Petr Konstantinovich Rashevsky (on his 70th birthday)

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1977 September - October vol. XXXJI, item 5 (197) Success of Mathematical Sciences Mathematical Life In the USSR, Peter Konstantinovich Rashevsky was born in 1907 in Moscow; in the scientific career of Rashevsky (to the seventieth anniversary of his birth) July 27, 1977 was 70 years since the birth of the famous Soviet mathematician, the outstanding geometer Peter Konstantinovich Rashevsky was born in 1907 in Moscow; in the original P.

His scientific work 30-40 years was described in detail in the article dedicated to the 50th anniversary ^ see UMN 13:1 (79) (1958), pp. 225-231, and the period 1957-1967. reflected in the article, dedicated to his 60th anniversary (see UMN 23:1 (1968), pp. 229-234). Therefore, we focus on the activities of P. K. Rashevsky for the last 10 years, 1967-1977. The works of P. K. Rashevsky are distinguished by a wide variety of topics, depth of analysis, freshness of tasks.

All this fully applies to the works of 1967-1977, devoted mainly to the following issues: the problem of describing tensors that allow for this group of invariance; global properties of homogeneous spaces and their implementations (for example, in the form of projective models); algebro-topological properties of homogeneous spaces and groups of L and; the theory of group representations and Lie algebras. Much attention is paid to questions of mathematical physics (for example, problems of quantum electrodynamics) In the work of P. K. Rashevsky, a large place was occupied by the classical topic [68], the following problem belongs to them.

Let the Sp (or Ep (K), where K is an algebraically closed field of characteristic zero) set a system of tensors {Z} that allow for some group of invariance ®, and let a certain tensor W be given that allows the same group of invariance. The question is: how is the tensor W and the tensors of the system {Z} connected? It turns out that in some natural assumptions the answer exists and is formulated in terms associated with the system {Z}. This result allowed to describe the structure of many spherical functions on the same space, in terms, Ps, in terms of the same spaces.

Rashevsky has only recently completed a large study summarizing his work in the field of representation theory from 1963 to 1977. Let us give a brief overview of this cycle of results. In the work “Associative supershell of Lie algebra and its infinite-dimensional representations in the spaces of analytical sprouts” (DAN 151:4 (1963), pp. 778-780), a new object is being built – the associative supershell of Lie algebra (see also Proces.

206 MMO 15 (1966), pp. 3–54). Let f be Lie algebra over C, U(@>) be its wrapping algebra, i.e., associative shell. Through G we denote a complex Lie group with algebra f. In U((&) a topology is introduced using an explicitly specified set of norms. As a result of replenishment of U(%) in this topology and the natural continuation of algebraic operations in U($) - a supercarbohydrated (F&) - a topographicalgebracial, which is called by F&($)(($) (a) (abolic).

F($) is a complete reflexive counting space; conjugated to F((&) space F' is identified with the space of all holomorphic sprouts in the unit of group G. Elements £/(($) can be forged as left-invariant differential operators on G, and elements from F(@>) - as differential operators, generally speaking, of infinite order, acting in the space of shoots F'. A regular representation of F((&) in the space of shoots F' turns out there is a mutually unambiguous correspondence between the representations of this in the closed F($) ideals of F(%).

A closed subspace in F((&) is a right ideal if and only if it is an orthogonal addition to the subspace in F', an invariant with respect to the right infinitezymal shifts. The algebra F(((($)) introduced by P. K. Rashevsky was studied and compared with other topological algebras in the works of G. L. Litvinov, V. V. Varfolomeev, Goodman. The task of describing closed ideals BF(@) is very difficult for estimating functions in this case, where ⁇ 1(C) is satisfied with ⁇ 1 / ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , where ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇

The satisfactory estimation (1) for any 8 function forms a topological algebra. Operations are ordinary addition, multiplication of functions. The topology is given by the norms II / lie = Cmin DE ^min = lower edge of the constants C(f, 8) satisfying (1) at, data /, 8. The conjugated space F' is identified with the space of analytic sprouts q>(z) from one complex variable z. Theorem. Let Bg, b2, ” , ” then the sequence of J(b) , if the number of the root of the number of the j(b) |2.

(with a multiplicity not lower than their multiplicity, in the sequence {bp}), form a nontrivial closed ideal in F. Any nontrivial closed ideal in F can be obtained in the specified way. Corollary 1. Let /&"p is an ideal consisting of all functions /(£) F, turning to zero at the point q with a multiplicity not less than n. Such an ideal is primary in F, i.e. is contained in only one minimum ideal. It turns out that any nontrivial closed ideal in F is an intersection of no more than countable family ideal at 2 consequences.

The general view of the nontrivial closed subspace in F'r d invariant with respect to the operator is described as follows: this is the minimum closed-U/Z that subspace in F' stretched on the sprouts. (3) ebzf zebz, . . . . , zn~xebz, where Kommersant runs through the sequence {6g-} described in the the the theorem, n is the multiplicity of the number b in this sequence. Claims of spectral synthesis, like the consequence of the zebz zebz, . . . . . . . . .

Raszewski not only proved the assertion that “any

In the USSR, 207, the ideal is the intersection of primordial ideals," but also explicitly indicated the condition (2), necessary and sufficient to describe those sets of primordial ideals that give rise to a non-trivial ideal of a general form. In the theory of unitary representations, the analogy may be the difference between the proof of "Plancherel's abstract theorem" and the calculation of a specific measure of Plancherel for this group. The appearance of the described results stimulated the cycle of work of I.

Krasichkov-Ternovsky, where for a wide class of topological algebras formed by whole analytic functions of one complex variable that satisfy the growth conditions, a complete description of closed ideals in the spirit of P. K. Rashevsky's theorem is obtained. In P. K. Rashevsky's work "Description of closed invariant subspaces in some invariant spaces" (1977), the space L% of all possible odd measurable complex-valued functions f(x) on a real line satisfying the condition: oh2(h) = I | 2°(x)(h) = I(h = I(x)(x)(x = | 2°(x)(x)(x)(x)(x)(x)(2(x)

Then b% is the unification of the expanding sequence about Hilbert spaces L|. The subspace E* in L\ is called closed if E* p) b£ is closed in L£; \jk. P. K. Rashevsky described all closed nontrivial under spaces in L|, invariant with respect to the next generalized shift in the sense of Delsart-Levitan for all a £ R: (5) /(*)-+ y (/(*+a) + f(z-a)).

Any closed non-zero eigenspace E* a b% that withstands transformations (5), mutually uniquely corresponds to a finite or counting set of complex numbers {sp} (possibly with finite multiplicities \ p = ap - ibn, ap ^ 0 and bp ^ 0 at ap = 0), provided that this set satisfies the ratio: Y/1 > 0 for those p = ap - ibn, in which \ bp | < Z, after their renumbering in order of growth, ap - ^6 under the condition that {\n} meets {\n} {\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

stretched with a closure in b\ on the functions: (7) {xPsmcnx}, {xPcosCnx}, where Vcn> P runs even, a q - odd from the number of values 0, 1, . • ., gp - 1; Gp - multiplicity of p in {CJ}. Prisp = 0 multiplicity is even, equal to 2gp and instead of (7) take x, x3, . . . . . . 2g - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rashevsky to solve equivalent problems on the description of invariant subspaces in spaces of analytic functions with respect to some transformations. The resulting series of equivalent problems is solved with the help of fine analytical techniques. P. K. Rashevsky completely solved the problem of describing all closed bilateral invariant subspaces in LJ at p > 2. For other functional spaces on SL(2; R), a similar problem was solved in the cycle of works by L. Ehrenpreis and F. Mautner. P. K. Rashevsky not only considers other functional spaces, but solves the problem in other terms and solves

208 MATHEMATIC LIFE In the USSR, other methods that have brought the description of invariant subspaces in S.H. to complete. In addition, there is reason to expect that the methods of P. K. Rashevsky, and especially his reduction of the problem of describing invariant subspaces in S.H. to the corresponding problem for a generalized shift, allow you to advance in solving the problem for other semi-simple groups. P. K. Rashevsky for many years has been teaching special courses in representation theory at the Faculty of Mechanics and Mathematics, but only the latest achievements from the Faculty of Geometrics.

Although much attention has been paid to the topology of Lie groups in recent years by a wide variety of scientific schools, the Soviet mathematical literature lacks a comprehensive overview that provides clear evidence of the foundation of mental results from the theory of Lie cohomology and homogeneous spaces. This review was created by P. K. Rashevsky ([71]) This work not only contains the complete" known information about the material cohomologies of homogeneous spaces, but also includes a new, thoughtful presentation of the basic methodology that allows it (at least in principle) to calculate cohomologies of homogeneous space (abodity above R).

In the theory of symmetric spaces (for example, in their classification), a large role is played by the fact that the set of fixed points of involutive automor of a single-connected Lie group is connected. P. K. Rashevsky found ([73], [76]) that this fact is a special case of a very general statement associated with the deepest Lie groups, according to which the plurality of all points of a single-connected Lie group, fixed with any type of compact group, is always an automorphic, if it is a compact group.

For whom pact groups this statement was also proved by A. Borel. In [76] there is an unexpected result relating to the introduced by P. K. Rashevsky "autonomous covariant differentiation" - an operation defined on any smooth manifold and processing into itself the so-called supertensors. the Existence of such differentiation allows to introduce the concept of closed supertensors and to build "supertensor cohomologies" of diversity, which presuppose nothing for their definition except the assignment of a smooth structure on the diversity.

Rashevsky, the depth of his scientific penetration into the essence of the problem; all these qualities have had a great influence on many mathematicians who are either direct students of P. K. Rashevsky or participants in numerous seminars, which for many years led by P. K. Rashevsky. P. K. Rashevsky has educated a large number of specialists-geometers of our country; his students work in many scientific centers of the Soviet Union. P. K. Rashevsky sensitively reacts to new scientific trends in the field of education of young mathematicians: he is the initiator of the introduction of fundamentally new courses in the teaching system at the Mechanics and Mathemics Faculty of Moscow State University.

Among such courses, the Geometry course, which is now taught in junior courses for mathematics and mechanics, accumulates not only basic geometric facts, knowledge of which has been shown to be absolutely necessary for graduates of the faculty, but also the main scientific trends of recent years.

No article about Pyotr Konstantinovich Rashchevsky could be considered complete, if not to note in it some features of the personality of Pyotr Konstantinovich: the utmost scientific honesty, friendly attention to others (and especially young people looking for their way in science), in short, all that creates the charm of the personality of Pyotr Konstantinovich and forever attracts to him people who had the opportunity to communicate with him.

209 list of printed work in the USSR by P. K. Rashevsky* [68] On the structure of tensors admitting this group of invariance, DAN 177:2 (1967), 275-276. [69] Studies on differential geometry in the Moscow University in the Soviet period (to the 50th anniversary of Soviet power). Vestn. MSU, Mathematics, Mechanics, No 5 (1967), 12-23 (sovietary 25.69.), 201 MN, 290, TN, Ts, 270, Ts, and Ts, 771.

[73] On the Connectivity of the Multiple Points of the Lie Group, Unmoving in Its Automorphism, Funk. Analysis 6:4 (1972), 97–98. [74] On the Dogma of the Natural Series, UMN 28:4 (1973), 243–246. [75] Theorem on the Connectivity of the Subgroup of a Single-Connected Lie Group Permutual with Any Automorphism, Proceeds of MMO 30 (1974), 3–22. [76] Supertensor Analy of the De Rama Complex and the Pontryagin Classes, Processes of the Seminar named after Imbriagin, 243–246 (1977), 207, 207 ), Functs of the functional spaces (1977), Funcs, 207, 207, 207, 177, 177, 177, 177, 177, 177, 177, )

(x) The beginning of the list is published in UMN 13:1 (1958), 225-231; UMN 23:1 (1968), 229-234.