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Geometry, Topology, and Analysis
Geometric research at the Sobolev Institute
of Mathematics is carried out in the direction established by Academician
A. D. Alexandrov.
This direction is characterized by choice of the problems concerning various
geometric objects, study "in the large" characteristics of a
given geometric object. Geometrical studies at the Institute deal with
Alekxandrov's theory of surfaces, extremal problems of isoperimetric type,
metric spaces of curvature bounded above, the theory of irregular curves,
the theory of Riemann spaces of curvature bounded below, etc.
Topological studies began with the arrival of representatives of the
Moscow topological school V. I. Kuz’minov, I. A. Shvedov,
and L. N. Ivanovskii at Novosibirsk in the 60-s. They are connected
with dimension theory, homological algebra, the theory of Lp-cohomologies,
algebraic topology, topology of small dimensions, and braid groups on
3-dimensional manifolds.
Principal directions of research in mathematical and functional analysis
are connected with the names of Academicians M. A. Lavrent’ev,
S. L. Sobolev, and L. V. Kantorovich.
M. A. Lavrent’ev is a founder of the theory of spatial quasiconformal
mappings, and S. L. Sobolev founded the theory of functions
with weak derivatives. These fields intertwine both the methods of research
and applications, which include the theory of Sobolev spaces, the theory
of mappings with bounded distortion, stability theory for classes of mappings,
analysis on nonholonomic structures, etc.
Research in functional analys which is being developed at the Institute
is originated by Academition L. V. Kantorovich, a Nobel prize
winner. The problems under study belong to nonstandard analysis, the theory
of dynamic systems, the theory of integral operators, and the theory of
ordered vector spaces.
The Department of Geometry and Analysis is headed by Academician Yu. G. Reshetnyak
who has founded a scientific school of geometry, topology, and quasiconformal
analysis in Siberia. The Department consists of four laboratories as follows:
geometry and theory of functions of real variables (headed by Yu. G. Reshetnyak),
topology and chronogeometry (headed by V. I. Kuz’minov), functional
analysis (headed by S. S. Kutateladze),
and mathematical methods in chemistry and biology (headed by V. A. Toponogov).
The Department works in close cooperation with the Chair of Calculus
of the Faculty of Mathematics and Mechanics of Novosibirsk
State University (the Chair is occupied by Academician Yu. G. Reshetnyak),
the Chair of Geometry and Topology (occupied by Professor V. I. Kuz’minov),
and the Chair of Higher Mathematics (occupied by Professor V. V. Ivanov).
There are over twenty doctors of science among the graduates supervised
by these chairs.
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