Publications and preprints - 2001


Victor M. Buchstaber, Elmer G. Rees.
The Gelfand map and symmetric products. 
English:  ps (gzipped)
9 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math.CO/0109122

Abstract. If A is an algebra of functions on X, there are many cases when X can be regarded as included in Hom(A,C) as the set of ring homomorphisms. In this paper the corresponding results for the symmetric products of X are introduced. It is shown that the symmetric product Symn(X) is included in Hom(A,C) as the set of those functions that satisfy equations generalising f(xy)=f(x)f(y). These equations are related to formulae introduced by Frobenius and, for the relevant A, they characterise linear maps on A that are the sum of ring homomorphisms. The main theorem is proved using an identity satisfied by partitions of finite sets.


Petr G. Grinevich, Sergey P. Novikov.
Topological charge of the real periodic finite-gap sine-Gordon solutions. 
English:  ps (gzipped)
30 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math-ph/0111039

Abstract. An effective description of the inverse spectral data corresponding to the real periodic and quasiperiodic solutions for the sine-gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proved. As it was understood already 20 years ago, it is very hard to extract any formula for this quantity from the Theta-functional expressions. A new method was developed by the authors for this goal. In the appendix 3 an analog of the Fourier-Laurent integral transform on the Riemann surfaces is defined, based on the constructions of the Soliton Theory. It is a natural continuous analog of the discrete Krichever-Novikov bases developed in the late 80s for the needs of the Operator Quantization of String Theory.


Dmitri V. Millionschikov.
Cohomology of nilmanifolds and Gontcharova's theorem. 
English:  ps
Russian:  ps
5 pages
Published in: Fern�ndez, Marisa (ed.) et al., Global differential geometry: the mathematical legacy of Alfred Gray, pp.381-385 Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001.

Sergey P. Novikov.
A note on the real fermionic and bosonic quadratic forms: their diagonalization and topological interpreation. 
English:  ps (gzipped)
9 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math-ph/0110032

Abstract. We explain in this note how real fermionic and bosonic quadratic forms can be effectively diagonalized. Nothing like that exists for the general complex hermitian forms. Looks like this observation was missed in the Quantum Field theoretical literature. The present author observed it for the case of fermions in 1986 making some topological work dedicated to the problem: how to construct Morse-type inequalities for the generic real vector fields? This idea also is presented in the note.