorn had thought that yes! 1975-12. Does every real-valued function have a real Modification (with \i real critical points)? 1975-13. What is the minimal number of critical values obtained by a perturbation of a critical point of multiplicity \i with (i Morse critical points? Conjecturally it is n + 1, where n is the number of variables (or corank).

Investigate the bifurcations of type D5 in the 3-space topologically {the problem has been studied by V. I. Bakhtin). 1978-20 The Problems 37 1978-20. Investigate the singularities of bicaustics of type D$, up to diffeomorphisms. 1978-21. Investigate the process of sweeping the bicaustic D\, up to equivalence (strong equivalence): given three smooth curves (p,-: (M, 0) —>• (M2,0) starting from 0 with the same velocity v ^ 0, in all other generic. The equivalence is provided by the diagrams (R,0) - ^ ^ (R 2 ,0) (where the diffeomorphisms x and h are independent of /); the strong equivalence is: x(t) = t + const.

1980-6. ). 1980-7. Construct a theory of caustic cobordisms (different from that of Lagrangian cobordisms). 1980-8. In the theory of singularities (e. , critical points of functions), why is the codimension in the real case the same as in the complex case? Compare with the R- and C-modality and with the {co)dimension of the prolonged self-intersection line of the swallowtail or the umbrella. 1980-9. Apply mixed Hodge structures to real algebraic geometry. For example, for estimation of topological invariants of real Morsifications, and for investigation of the topology of discriminants.