A Guide to the Russian Academy of Sciences - University of Texas ...

3. Department of Algebra: The Department of Algebra was organized and for a long
time headed by B. N. Delone. Due to his influence, the Department is still active in
the theory of algebraic numbers and diophantine equations. Since 1960, the Head
of Department has been I. R. Shafarevich. Since about the same time, the
Department became active in algebraic geometry. The Department staff includes: A.
I. Kostrikin, A. N. Parshin, A. N. Turin, V.A.Kolyvagin, V. N. Nikulin, S. A.
Stepanov. At the Department, A. I. Malte, S. A. Chunikhin, I. M. Gel'fand, S. P.
Demushkin, S. P. Novikov, A. I. Lapin, and M. M. Kapranovalso worked at
different times.
There are six main topics actively developed in the Department, namely:
The arithmetic of fields of algebraic numbers. The most famous results obtained
here are the solution of the inverse problem of Galois theory for solvable groups,
the investigation of the Galois groups of fields of p-adic numbers, the solution of
the problem of class field tower.
The theory of Lie groups and algebras. At the time when A. I. Maltsev worked at
the Department he obtained results on semi-simple subgroups of semi-simple Lee
groups, the theory of nimanifolds of the Maltsev completions of groups. When I.
M. Gel'fand worked at the Department, the Gelfand-Naimark theory of infinite
dimensional representations of classical Lee groups was developed.
The theory of finite groups and modular Lee algebras. The limited Bernside
problem was solved and the basis of the theory of simple Lee p-algebras was
constructed.
The arithmetic of algebraic manifolds: the theory of elliptic curves, the
multidimensional class field, the arithmetic of algebraic and arithmetic surfaces,
``the Arakelov geometry''.
Algebraic geometry: the solution of the Luroth problem, the theory of K3 type algebraic
surfaces.
Problems of mathematical physics connected with algebraic geometry: instant
on theory, gauge fields theory, super-geometry. Of course some investigation do
not fit in the outlined directions. Recently also new directions of investigations
appeared: problems of homological algebra; the classification of multidimensional
algebraic manifolds; the theory of nilpotent algebras; and the study of the Galois
groups of the algebraic closure of fields of a finite type.
Many young mathematicians have been trained at the Department. The present members of
the Department have advised more then 75 candidates and 30 doctors of Physical-
Mathematical sciences.
4. Department of Geometry and Topology: The Department of Geometry and
Topology at the Mathematical Institute was organized in 1939 and the first Head of
Department was L. S. Pontryagin. P. S. Aleksandrov joined the Department at that
time.
Topology became an independent branch of science and the organization of the Department
was a reflection of it. The main topics investigated at the Department were
differential topology (L. S. Pontryagin) and general algebraic topology (P. S.
Aleksandrov). Even though the Department was rather small, investigations in all of
the main directions of the development of topology were carried on.
After the War, a large number of talented mathematicians joined the Department. They were
L. V. Keldysh, (Keldish) who actively studied problems of geometric topology,
and L. S. Pontryagin's and P. S. Aleksandrov's students N. A. Berikashvili, V.
G. Boltyanskii, R. V. Gamkrelidze, M. M. Postnikov, K. A. Sitnikov . At the
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