Vagner's interests were, at this time, mainly in theoretical physics. He was particularly attracted by the theory of relativity and he now asked Igor Tamm, a professor in Moscow, if he would supervise his doctoral studies in that topic. Certainly Tamm was a good person to talk to - he was educated at the University of Edinburgh, then moved to the Moscow University, graduating in 1918. In 1928 Tamm spent a few months with Paul Ehrenfest at the University of Leiden. As a mark of his quality we note that Tamm was Nobel Laureate in Physics for the year 1958. Although Tamm was very interested in the theory of relativity, he was not allowed to have students in this field. Of course, this was a political decision by the Soviet government who had decided that relativity was not a proper science. Tamm had to supervise students on the physics of metals but after having discussions with Vagner he clearly saw how the young student had his heart set on studying relativity. Tamm told Vagner (see ):-
I hope this craziness will pass over. I can wait but you are young and you can't wait, these are your best years. Go and study differential geometry under Professor Kagan. The very spirit of modern geometry is close to that of relativity. They [the Government] think they understand physics and tell us physicists what to do. However, even they don't dare to tell mathematicians what to do. The atmosphere is better in mathematics.So Vagner tool Tamm's advice and approached Benjamin Fedorovich Kagan. In 1922 Kagan had moved from Odessa to Moscow when the Department of Differential Geometry was founded at Moscow State University. Kagan was the first Head of Department and he founded an important School of Differential Geometry there. Vagner became his student in 1932 and wrote a thesis on the differential geometry of non-holonomic manifolds for his Candidate's Degree (equivalent to a Ph.D.). In an unprecedented move, the Scientific Council of the Physics and Mathematics Faculty of Moscow University decided that the quality of his thesis was so high, so far beyond that required for the Candidate's Degree, that in 1935 he was awarded the degree of Doctor of Science (equivalent to the German habilitation). Vagner's first paper Sur la géométrie différentielle des multiplicités anholonomes Ⓣ was published in 1935 followed, remarkably, by eight papers in 1938 including A generalization of non-holonomic manifolds in Finslerian space.
Vagner was appointed to the Chair of Geometry at Saratov University after the award of the degree of Doctor of Science and he continued to work there until he retired in 1978. Boris Schein writes about his research in :-
Vagner started his research activity at the time when differential geometry was rapidly developing and providing a part of mathematical apparatus for general relativity. At that time quite a few people believed that the importance of new geometric theories, having a general scientific character, transcended mathematics. All Vagner's research is connected with differential geometry and algebraization of its foundations. Algebraic systems considered by Vagner were usually related to differential geometric structures. His research activities were connected with the Seminar on Vector and Tensor Analysis at Moscow University.Among Vagner's early papers we mention Differential geometry of non-linear non-holonomic manifolds in the three-dimensional Euclidean space (1940), The geometry of an (n-1)-dimensional non-holonomic manifold in an n-dimensional space (Russian) (1941), Geometric interpretation of the motion of non-holonomic dynamical systems (Russian) (1941), On the problem of determining the invariant characteristics of Liouville surfaces (Russian) (1941), and On the Cartan group of holonomicity for surfaces (1942). He published a major 70 page paper General affine and central projective geometry of a hypersurface in a central affine space and its application to the geometrical theory of Carathéodory's transformations in the calculus of variations (Russian) in 1952. This is described by Struik as follows:-
The first sections deal with the contact of arbitrary order of m-dimensional surfaces in a central-affine En and its osculating hypersurfaces of given order and class. Then the influence of projective transformations in the En is studied. The fourth section is a discussion of the hyperquadrics of Darboux. The next sections deal with the affine and central-projective normals of a hypersurface and the general theory of hypersurfaces in a central affine En under transformations of the affine and central-projective group. The theory is applied to affine hyperspheres (all normals through one point) and hyperquadrics (Darboux tensor vanishes). The paper ends with the general theory of curves under the same groups.Vagner published the book Geometria del calcolo delle variazioni Ⓣ in Italian in 1965 in which he gave a systematic treatment of his own approach to the geometry of the calculus of variations, which he developed during the years 1942-1952.
The quote by Schein above indicates how geometry led Vagner to study algebraic systems. Let us quote Vagner's own words from the paper The foundations of differential geometry and modern algebra (Russian) (1963):-
For contemporary differential geometry the concept of group is quite insufficient for the examination, from an algebraic point of view, of the basic concepts of the corresponding geometrical theories. Moreover, algebraic problems arising in investigations concerning the foundations of contemporary differential geometry require the study of special algebraic systems which at present are not very seriously discussed.This led Vagner to investigate inverse semigroups which he was the first to introduce (although he did not use that name) in the paper Generalised groups (Russian) paper in 1952. He also published On the theory of partial transformations (Russian) in 1952 and then the major 90 page paper The theory of generalized heaps and generalized groups (Russian) in the following year. Schein writes:-
In my opinion, even now, almost 30 years after its first publication, this paper is grossly under-estimated (probably because a good many of researchers in inverse semigroups cannot read this paper in the original Russian). This paper contains a wealth of results, some (but only some!) of which have been rediscovered later (importance of the natural order relation, the smallest group congruence, the greatest idempotent-separating congruence, etc.). Now the theory of inverse semigroups is one of the most vital and important parts of semigroup theory.Schein gives a very valuable description of Vagner as a person in :-
Personally, Vagner was a very agreeable man. What always surprised his interlocutors was his vast erudition. He could read in practically all European languages and even managed somehow to get books in foreign languages. Even specialists sought Vagner's advice in various questions of philosophy, history, linguistics, literature.
I remember how, when I was a sophomore (and later), I was always accompanying Vagner, who used to walk home after his classes, and I visited his bachelor's apartment almost weekly. The amount of topics we discussed was innumerable. It was from him that I heard first about Freudian psychology (a taboo topic; S Freud's books were on the index and could not be borrowed in the libraries), K Menger's dimension theory (luckily, Menger's works might be read and I did that), the Vienna Circle of philosophers, R M Rilke's poetry, German expressionist fiction, Schopenhauer's philosophy, comparative studies of Indo-European languages, and many many other things. I often wondered how it was in the pre-Gutenberg era, before printing presses were invented. I imagine that then such conversations as ours were the principal way of transfer of the knowledge from teachers to their disciples.
And also Vagner was a very decent man. To some people this statement may sound shallow, trivial, almost ridiculous. Blessed are the people who have never lived in circumstances when common human decency almost amounts to a heroic deed.
Article by: J J O'Connor and E F Robertson
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