Petr Konstantinovich Rashevsky (on his 60th birthday)

Russian original (PDF)

1968 YAHVAR-February vol. XXIII, item 1 (139) SUCCESS OF MATHEMATICAL LIFE IN THE USSR PETR KONSTANTINOVICH (on the sixtieth anniversary of his birth) July 27, 1967 was the 60th anniversary of the birth of the famous Soviet mathematician Pyotr Konstantinovich Rashevsky. Scientific creativity of Petr Konstantinovich is inexibly connected with the Moscow State University, where he began his career as a graduate student (1928), he was a professor of geology (1933) and a PhD student (1923) in Moscow State University.

In an article written on the occasion of the 50th anniversary of P. K. Rashevsky (UMN 13, item 1 (79) (1958), 225-231), the life path of P. K. Rashevsky is outlined and the first articles of P. K. Rashevsky described in detail his scientific work of the 30s and 40s. Given this, we consider it possible to limit the scope of this article to the activities of P. K. Rashevsky in the 50s and 60s. The scientific appearance of P. K. Rashevsky is characterized by an extreme diversity and breadth of scientific interests. The first articles of P. K. Rashevsky were related to the shadow of geometry.

However, they have been followed by works on differential equations and quantum statistics, Lie group theory and quantum electrodynamics, generalized function theory and homogeneous space geometry. In recent years, P. K. Rashevsky's scientific interests have focused on problems related to the study of Lie groups and homogeneous pro-travels; of the 25 papers he published since 1950, this topic has been devoted to 13. The series of papers [50], [51], [59] contains a description of all finite linear representations of Lie groups with a nilpotent radical, and, as a special case, all of which are realized in the latter.

systems of magnitudes of the form Zj\\\\\'.jvq, where each "superindex" / or / runs all possible combinations with repetitions (c), (iu i2), . . . . ., (h, &2, . . . . ., h) from ordinary indices 1, 2, . . . . ., n taken in the amount of 1, 2, . . . . . . . ., v;

230 MATHEMATIC LIFE In the USSR, this refers to the differential group DV1 operating on a multi-dimensional image n in the vicinity of any point. It has been shown that supertensors exhaust all linear differentiation cial-geometric objects. Again, to the question of linear representations, P. K. Rashevsky returns in the works [64], [66], but on a completely different basis. In these works, he introduces some new construction - the "associative differentiative differentiation of cial-geometric objects."

This is an asso ciative algebra, which is a replenishment (according to a certain system of norms) of the ordinary associative shell (or, what is the same, the universal envelope of algebra) U(G) of the Lie G algebra. Polynomials from the basic vectors of Lie 6? algebra, of which U(G) consists, turn into power rows with certain restrictions on the growth of coefficients: roughly speaking, these power series correspond to entire analytical functions from g complex arguments zu z + z + z .

Thus, the supershell F(G) is as if an intermediate link between the algebra of the roy Li G and the corresponding Li ^ group, combining the features of algebra and analysis. If you interpret the elements of the Li G algebra as operators of left shifts on the Lie group */, then each element / £ F(G) becomes a “dif ferential operator of infinitely large order” acting on &. Exploring the resulting representation of the algebra F(G), P. K. Rashevsky establishes a natural correspondence between his unrepresented representations of the left shifts on the Lie group * / / / / / G group and the maximum closed ideals.

The study of the above “regular” representation of the algebra F(G) should be considered a natural intermediate step before turning to the more difficult and important task of studying any infinite-dimensional representations of the algebra F(G), and with it the representations of the Lie group.

In the second of them, the conditions under which the homogeneous space M = 'SISB (S - a group of motions, SB - a stationary subgroup) allows a projective model are clarified; the latter means that M is realized as a sub-manifold in projective space, and & - as a sub-group of a projective group translating this sub-manifold into itself. A significant achievement is the research of P. K. Rashevsky on the theory of symmetric spaces. These studies are stimulated by the fundamental works of E. Cartan; we note in passing that a number of E. Cartan's articles in the Russian translation of the projective group, which is published in the collection of Liebrymmetrical spaces in the book "Kras" by the book "Krazemmetrical space".

The theory of symmetric spaces (Riemannian or affine connectivity) was a true discovery in geometry; the richness of connections with algebra, functional analysis, the theory of functions of a complex variable, etc., has been discovered, perhaps, in geometry, nothing equal.

231 Symmetrical space in the sense of E. Cartan is a space of affinity connection without torsion, the tensor of curvature of which is maintained in parallel transfer. The name "symmetric" is associated with one important geometric property of such spaces, which can be taken as a definition: geodetic symmetry with respect to any point is the automorphism of space, that is, such a transformation at which the given affinity of connection passes into itself.

Rashevsky introduced a new class of spaces, which are spaces of affinity torsion, in which, in parallel transfer, both the curvature tensor and the torsion tensor are preserved; these spaces he called symmetric spaces with torsion and dedicated to them work [43], [45], [46]. The last two of them, published under the name "On the geometry of homogeneous spaces," have a special meaning. In them, for a very wide class of homogeneous spaces, the uniform method reveals the geometry inherent in these spaces, this connection with spaces, namely, this geometry, this connection with space, and in general, it turns out to be the geometry.

Unfortunately, these works by P. K. Rashevsky were not properly appreciated at the time; his results were rediscovered by K. Nomizu and have since become one of the class spaces. The corresponding class of homogeneous spaces, called “reductive spaces”, has proved extremely useful for many studies on homogeneous spaces. P. K. Rashevsky’s work on mathematical questions of quantum electrodynamics stands out somewhat separately. People who know Peter Constanti novicz, know how much energy, mental and intellectual energy P. K. Rashevsky has invested in studying these questions.

However, experts also know how difficult this branch of science is, stubbornly not given to the efforts of the most sophisticated minds. And although P. K. Rashevsky’s ideas in this field have not been implemented, a well-written review [60] can serve as a well-known compensation, in which the problem of substantiating quantum electrodynamics was brought to the point that is only possible at the moment.

Rashevsky can serve as an example of a thoughtful exposition, and his large reviews [48], [53], [60] can be quite considered as complete chapters of textbooks in the relevant disciplines. The same high methodological and literary merits have the books of P. K. Rashevsky, which fully reflect the clarity of the scientific thought of their author, unquestionably related to the “classes of sicas” (not “romantics”), if we proceed from the well-known classification of scientists belonging to V. Oswald.

232 MATHEMATIC LIFE In the USSR, as the head of the Department of Differential Geometry of Moscow State University, the driver of scientific seminars and the editor-compiler of the “Proceedings of the Seminar on Vectorial and Tensor Analysis” at Moscow State University. It is well known that geometry is currently experiencing a certain crisis associated with the once fascination with the concepts of local differential geometry and the penetration of new points of view, dictated to some extent by topology and related to the global study of differential-geometric objects.

Rashevsky is not inclined to turn a blind eye to modern trends in the teaching of geometry. He made detailed proposals for restructuring the uni versithete course of geometry on the basis of the theory of smooth manifolds and modern approaches to differential-geometric structures. From the author of well-deserved textbooks on differential geometry and Riemannian geometry, reflecting the pedagogical attitudes of the previous period, such a speech undoubtedly required some courage.

The same tendency to “modernize” the activities of the department can be found in the dissertation topics of postgraduate students of the Department of Differential Geometry, in the lists of special courses read by the Department. It seems instructive to us that the comparison of the table of contents of the following volumes of the “Proceedings of the Seminar on Tensor and Vector Analysis at MSU” with the table of contents of the first volumes, published under the editorship of V. F. Kaga on, almost completely disappeared articles on local differential geometry, and their place was taken by the same-geneous geometry in the modern understanding of these words.

This theme prevails in the activities of the head of the seminar P. K. Rashevsky on the theory of homogeneous spaces on fur of the Moscow State University Faculty of Mathematics, which was a real school for many young and not very young geometers. For the participants of this seminar, scientific principle, deep understanding of mathematical problems and merciless clarity of thought of his leader. We wish Peter Konstantinovich further success in his many facet scientific, pedagogical and literary activities.

8, 82-92. 44. On the pair of connections on ^-dimensional surfaces in a 2p-dimensional layered space, Proceedings of Seminary Vector and Tension Analysis, item 8, 301-313. *) The beginning of the list is placed in UMN 13, item 1 (79).

MATHEMATIC LIFE In the USSR 233^ 1951 45. On the geometry of homogeneous spaces, DAN 80, 169-171. 1952 46. On the geometry of homogeneous spaces, Proceedings of Seminary Vector and Tensions. analysis, note 9, 49-74. 1953 47. Rimanian geometry and tensor analysis, M., Gostehizdat, 635 pages. (ed. 2-1964; ed. 3-1967 48. On some major theoremic groups of Liema, 581-97, 540 AN, 4000, 4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

1955 Theory of Spinors, UMN 10, pp. 2, 3-110. 54. Multidimensional B-functions and differential-geometric objects, UMN 10, pp. 4, "145-152. 1956 55. Course of differential geometry, ed. 4, M., Gostechizdat, 420 pages 56. Linear semisimple group as a group of invariance of the fourth valence tensor, Processes of vector analysis III., 1956, "II", "Tech" and "Tech" from 10, M., 57, Moscow, 5-1, Moscow, Russia, Russia, 142, Russia, Russia, Russia, 1 1 1 1 1 1 1 1 1 - .

On Linear Representations of Differential Groups and Lie Groups with Nilpotent Radical, Proceedings of the Moscow Mathematical Island 6, 337–370. 1958 60. On the Mathematical Basics of Quantum Electrodynamics, UMN 13, pp. 3, 3–110. 1960 61. Geometry and its Axiomatics, Mathem. Enlightenment, pp. 5, 73–98. 62. On Projective Representations of Homogeneous Spaces, Mathem. 50 (92): 2, 171–202.63.1961.

234 MATHEMATIC LIFE In the USSR 1963 64. Associative supershell of Lie algebra and its infinite-dimensional representations in the space of analytical sprouts, DAN 151, No. 4, 778-780. 1965 65. On closed ideals in one countable algebra of entire analytical functions, DAN 162, No. 3, 513-515. 1966 66. Associative supershell of Lie algebra, its regular representation and ideals, Proces Mosk., No. 275-17, DM., No. 775, D., No. 776, D. - D. D., No. 177, No. ^7, D. - D. D. D. ^7, No. D. D., No., No. 177, No. D. D. D. D. D. D., No. D., No. 275, No. D. D., No. 775, D. D. D. D. D. D. D. D., No., No., No. 2