A.T.FOMENKO

       MATHEMATICAL IMAGES IN THE REAL AND UNREAL WORLD

                     ABOUT THE AUTHOR

   Prof. A.T.FOMENKO (born 1945), is a Full Member of the Russian

Academy of Sci., Dr. of Sci. (Math. and Phys.), Moscow State

University (Moscow), and Head of the Department of Differential

Geometry and Applications in Moscow University.

   He is a distinguished mathematician and a well-known specialist

in the fields of geometry, Hamiltonian mechanics, the calculus of

variations, computer geometry, and algorithmical problems in pattern

recognition. He was a winner of the Award of the Moscow Math.Soc.

(1974), the Award of the Presidium of USSR Acad. of Sci.

in mathematics (1987), and of the State Award of Russia

(in mathemativs) (1994).

   He has obtained fundamental results in the theory of minimal

surfaces and solved the multidimensional Plateau problem -

the existence of a globally minimal surface in each spectral

bordism class. He created a theory of topological classification of

integrable Hamiltonian systems . This theory was applied to the

important problem of classification of isoenergy surfaces for

integrable dynamical systems (which arise for example in

celestial mechanics and in the theory of motion of a rigid body).

Fomenko obtained the complete classification theory forthe

bifurcation of solutions in integrable Hamiltinian systems.

He also developed a new empirico-statistical method for

the analysis of narrative texts (e.g. chronicles).

   Among the well-known books written by A.T.Fomenko are:

"Variational Principles in Topology (Multidimensional Minimal

Surface Theory)", "Differential Geometry and Topology",

"Variational Problems in Topology (The Geometry of Length, Area

   and Volume)", "Homotopic Topology" (together with D.B.Fuchs

and V.L.Gutenmacher), "Symplectic Geometry. Methods and

Applications", "Basic Elements of Differential Geometry and

Topology" (together with S.P.Novikov), "Modern Geometry"

(together with S.P.Novikov and B.A.Dubrovin), "A Course in

Homotopic Topology" (together with D.B.Fuchs), "Minimal

Surfaces and Plateau's Problem" (together with Dao C.T.),

"Methods of Statistical Analysis for Narrative Texts and Applications

to Chronology".

   These books were translated into English by Springer-Verlag,

Plenum, Reidel (Kluwer), Gordon and Breach, American Math.Soc.

   The following books were published originally in English:

"Plateau's Problem (vol.1 - Historical Survey, vol.2 - The Present

   State of the Theory" (Gordon and Breach),

"Integrability and Nonintegrability in Geometry and Mechanics"

   (Kluwer Acad.Publ.),

"Integrable Systems on Lie Algebras and Symmetric Spaces" (together

with V.V.Trofimov) (Gordon and Breach).

   A.T.Fomenko has also the talent for expressing abstract mathe-

matical concepts through artwork. Since the mid-1970s, Fomenko has

created more than 280 graphic works. Not only have his images

filled pages of some of his own books in geometry, but they have

also been chosen to illustrate books (of many mathematicians) on

other subjects, such as statistics, probability, number theory and

so on. In addition, his works have found their way into the

scientific and popular press and have been displayed in more than

100 exhibits in the Russia, USA, Canada, the Netherland, India and

much of Eastern Europe.

   In 1990 the American Math.Soc. published the book by Fomenko:

"Mathematical Impressions" containing 84 reproductions of works by

Fomenko.

   "What's interesting about his work to me is that it shows the

impact of certain mathematical ideas", says William Thurston, Profes-

sor of mathematics at Princeton University. "It excites your imagi-

nation. It's interesting to look at and think about. It's not designed

to be just a straightforward communication of a simple idea, but to

stir up your imagination - which it does. And in that sense it is

very good and successful. I think it's a very effective way of commu-

nicating mathematics." (Insight Magazine, April 30, 1990).

   Springer-Verlag published the English translation of the book:

A.T.Fomenko "Visual Geometry and Topology". This mathematical book

also contains 50 reproductions of graphic works of Fomenko.

   The book ("Mathematical Images in the Real and Unreal World")

is quite different from the books listed above and contains the total

collection of Fomenko's works with 1/2-mathematical and 1/2

philosophical comments. The works are organized in chapters correspon-

ding to different branches of mathematics, history and philosophy.

The book is unique event in mathematical literature and does not have

analogues.

   The epilogue to this book is written by famous mathematician

Yu.I.Manin.

                      ABOUT THE BOOK

   The book is published in Russia, Moscow (in Russian),

Moscow University Press, 1998.

   The book contains 208 reproductions of works by Fomenko (172 in

black and white and 36 in color). In the accompanying captions,

Fomenko explains the mathematical motivations behind the illustrations

as well as the emotional, historical, or mythical subtexts they evoke.

The illustrations carry the viewer through a mathematical world

consisting not of equations and dry logic, but of intuition and

inspiration. Stimulating to the imagination and to the eye, these

works can be interpreted and appreciated in various ways - methema-

tical, aesthetic, or emotional. The commentary to each graphic work

consists of two parts: 1) mathematical explanation, 2) non-mathe-

matical associations connected with the theme of the drawing (his-

torical, mythological et cetera). This second part of comments is

unique in mathematical literature and at first appears in the present

book. Each commentary has the volume approximately 1 page (type-written

manuscript). The book is oriented on the wide audience and is intented

for students, mathematicians, physists and all readers who is interested

in visualization of modern mathematical ideas and their connection

with general human concepts.

   The volume of the book: 210 pages (manuscript), 172 graphic

works (black-white, half-tone, the size: 30 x 40 cm, 36 color works;

each print needs in 1 individual page. Thus, the total volume is as

follows: 210 + 208 = 418 pages.

   C o n t e n t.

Introduction. Associative-visual thinking in modern mathematics.

1. Images in general and algebraic topology. Simplicial and cell

   complexes.

2. Images in geometry and topology of smooth manifolds.

3. Images in mathematical analysis. Functions on the manifolds,

   algebraic surfaces and singular points.

4. Images in mathematical physics, mechanics, differential equations.

5. Images in the calculus of variations, differential equations,

   group theory and crystallography.

6. Images in computer geometry. Algorithmical problems of recogni-

   zability. Mathematical statistics and probability theory.

7. Geometrical images in the novel of M.A.Bulgakov "Master and

   Margaret".

8. Images in the general mathematical concepts.

9. Images in color.

10. APPENXIX

    Yu.I.MANIN.

    About graphic works of Fomenko.