Publications and preprints - 2002
Victor M. Buchstaber, Dmitri V. Leykin.
Lie algebras associated with sigma-functions, and versal deformations.
Russian: ps
2 pages
Published in: Russian
Math. Surveys 57 (2002), no.3.
Victor Buchstaber, Elmer Rees.
Applications of Frobenius n-homomorphisms.
Russian: ps
2 pages
Published in: Russian
Math. Surveys 57 (2002), no.1, 148-149.
Ivan A. Dynnikov.
A new way to represent links, one-dimensional formalism and untangling
technology.
English: ps (gzipped)
41 pages
Published in: Acta Appl.
Math 69 (2002), no.3, 243-283.
Abstract. An alternative link representation different from
planar diagrams is discussed.
Isotopy classes of unordered nonoriented links are
realized as central elements of a monoid presented explicitly
by a finite number of generators and relations.
A new partial algorithm for recognizing is constructed.
Experiments show that the algorithm allows to untangle
unknots whose planar diagram has hundreds of crossings.
Here `to untangle' means `to find an isotopy to the circle'
Ivan A. Dynnikov.
On a certain Yang-Baxter map and Dehornoy's ordering.
Russian: ps
2 pages
Published in: Russian
Math. Surveys 57 (2002), no.3.
Ivan A. Dynnikov.
Finitely presented groups and semigroups in knot theory.
Russian: ps (gzipped)
tex
English: ps (gzipped)
tex
Published in: Proceedings of the Steklov Institute of
Mathematics 231 (2001).
Abstract. In this paper we construct finitely presented semigroups
whose central elements are in one-to-one correspondence
with isotopy classes of non-oriented links in three-space. Solving the
word problem for those semigroups is equivalent to solving
the classification problem for links and tangles.
Also, we give a construction of finitely presented groups
containing the braid group as a subgroup.
Dmitri V. Millionschikov.
Graded filiform Lie algebras and symplectic nilmanifolds.
Russian: ps (gzipped)
English: ps
20 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math.RA/0205042
Published in: Russian
Math. Surveys 57 (2002), no.2, 422-424.
Dmitri V. Millionschikov.
Cohomology with local coefficients of solvmanifolds and
Morse-Novikov theory.
Russian: ps
English: ps
11 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math.DG/0203067
Published in: Russian
Math. Surveys 57 (2002), no.4.
Victor Nistor, Evgenij Troitsky
An index for gauge-invariant operators
and the Dixmier-Douady invariant.
English: ps (gzipped)
28 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math.KT/0201207
Abstract. Let $\GR \to B$ be a bundle of compact Lie groups
acting on a fiber bundle $Y \to B$. In this paper we introduce
and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$.
These groups satisfy the usual properties of the
equivariant $K$-theory groups, but also some new
phenomena arise due to the topological non-triviality of the
bundle $\GR \to B$. As an application, we define a
gauge-equivariant index for a family of elliptic operators
$(P_b)_{b \in B}$ invariant with respect to
the action of $\GR \to B$, which, in this approach, is an element of
$K_\GR^0(B)$. We then give another definition of
the gauge-equivariant index as an element of
$K_0(C^*(\GR))$, the $K$-theory group of the Banach
algebra $C^*(\GR)$. We prove that
$K_0(C^*(\GR))\simeq K^0_\GR(\GR)$ and that the two
definitions of the gauge-equivariant index are equivalent. The algebra
$C^*(\GR)$ is the algebra of continuous sections of a
certain field of $C^*$-algebras with non-trivial
Dixmier-Douady invariant. The gauge-equivariant $K$-theory
groups are thus examples of twisted $K$-theory
groups, which have recently proved themselves
useful in the study of Ramond-Ramond fields.
Sergey P. Novikov.
On the exotic De-Rham cohomology.
Perturbation theory as a spectral sequence.
English: ps (gzipped)
18 pages
arXiv.org e-Print archive: http://arXiv.org/abs/math-ph/0201019
Abstract. This work is dedicated to some new exotic
homological constructions associated with the different Morse-type
inequalities for differential forms and vector fields.
It contains also survey of ideas developed by the present
author in 1986 for this goal.
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