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

Upper and lower estimates of complexity of proofs and their transformations in axiomatic
theories were obtained. For formal arithmetics the problem of construction of a
statement of some proof in the general case on the basis of its short enough proofs
in several particular cases was solved.
Methods of calculations on computers with controlled accuracy were developed. On the
basis of these methods a solution of D. Polya's problem about a proof without the
Riemann hypothesis of Turan's inequalities for Taylor's coefficient of the Riemannfunction
was found.
2. Laboratory of Number Theory: Staff: A. N. Andrianov, (Head of Laboratory), V.
A. Gritsenko, A. V. Malyshev, N. V. Proskurin, B. F. Skubenko, Iurii G.
Teterin, and O. M. Fomenko. Structural and analytic properties of solutions of
quadratic Diophantine equations were studied systematically. An algebraic
technique was developed, the technique of parabolic extensions of the Gekke rings,
which permitted to construct the basis of the theory of zeta-function of symplectic
group and to discover multiplicative expansions of integer solutions of quadratic
Diophantine equations in elementary solutions.
In the geometry of numbers fundamental results on integer minima of products of linear
polynomials were obtained and the well known Minkowski hypothesis was proved
for the dimension five.
3. Laboratory of Algebra: Staff: B. B. Venkov, (Head of Laboratory), V. V.
Ishkhanov, A. N. Kirillov, B. B. Lur'e, I. A. Panin, A. I. Skopin, and A. A.
Suslin. The problems of immersion in the Galois theory were investigated,
universally solvable immersion problems were studied, and a systematic study of
immersion problems with non-abelian kernel of order p was started.
A classification of all 32-dimensional even unimodular lattices was given. Representations
of quantum groups were studied with applications to the theory of combinations
and quantum field theory. The Milnor and K-theory of local fields as well as
connections between quadratic forms and the Galois cohomologies were studied.
K-theory and K-cohomology of different manifolds (classical groups, Grassman
manifolds, spaces of generalized flags) were calculated.
A computer study of transmetabelian groups of exponent 9 was carried out.
4. Laboratory of Geometry and Topology: Staff: O. Ya. Viro, (Head of
Laboratory), A. D. Aleksandrov, Iurii D. Burago, A. I. Degtyarev, V. A. Zalgaller,
A. A. Ivanov, N. V. Ivanov, V. G. Turaev. Topological and homotopic properties
of real algebraic manifolds were studied.
New algebraic and topological invariants of small dimension manifolds and low
dimensional knots and links were investigated. New topological invariants of threedimensional
manifolds of statistical sums type and the corresponding topological
quantum field theory of dimension 2+1 were constructed.
Algebraic aspects of bitopological structures and modular Teichmtler groups were studied.
A study of geometry of metrized manifolds and of geometric inequalities was carried out.
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