Geometrical imagination and intuition play � very important role in modern mathematical studies, especially in those connected with mathematical physics, geometry and topology. In many profound mathematical works devoted to very complicated problems (for example, multidimensional geometry, variational calculus, etc.), "visual slang" of two- and three-dimensional geometry is widely used: "cutting � surface", "pasting the leaves of � surface", "pasting � cylinder", "turning � sphere inside out", "mounting � handle", etc. These terms have been called forth by necessity rather than mathematicians' caprice, because their use in the mathematical mentality of professionals is absolutely necessary for
obtaining many extraordinarily complicated results. It is � rather common situation that the proof of � mathematical fact can at first be "seen" using some mathematical terms, and then, (according to this visual idea) be shaped into � logically self-consistent speculation. Sometimes this process is very difficult and requires great intellectual effort. However, these efforts are repaid in � psychological way by the beautiful and clear picture formed in � researcher's mind. This picture convinces the researcher that he is on the right track. Therefore, the beauty of � geometrical image can often serve as � criterion for choosing the optimum way for further formal proofs.
Each professional mathematician has � "personal" set of ideas regarding the internal geometry of his intrinsic mathematical world, as well as visual images associated with certain abstract mathematical concepts (including those of algebra, number theory, mathematical analysis, etc.). It isvery interesting that different mathematicians often use similar visual images to express the same abstract ideas. In many cases, they are very difficult to draw on � sheet of paper. These graphic pieces presented to the reader are an attempt to make � "photo from the inside" of the complicated mathematical world, which is saturated with images and ideas of modern geometry (this term being understood in � wide sense). These graphic works are based on either specific mathematical constructions, ideas, and theorems, and serve as illustrations of actual mathematical objects and processes, or, they reflect different ways of perceiving certain abstract mathematical ideas, such as infinity, continuity, homeomorphism, homotopy, etc. From this viewpoint, one of the most important regions of
mathematics, topology, is in the most favorable position. Princi��lly, it deals with the properties of various objects that remain intact in their "deformations" (it would be better to stop on such � visual level, not going into more profound definitions). For example, when an elastic sphere, �.g. � balloon, is deformed, the lengths of lines drawn on it are certainly changed but � very important property is still ���served: any closed loop drawn on the sphere can be reduced to � point. This is an example of the simplest topological properties of � sphere, which distinguishes it from, for example, � torus. This geometrical object can have some closed loops drawn on it (for example, parallels and meridians) that cannot be reduced to � point. � reader who wants to enter the depth of the "visual world" of modern mathematics, may address the book by �.Fomenko entitled, "Naglyadnaya geometriya i topologiya. Geometricheskie obrazyv ���l'nom mire." (Visual geometry and topology. Geometrical images in the real world), published by Moscow State University in 1990.
Author' s comments on the works by �.�.Fomenko1) � topological zoo. Geometry, topology, minimal surfaces. Some two-dimensional polyhedra used in topology, geometry, and minimal surfaces theory are painted. They provide an opportunity to visually demonstrate some non-trivial mathematical theorems. In the top right corner, � humorous scene is shown: � resuscitated polyhedron falls to pieces, it's components being shells (scorpions). The scorpion's tail bent toward its head clearly demonstrates this polyhedron's design. The proper way of pasting the shells to reconstruct the polyhedron is clearly seen. The process of turning � holed two-dimensional torus inside out is portrayed in the central gallery of � medieval castle (the exhibition is inside it). It turns out that if such � holed torus is turned inside out (by � homeomorphism in � three-dimensional space), then � torus is again obtained. However, parallels and meridians of the initial torus exchange places after such
� procedure, the internal surface of the torus becoming the external one, and vice versa. In the left bottom corner is "Antoin's necklace", � well-known object in general topology. Beside it, on � lightened place, is � minimal surface (� soap film) bounded by � circle. This surface has � remarkable property: it can be continuously, without tearing, contracted to its boundary. Topologically speaking, the boundary circle of the film is its deformational retract. The striking feature of this example, belonging to J.F.Adams, is that the surface is modeled by � stable soap film, which closes up � wire loop in the three-dimensional Euclidean space. As can be seen from the picture, ���� � minimal surface can be obtained by pasting � usual Mobius' strip to � s�-called triple Mobius' strip. One extra (non-trivial) property of such � surface should be noted for professional topologists: it can be slightly reshaped into � "holed Bing's house" (i.�. � Bings' house, from which � small disk is removed). In other words, the "holed Bing's house" can be realized in the three-dimensioned
space as � stable soap film bounded by � wire contour homeomorphic to � circle. And in the center of the exhibition hall, � 2-adic solenoid, � topological object, is placed. 2) The star diagram (of Hertzsprung and Russel). Stellar astronomy. At the beginning of the 20th century, two astronomers (Hertzsprung and Russel) independently plotted � special stellar diagram in an attempt to find � correlation between the spectral class of � star and its luminousity. Along the horizontal axis, the sequence of the spectral classes was plotted, while the vertical axis corresponded to the absolute magnitudes of the stars, �� that their values decreased up the vertical axis, and, correspondingly, the luminousity of the stars increased in that direction. Therefore, each star is represented by � point on this diagram, and if there are many of them, one can try to find � correlation between the mentioned parameters. It turned out that the stars concentrated along � few lines (slightly diffused, but nevertheless, quite distinct), rather than being scattered chaotically. The so-called principal stellar sequence and the sub-dwarfs are sketched in our picture. In as-
tronomy, this diagram is nowadays termed the luminousity diagram. The sun is placed approximately in the center of the bent curve as shown in the picture. Althoundh the stars beside the "diffused curve" have different characteristics, they evidently have some common features. No common suggestion about the nature of such � correlation exists today. Perhaps, the stars had been formed in � common region of the Universe, or maybe they have nearly the same chemical composition, age, etc. 3) Fiber space. Topology of the manifolds. This picture represents � fibration, langential to � circle with � single corner point. The base of this fibration is � circle inserted into � two-dimensional plane. This circle has one singular point, in which the tangential line to the circle is not defined. The fibration layers are the tangential lines. The fibration, tangential to the circle, is homeomorphic to � two-dimensional cylinder (each tangential line can be rotated to the normal position to the circle inserted into the plane). This process is schematically portrayed in the picture: the tangential lines are rotated by different angles, one line "lagging behind" the
others. Since the tangential line is not defined at that singular point, the layers pictured are thinned as they approach this point; the circle looks like � "drop" hung by this singular point. To simplify the picture, the lines tangential to the right side of the circle are not presented. 4)Deformation of the Rlemann surface of an algebraic function. Theory of the algebraic functions. Presented here is � three- dimensional model of deformation of the Riemann surface of the algebraic function w=(z - a)(z - b)(z - c)(z- d)]1/2 in � four-dimensional Euclidean space R4, which is identified with � two-dimensional complex space �2. The Riemann surface of this function is homeomorphic to � two-dimensional sphere with one handle, i.e. to � two-dimensional torus (under the condition that all roots �, b, �, d of the 4th degree polynomial are different). From the viewpoint of the theory of algebraic functions, two-dimensional spheres can serve as the raw material for constructing this surface. Each sphere should be cut in two places, then the corresponding cur edges should be glued together. These actions will call forth � torus presented as two spheres connected by two tubes (cylinders)
(see the picture). This will be the picture if all four roots are simple ones (not multiple). The situation will be quite different, if the polynomial is deformeds� that its roots are approaching each other (i.�. when multiple roots appear in � limit). The Riemann surface is then also deformed in C2 so that vanishing cycles and singular points appear on it. As � result, the Riemann surface loses its smoothness in C2. An example of this deformation is presented in the picture. Two roots are approaching each other. Consequently, the size of the top sphere is decreased, while the bottom sphere is, on the contrary, swollen. For the reader's convenience, � part of the sphere surfaces is removed �� that the inner part of the spheres is shown. In � limit, � two-dimensional sphere with two identified points is obtained, which is the Riemann surface of the algebraic function w=(z - a)[(z- c)(z - d)]1/2 having � multiple root_�_. 5) The Poisson-Laplace theorem and Plateau's principles. Calculus of variations. � two-dimensional surface separating two physical media (�.g. gas-gas, gas-liquid, or liquid-liquid) is called the surface
of the interphase boundary.If this physical system is in an equilibrium, then, according to the Poisson-Laplace theorem, the interphase boundary has � ��nstant mean curvature. The �����ture value is proportional to the difference of the pressure of the media. Visual examples of these surfaces are soap' bubbles or soap films. Adjacent soap bubbles (which tend to press into one another) can form singular edges. These edges, consisting of singular points of the soap surface, look like � very complicated three-dimensional structure (� graph with many vertices). According to one of Plateau's principles, only three sheets of the soap film can meet at � stable singular edge. Besides, they must form equal angles of 1200. Only four singular edges can meet at � singular vertex (the angles between them must also be equal). � � soap bubble is very gingerly pierced through by � thin wire (see the picture), it will envelop the wire rather than burst. This process calls forth many different mathematical problems. For example, it would be interesting to describe the graphs formed by the singular edges of soap foam. 6) � heavy top floating in the Universe. Hamiltonian mechanics, symplectic geometry.
� heavy gyroscope is � rapidly spinning top (i.�. � solid body). The top axis has � remarkable property of conserving its orientation in space, independent of the motion of the apparatus (for example, � rocket) containing � gyroscope. It is an important instrument used in the orientation systems of various aircraft. The equations of motion of heavy solids are now studied by the methods of the theory of differential equations, symplectic geometry, algebra and topology. If � solid body has an "arbitrary shape", i.e. it has �� symmetry, its motion will be chaotic (i.�. it will make disorderly somersaults in space). 7) Anti-Durer. From the series, "� dialogue with the authors of the 16th century". � geometrical fantasy on general mathematical concepts. The picture was called forth by thoughts (from � mathematician's viewpoint) over the well-known engraving "Melancholy" by �. Durer. The attitude toward the scientific advances of the contemporaries has undergone great changes since Durer's time, three centuries ago. The
reader can understand the authir's opinion by comparing Durer's engraving with the author's work. For example, � magic square is shown in the top right corner of Durer's work. ����, the decimal decomposition of_� = 2,71828182845904523536 0287471352662497757247093 699959574966967_... is placed instead. This sequence of digits is placed in � square spiral which is untwisted clockwise from the center of the square (note the digit 2 in the center). Only 121 digits from the decimal decomposition of e are depicted. Such decimal decompositions (of some irrational numbers) are now used as random number generators in many statistical studies. The sequence of digits forming the decimal decomposition of e (or,, for example), serve as (in � sense that we will not elaborate on here) � random sequence. Today, with the help of computers, there are many 1000s of digits calculated from the decimal decomposition of and_e_. The reader can also find � picture of � separatrix diagram of � critical point of index I of � smooth function defined on � three-dimensional space (see the "bell" with the tongue-separatrix near the num-
ber e). The shape of the clouds in the sky over the ocean, also has � mathematical meaning. 8) �h� spines of two three-dimensional compact closed manifolds of minimal complexity. Hamiltonian ��d symplectic geometry, topology of three-dimensional manifolds, hyperbolic geometry The central mathematical objects in this graphic work are the two polyhedra rising above the horizon. They represent the two-dimensional neighborhoods of the one-dimensional skeletons in the spines of the two three-dimensional manifolds specified in the heading. � 3-manifold can be unequivocally (to � homeomorphism) reconstructed from its two-dimensional spine. These two remarkable manifolds are the first examples of isoenergy surfaces on which any Hamiltonian differential equation is non-integral, in the class of smooth integrals in the general position. The discovery of this fact was � result of combining the theory of complexity of 3-manifolds (S.V. Matveev) with the theory of topological classification of integral Hamiltonian equations (�.�. Fomenko). Later, these manifolds (they had been earlier studied by W. Thurston and J. Weeks
from another point of view), were found to have some other remarkable properties. For example, S.V. Matveev ��� �.�. Fomenko had found that these two manifolds had the minimal complexity in the class of all hyperbolic closed compact 3-manifolds, and, therefore, one of these manifolds probably had the minimal possible hyperbolic volume (of approximately 0.94...). The complexity of these manifolds is equal to 9, which can be checked just by counting the number of vertices of multiplicity 4 in the two-dimensional spines presented in our picture. On any 3-manifold of complexity less than 9, there exists at least one integral Hamiltonian differential equation. 9) Anti-Breugel. � work from the series, "Dialogue with the authors of the 16th century". The non-mathematical associations in mathematical images. This work was created after the famous engraving "Alchemists" by Pieter Breugel. It does not illustrate � specific mathematical theorem, but its construction consists of mathematical ideas and images. Th�reader is faced here with the idea of mathematical infinity (the row of cups filled with molten metal
receding to the horizon), dynamical flow, analytical functions (clouds in the sky), homeomorphism and homotopy (presented as the deformation of � human body), etc. The author treats this work as � reflection of the evolution undergone by scientific ideas since ������� time, three centuries ago. 10) Temptation of St. Anthony. The non-mathematical associations in mathematical images. This work was created after the well-known medieval legend about the temptation of St. Anthony. ����, the language of mathematical images is used. The Beltramy surfaces are shown (the funnel-shaped surfaces, on which the hyperbolic metric of Lobachevsky is realized; see the tubes in the top left side of the picture). The Beltramy surface is formed by rotating � flat curve of � specific shape around its asymptote. The ideas of mathematical infinity, homeomorphism, and continuous deformation are presented. Some famous problems that still wait for � solution are also remembered, �.g. the Poincare problem (whether the single-connected compact three-dimensional manifold is � standard sphere), and Fermat's theorem.
11) Mathematical infinity and its connection with geometry and topology. The algorithmic insolubility of the problem of manifold classification for dimensions greater than 3. General mathematical ideas. Mathematical infinity is ��vealed most clearly in geometry and topology. This can be explained not only by their specific features (they deal with visual mathematical images), but also by the fact that "geometrical infinity" appears in the very first steps of studying these disciplines. � concept as simple as � limit point of � set is based on the idea of infinity. One of the deepest manifestations of mathematical infinity is the algorithmic insolubility ofsome topological problems. Any smooth compact manifold (of any dimension) can be presented in the form of � simplicial complex. So, having drawn � table containing all such simplices, their faces and incidence coefficients, one can use this table to define � manifold, and regard the table as � manifold cade. ����, the problem of algorithmic classification of the manifolds of � given dimension arises. Namely, the problem is in ���1��� there exists an algorithm which operates according to � standard program
(which, in principle, can be realized on � computer) and provides an answer to the question: do two arbitrary codes define diffeomorphic (or homeomorphic) manifolds or not? The classification of the manifolds of dimensions 1 and 2 is quite simple (in particular, their complete list is well-known). The situation is cardinally different for the manifolds of dimension 3: today there is not only no "algorithm for their recognition and classification", but it is even difficult to formulate � verisimilar hypothesis about its existence. Starting with dimension 4, the picture begins to elucidate: it is proved' that the manifolds of this dimension cannot be classified (this is � strong mathematical theorem). And so, there existsno algorithm defined on the set of codes of all four-dimensional manifolds (and, indeed, on the set of codes of the manifolds of dimensions 5, 6, etc.), which answers the question: "Do two arbitrary input codes define diffeomorphic manifolds or not"? This graphic work is called forth by reflections on this problem. The reader can see different
methods of coding the manifolds and be filled with � sense of some type of mystery, which shrouds the ���� of three-dimensional manifolds. 12) Homeomorphism conserves many essential properties of � geometrical object. Topology. The 'idea of homeomorphism is one of the most important concepts of topology. It can be visually illustrated as � bilaterally continuous bijection (correspondence) that deforms objects made from the thinnest rubber. In such � deformation, "tearings" and "pastings" are forbidden. The picture helps to illustrate this idea. ����, one can see two human figures who resemble the central composition of Rembrandt's canvas, "Homecoming of � Prodigal Son". However, this famous image ����mes visible by �� means at once, since the composition is subjected to � homeomorphism that has distorted the initial painting beyond recognition. This is an important feature of the homeomorphism. It can change the metrical (easily recognized) properties of an object, stretch and compress it, and change the distances between different points. However, the topological properties (not so easily recognized) are preserved, �.g.
the number of "holes" in an object, the number of handles. 13) The theorem on fundamental groups of four-dimensional manifolds. Manifolds theory. This fundamental group measures the "number" of different non-equivalent closed loops that can be "drawn" on � given manifold. � natural question arises: "Which groups can be represented as fundamental groups of smooth manifolds?" For the manifolds of dimension 1 or 2, the answer is very simple. �llsuch groups have long ago been described and studied. For dimension 3, there are groups which cannot be represented as fundamental of � closed three-dimensional manifold. Starting with dimension 4, the situation changes. It turns out that any finitely-generated group (��. with � finite number of generators and � finite number of relations), can be represented as � fundamental group of � four-dimensional manifold that is smooth, compact, connected and closed. The picture shows one of the central methods used for proving thiswell-known topological theorem. First, the so-called "handles of index 1" should be constructed in order to realize the genera-
tors of the given group. The tube neighbourhoods of such handles are shown at the bottom of the picture in the form of � sequence of petals recedinginto the distance. Then � set of "handles of index 2" corresponding to the correlations in this group should be pasted to this space. One of these handles is shown in the top of the central part of the picture as � huge, dark, flexible object enveloping the "skeleton" constructed on the previous step. This body is shown to be flowing downward, which corresponds to the character of the executed operation; this figure should be pasted along � certain continuous map of its boundary. This map can deform the object significantly. 14) � step in � proof for the theorem of existence of globally minimal surfaces. Calculus of variations, minimal surfaces. Minimal surfaces are the surfaces of minimal volume. They can be modeled by the equilibrium interphase boundary. For example, the soap films bounded by closed wire contours (when they are pulled out from the soap solution), are minimal sur-
faces. The famous Plateau hypothesis in calculus of variations, states that any "contour" can be covered by a surface of minimal area (or by � minimal volume if many-dimensional surfaces are considered). At the beginning of the 20th century, this problem was solved for two-dimensional surfaces but for the many-dimensional ones, such � solution s� far is absent. However, profound theorems which solve the Plateau problem for the specific classes of the surfaces have been proved. This graphical piece illustrates one of the central steps in the proof of the theorem which solves the many-dimensional Plateau problem for the class ofso-called homological surfaces. In the process of minimization of the surface volume, thin "whiskers" may eventually grow, which practically does not affect the volume (or area), but essentially changes its metrical properties and its position in the enveloping space. The picture shows that these "whiskers" can have � rather whimsical form. Their appearance may "spoil" the minimization process, so they should be "cut away". It is the neat mathe-
matical formulation of this process that turns out to be one of the most non-trivial moments of the proof. Eventually, one can manage to smooth over the surface and prove the existence of its limit (when the surface area approaches its minimum). 15) The problem of the algorithmic recognition of the standard three-dimensional sphere in the class of all three-dimensional manifolds. Computer and algorithmic topology. One of the most interesting problems in three-dimensional topology is in the recognition of � standard three-dimensional sphere in the class of all three- dimensional manifolds. � sphere is the simplest manifold, and no difficulties seem to be caused by the question of whether � given manifold is � sphere or not. However, an attempt to formulate � precise mathematical problem immediately calls forth very serious obstacles, because the necessary algorithm must deal with the "manifold codes". One manifold (including � sphere) can be represented by an infinite number of different codes (one object can be encoded in many ways). How can � computer find out if � given code represents the standard sphere? The problem of algorithmic recognition of the codes of � spe-
cific object within the set of all other codes is a difficult one. The reason is that one object (the sphere) can be "hidden" under various "masks" (codes). This can be conceived in the case of the homeomorphisms of an ordinary two-dimensional sphere. It is very djfficult to recognize that the complex figure in the foreground is, in reality, the usual sphere which has been subjected to � rather complex homeomorphism. The reader can assess the complexity of the recognition problem by this concrete example. 16) The vortices problem. Hamiltonian mechanics, symplectic geometry, differential equations. Many fundamental laws of physics and mechanics can be formulated through Hamilroniandifferential equations. The problem of finding their solution (i.�. integration) is one of the most topical problems in modern physics and geometry. This graphic piece is based on geometrical images connected with the vortices problem. � vortex is � singular point in the flow of fluids, in whose vicinity fluid begins to rotate, as shown in the picture. The fluid particles move along the spirals toward the vortex center or away from it. The vortices formed in the real at-
mosphere (including hurricanes and typhoons) are familiar to everybody. One of the mathematical models used in this field is the flow of fluids along � two-dimensional surface. If several vortices are formed in the fluid, they will interact with each other, moving along the surface by rather complex trajectories. The interaction of at least two vortices is shown in the picture. In some specific cases, "the vortices problem is integral" (the
motion of the vortices' centers can be described by some "formulae"). 17) Singular points of the algebraic surfaces. Algebraic geometry. The landscape portrayed in the graphic piece is "woven" from � large number of different examples of algebraic surfaces. An algebraic surface can be set into the three-dimensional space by � polynomial equation (i.�. as � level surface of � polynomial equation). The singular points of such surfaces (the points of singularity of the surfaces) are of utmost importance. These singularities can be (though this problem is � very difficult one) classified. Typical singularities resemble spikes, beaks, blades (some of them can be seen in the picture). These points appear quite naturally in geometrical optics, in the problem of propagation of wave fronts, in the theory of minimal surfaces, etc. 18) The spatial �-bodies problem in celestial mechanics. Celestial mechanics and geometry. In the first approximation, it can be assumed that the real planets of the solar system, asteroids, etc., move in one plane called the ecliptic plane. And it
may be assumed that the center of mass in this system coincides with the sun. The motion of this system is regulated by the Newtonian potential, according to the laws of classical mechanics. The evolution of the whole _n_-bodies system is determined by the initial conditions; it is necessary to set the positions of the gravitating masses and their velocities at the initial moment. The general solutions in these types of equations are known to be very complex. For example, according to the Poin-
care theorem, the system does not allow additional analytical motion integrals. The problem of describing the three-dimensional motion of_n_ bodies, which are not situated in one plane, is even more difficult (see the picture). Each body is depicted as � heavy bird soaring in space not according to its own desires but in accordance with its mutual attraction to the other birds. The whole system is rotated around � common (approximately) center of mass, but some bodies can go away from the center, while others can approach it (or even "fall upon" the center of mass), etc. From the mathematical viewpoint, the _n_-bodies problem is one of the most interesting ones, because it lays on the border of such sciences as geometry, celestial mechanics, theory of differential equations, etc. 19) The Jacoby fields and conjugate points on the geodesic lines. Calculus of variations and geometry. The landscape consists of several mathematical objects. The central one is the fluorescent sphere with two "wings" at the top left. The border of this luminous cloud consists of several curved arcs resembling � bent bow. The whole figure can
be obtained as � result of sequential bending of the bow. This picture is familiar to all scientists dealing with the idea of conjugate points along the geodesic lines. Geodesic lines are lines of minimum length (local). And as portrayed in the picture, after the infinitesimal bending of � geodesic line, which conserves its main property, "sectors" are formed. In such bendings, some points of � geodesic line may remain immobile. These are the conjugate points. They can be formally determined as the points where the Jacoby fields (defined along the geodesic lines) approach zero. There can be several Jacoby fields. Their number is the index of the geodesic line. The light disk or ball in the center of the sector represents the ball, ortogonal to the geodesic line in its middle (between two adjacent conjugate points). It consists of the vectors of all the Jacoby fields which can be set along the given geodesic line. The dimension ofsuch � ball (disk) is equal to the geodesic index. 20) Minimal cones ���� ��� minimal manifolds. Topology of the minimal surfaces. This piece is saturated with various mathematical images.
Among these, we would like to note the singular points of analytic functions: the billiard problem (the problem of ideal balls in motion in the areas of various shapes, the balls being reflected from the wall in accordance with the classical reflection law: the angle of incidence is equal to the angle of reflection); the operation of cutting � surface (and the inverse operation of pasting the cut edges) etc. But the central image is an infinite sequence of bent cones portrayed as marquees or funnels turned upside down. The vertex of each cone is furnished with � figure corresponding to the type of singularity of this point. Minimal cones naturally appear in the theory of minimal surfaces. These minimal cones approximate these minimal surfaces within the vicinity of the singular points. The types of such cones are studied in the special division of calculus of variations. As the "cones' base", not only can the usual sphere be used, but also far more complicated manifolds. The ���� ���plicated � manifold, themore complicated the cones's vertex. 21) Gauss curvature and the mean curvature of the surface. Theory of surfaces. One can distinguish the regions, which consist of points, where the Gauss cur-
vature is positive (cap), negative (saddle), or zero (cylinder surface). For example, the right vertical part of the surface consists of points where the Gauss curvature is negative. On the whole, the object's structure ��sembles � helicoid, � surface formed by sliding � rotating straight line along another, immobile, straight line (which is perpendicular to the first one). � surface similar to � screw surface is formed (each point moves along � screw spiral). ����, the surface is partitioned into layers which set the fibrarion structure. Brief information about the author.Fomenko, Anatoly Timofeevich, Doctor of Sciences (Mathematics), Professor of the Department of Higher Geometry and Topology, Faculty of Mathematics and Mechanics, Moscow State University. � well-known expert in the field of geometry, including computer topology, calculus of variations, and Hamiltonian mechanics. It was Fomenko who solved the famous spectral multidimensional Plateau problem in the theory of minimal surfaces.
�� created the theory of topological classification of the Hamiltonian differential equations. Fomenko is � prize-winner of the Moscow Mathematical Society (1974) and of the Presidium of the Academy of Sciences of the USSR (1987). �� is also the
author of ���� than 130 mathematical works and 11 books (monographs, textbooks) in mathematics and its applications. All these books have been translated into English by principal foreign scientific publishers. In 1991 he became correspondence member of the Academy of Sciences of the USSR.
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