# A. S. MishchenkoLomonosov Moscow State University | MSU · Faculty of Mechanics and Mathematics

A. S. Mishchenko

PhD, Professor

## About

220

Publications

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Introduction

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September 1968 - present

## Publications

Publications (220)

A geometrical approach to the analysis of a data bank containing information on space structures of proteins is suggested. As a result of the analysis, several relationships concerning possible conformations are found, and also there is obtained a list of polypeptides whose structures differ essentially from typical concepts of the 3D-structure of...

In this paper, the functorial property of the inverse image for transitive Lie algebroids is proved and also there is proved the functorial property for all objects that are necessary for building transitive Lie algebroids due to K. Mackenzie—bundles L of finite-dimensional Lie algebras, covariant connections of derivations ∇, associated differenti...

In this paper we investigate the existence and classification of couplings between Lie algebra bundles and tangent bundles. We first give a sufficient and necessary condition on the existence of coupling between Lie algebra bundle (LAB) and the tangent bundle; i.e., we define a new topology on the group Autg of all automorphisms of the Lie algebra...

In this paper we investigate the existence and classification of couplings between Lie algebra bundles and tangent bundles. We first give a sufficient and necessary condition on the existence of coupling between Lie algebra bundle (LAB) and the tangent bundle; i.e., we define a new topology on the group of all automorphisms of the Lie algebra , den...

Recent work of Mishchenko and Morales Meléndez (arXiv:1112.2104 [math.AT], 2011) has shed new light on the classical exact sequence of Conner and Floyd in G-equivariant bordism for a finite group G. This paper is primarily an exposition of the results obtained there as specialized to finite groups. For a countable discrete group G, there is a brief...

Let g be a finite-dimensional Lie algebra and L be a Lie algebra bundle (LAB). A given coupling Ξ between the LAB L and the tangent bundle TM of a manifold M generates the so-called Mackenzie obstruction Obs(Ξ) ∈ H 3 (M; ZL) to the existence of a transitive Lie algebroid (K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, 2005, p. 27...

A geometrical approach to the analysis of a data bank containing information on space structures of proteins is suggested. As a result of the analysis, several relationships concerning possible conformations are found, and also there is obtained a list of polypeptides whose structures differ essentially from typical concepts of the 3D-structure of...

We prove that the homotopy classification is reduced to the
construction of the final space in the form of the classifying space BG,
where G is the group Aut(g)^δ of automorphisms of the adjoint Lie algebra
g with new topology thinner than the classical topology.
The description of the classifying space B g is reduced to classification
of coupling...

In our previous paper (arXiv:1306.5449) we have given a sufficient and
necessary condition when the coupling between Lie algebra bundle (LAB) and the
tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition
7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a
new topology on the group $\Aut(\rg)$ of all a...

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝn
to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus \(\mathbb{T}^{2n} \). This reduction en...

The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$
(K.Mackenzie,2005, Definition 7.2.2) plays the crucial role in the
classification of the transitive Lie algebroids for Lie algebra bundle $L$ with
fixed finite dimensional Lie algebra $\rg$ as a fiber of $L$. Here we give a
necessary and sufficient condition for existence such...

This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to...

In this work the equivariant signature of a manifold with proper action of a
discrete group is defined as an invariant of equivariant bordisms. It is shown
that the computation of this signature can be reduced to its computation on
fixed points sets equipped with their tubular neighborhoods. It is given a
description of the equivariant vector bundl...

The talk was done at the International Conference "Analysis, Topology and
Applications", Harbin, China, 23.08.2011. Transitive Lie algebroids have
specific properties that allow to look at the transitive Lie algebroid as an
element of the object of a homotopy functor. Roughly speaking each transitive
Lie algebroids can be described as a vector bund...

D.Sullivan (1977) (see also the book by H.Whitney "Geometric Integration
Theory",1957) considered a new model for underlying cochain complex for
classical cohomologies with rational coefficients for arbitrary simplicial
spaces that gives the isomorphism with classical rational cohomologies. We
apply the key ideas developed by K.MacKenzie (2005) and...

We investigate orthonormality-preserving, C⁎-conformal and conformal module mappings on full Hilbert C⁎-modules to obtain their general structure. Orthogonality-preserving bounded module maps T act as a multiplication by an element λ of the center of the multiplier algebra of the C⁎-algebra of coefficients combined with an isometric module operator...

It is well-known that distinct biological indices (analytes) have distinct
variability. We try to use some mathematical algorithms to pick out a set of
blood parameters which give an opportunity to retrieve the initial volume of
the blood spotted, and use it to calculate exact concentrations of analyts
interesting to a physician. For our analysis w...

This paper precludes the proceedings of the conference “Geometry and Operator Theory” dedicated to N. Teleman’s 65th birthday
celebration (Ancona, September 2007)

The Fredholm representation theory is well adapted to the construction of homotopy invariants of non-simply-connected manifolds by means of the generalized Hirzebruch formula [σ(M)] = 〈L(M)chA
f*ξ, [M]〉 ∈ K
A
0(pt) ⊗ Q, where A = C*[π] is the C*-algebra of the group π, π = π
1(M). The bundle ξ ∈ K
A
0(Bπ) is the canonical A-bundle generated by the...

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ([10], 1972) and Gromov ([4], 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In [6], we gave a signature op...

We give a description of the vector $G$-bundles over $G$-spaces with quasi-free proper action of discrete group $G$ in terms of the classifying space.

The signature of the Poincaré duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W=V⊕V∗ where V is abstract linear space wi...

In memory of Yurii Petrovich Solovyev - Volume 2 Issue 2 - A. S. Mishchenko, Th. Yu. Popelensky, E. V. Troitsky

It is well-known that bounded operators in Hilbert C *-modules over C *-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C *-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.
Classical Fredhol...

A -algebra servicing the theory of asymptotic representations and its embedding into the Calkin algebra that induces an isomorphism of -groups is constructed. As a consequence, it is shown that all vector bundles over the classifying space that can be obtained by means of asymptotic representations of a discrete group can also be obtained by means...

In this paper natural generalizations are developed of the theory of elliptic operators invariant under the action of a C*-algebra. The theory of compact and Fredholm operators acting in spaces of the type of a Hilbert space over a C*-algebra is developed. A formula of the Atiyah-Singer type for elliptic operators over a C*-algebra is developed.
Bi...

A full description of the ring of unitary bordisms with the action of
Zp and of the ring of all
admissible collections of fixed submanifolds under the action of Zp is given in terms of generators
and relations. The calculations are based on a special choice of polynomial generators in the ordinary
ring of unitary bordisms and Poincaré duality in b...

A description of Fredholm representations as a particular case of graded representations is given. For graded representations of Banach algebras Hirzebruch type formulas are deduced by the method of bordism theory and the theory of bundles of algebraic Poincaré complexes.
Bibliography: 13 titles.

The homotopy invariance of the higher signatures of nonsimply connected manifolds is proved in this paper. The method of proof is based on the study of absolute invariants of nonsimply connected manifolds similar to algebraic K-theory and on the construction of an analog to intersection theory for Poincaré complexes.

We prove a formula that expresses the square of the statistical sum for the two-dimensional Ising model for an arbitrary plane lattice in terms of the determinant of a matrix similar to the Kac-Ward matrix.

We present an analytic definition for the relative torsion for flat C * -algebra bundles over a compact manifold. The advantage of such a relative torsion is that it is defined without any hypotheses on the flat C * -algebra bundle. In the case where the flat C * -algebra bundle is of determinant class, we relate it easily to the L 2 -torsion as de...

These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica,
Poland, “Geometry and Topology of Manifolds”, May, 2–8, 2004.

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [2]. Using a natural construction of [1], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles ov...

The Evens-Lu-Weinstein representation (Q
A
, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q
Aor, Dor) by tensoring by orientation flat line bundle, Q
Aor=QA⊗or (M) and D
or=D⊗∂
Aor. It is shown that the induced co...

The Evans-Lu-Weinstein representation (QA,D) for a Lie alge- broid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to Qor A ,D or by tensoring by orientation flat line bundle, Qor A = QA or (M) and Dor = D @or A. It is shown that the induced c...

In contemporary Russian, the word “model” is often associated with a showing of fashionable clothes, that is, with something purely external having no more purpose than to decorate the real human essence. Here, we discuss the very opposite aspect of this notion. When speaking of mathematical models, we imply a speculative construction designed to e...

The complete version of this abridged paper has been published and reviewed in [Sb. Math. 194, No. 7, 1079–1103 (2003); translation from Mat. Sb. 194, No. 7, 127–154 (2003; Zbl 1074.58008)].

It is proved that for any transitive Lie algebroid L on a compact oriented connected manifold with unimodular isotropy Lie algebras and trivial monodromy the cohomology algebra is a Poincaré algebra with trivial signature. Examples of such algebroids are algebroids on simply connected manifolds, algebroids such that the outer automorphism group of...

A class of Lagrangian manifolds is presented, together with a groupoid of symplectic transformations preserving the class of Lagrangian manifolds, for which the local property of stability is related to the global property to define an essential point on the corresponding Lagrangian manifold.

The signature of the Poincare duality of compact topological manifolds with local system of coecients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V ' V ⁄ where V is abstarct linear space...

We develop a theory of almost algebraic Poincaré complexes to write an analog of the Hirzebruch formula with nonflat coefficients for combinatorial manifolds.

An almost representation of a group is a map from this group into the unitary group of a Hilbert space, such that the group relations hold only approximately. We give a survey of the recent results on almost representations and on their relations to asymptotic representations. Applications to K-theory of classifying spaces are also discussed.

The -particle problem of the Schrodinger-Laplace-Beltrami equation on a manifold with an arbitrary interaction potential between particles is studied. A pseudodifferential operator on the manifold is obtained that describes the energy level of the Hamiltonian for a self-consistent field. The equations for a quasi-particle are the variational equati...

If one has a unitary representation p : π →U(H) of the fundamental group π1(M) of the manifold M, then one can do many useful things:
1)
construct a natural vector bundle over M;
2)
construct the cohomology groups with respect to the local system of coefficients;
3)
construct the signature of manifold M with respect to the local system of coefficie...

In the last few years the use of geometric methods has permeated many more branches of mathematics and the sciences. Briefly its role may be characterized as follows. Whereas methods of mathematical analysis describe phenomena ‘in the small’, geometric methods contribute to giving the picture ‘in the large’. A second no less important property of g...

To extend the study of the properties of the vector bundles we need further geometric ideas and constructions. This chapter is devoted to the most frequently used constructions which lead to deeper properties of vector bundles. They are Bott periodicity — the main instrument of the calculation of K-theory, linear representations and cohomology oper...

In this chapter we shall give some methods for describing the K-groups of various concrete spaces by reducing them to a description in terms of the usual cohomology groups of spaces. The methods we discuss are spectral sequences, cohomology operations and direct images. These methods cannot be used universally but they allow us to use certain geome...

In the section 1.2 of the chapter 1 (Theorem 3) it was shown that if the base of a locally trivial bundle with a Lie group as structure group has the form B x I then the restrictions of the bundle to B x {0} and B x {1} are isomorphic. This property of locally trivial bundles allows us to describe bundles in the terms of homotopy properties of topo...

Here we describe some problems where vector bundles appear in a natural way. We do not pretend to give a complete list but rather they correspond to the author’s interest. Nevertheless, we hope that these examples will demonstrate the usefulness of vector bundle theory as a geometric technique.

The definition of a locally trivial bundle was coined to capture an idea which recurs in a number of different geometric situations. We commence by giving a number of examples.

In this chapter we describe some of the most fruitful applications of vector bundles, namely, in elliptic operator theory. We study some of the geometrical constructions which appear naturally in the analysis of differential and pseudodifferential operators on smooth manifolds.

We prove that for matrix algebras $M_n$ there exists a monomorphism $(\prod_n M_n/\oplus_n M_n)\otimes C(S^1) \to {\cal Q} $ into the Calkin algebra which induces an isomorphism of the $K_1$-groups. As a consequence we show that every vector bundle over a classifying space $B\pi$ which can be obtained from an asymptotic representation of a discrete...

A notion of family of Fredholm representations controlled at infinity and a new topology of the space of Fredholm representations which differs from the Kasparov one are introduced. These are used to obtain a new proof of Novikov’s conjecture on the homotopy invariance of higher signatures for complete non-positively curved Riemannian manifolds. Th...

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