Alexander Mishchenko | Lomonosov Moscow State University (original) (raw)

Papers by Alexander Mishchenko

Research paper thumbnail of Adequacy of Mathematical Models in Control Theory, Physics, and Environmental Science

Technological Concepts and Mathematical Models in the Evolution of Modern Engineering Systems, 2004

In contemporary Russian, the word “model” is often associated with a showing of fashionable cloth... more 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 express the real essence of a phenomenon and the causes of the processes in question. It must be clearly perceived that all natural phenomena are interrelated. The modest abilities of the human intellect do not allow taking all relations into account. But, luckily, some relations are strong, and others are vanishing. So we have a possibility to reveal the main acting forces, discarding all that are of secondary importance. This is the common basic paradigm of all natural sciences, if we exclude their purely descriptive aspects. This paradigm preordains the leading role of mathematics in the process of designing and exploring models.

Research paper thumbnail of K-theory over C * -algebras

Geometry and Topology of Manifolds, 2007

Research paper thumbnail of K-theory from the point of view of C *-algebras and Fredholm representations

Central European Journal of Mathematics, 2005

These notes represent the subject of five lectures which were delivered as a minicourse during the

Research paper thumbnail of Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

Central European Journal of Mathematics, 2004

The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied i... more 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 cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid coho...

Research paper thumbnail of Relations between asymptotic and Fredholm representations

Research paper thumbnail of Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерности

Matematicheskie Zametki, 2015

Общероссийский математический портал А. А. Арутюнов, Редукция нелокальных псевдодифференциальных ... more Общероссийский математический портал А. А. Арутюнов, Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерно

Research paper thumbnail of In memory of Yurii Petrovich Solovyev

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2008

was born on 8 October 1944 in Krasnoyarsk, a city in south central Siberia, on the Yenisei river.... more was born on 8 October 1944 in Krasnoyarsk, a city in south central Siberia, on the Yenisei river. In 1948, his family moved to Bryansk, a city in Russia which is 369 kilometers southwest of Moscow. He did his primary and secondary school education there. In 1963, he graduated with distinction from Bryansk Technical College of Mechanical Engineering. After three years of service in the Soviet Army, he entered the Mechanics and Mathematics Department of the Moscow State University in 1968. From that point on, his life was closely tied to the university. In 1973, he graduated from the Department of Mechanics and Mathematics with distinction. His diploma thesis [1] was published two years later while he was already working on his Candidate of Sciences thesis. In 1976, he defended this thesis and received his Candidate of Science degree, which in the Soviet Union was the equivalent of a Ph.D. degree. After that he became an assistant professor in the Department and from 1983 till 1992 he was a senior researcher. In 1988, he defended his Doctoral thesis and received his Doctor of Science degree. In the Soviet Union this degree was much higher than the Candidate of Science degree and relatively few mathematicians received it. In 1992, he was appointed Professor at the Department of Mechanics and Mathematics.

Research paper thumbnail of Hypergraph partitions

ArXiv, 2019

We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous op... more We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous optimization problem.

Research paper thumbnail of Geometric description of the Hochschild cohomology of group algebras

Topology, Geometry, and Dynamics, 2021

There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eile... more There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space B G BG (or the Eilenberg–MacLane complex K ( G , 1 ) K(G,1) ). Note that the space B G BG can be interpreted as a classifying space of the groupoid of the trivial action of the group G G . The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra R [ G ] R[G] and their derivative functors Ext ⁡ ( R [ G ] , M ) \operatorname {Ext}(R[G],M) , for which no geometric interpretation has been known so far. The key point for calculating the Hochschild cohomology H H ∗ ( R [ G ] ) HH^*(R[G]) is the new groupoid G r Gr associated with the adjoint action of the group G G . For this groupoid, the classical cohomology of the corresponding classification space B G r BGr with the finiteness condition for t...

Research paper thumbnail of Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism

Russian Journal of Mathematical Physics, 2015

Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algeb... more 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 bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. Due to to K.Mackenzie ([1]) the construction can be managed as a homotopy functor T LAg from category of smooth manifolds to the transitive Lie algebroids. The functor T LAg associates with each smooth manifold M the set T LAg(M) of all transitive algebroids with fixed structural finite dimensional Lie algebra g. Hence one can construct ([4],[5]) a classifying space Bg such that the family of all transitive Lie algebroids with fixed Lie algebra g over the manifold M has one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. It allows to describe characteristic classes of transitive Lie algebroids from the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with that derived from the Chern-Weil homomorphism by J.Kubarski([3]). As a matter of fact we show that the Chern-Weil homomorphism does not cover all characteristic classes from categorical point of view.

Research paper thumbnail of Almost flat bundles and almost flat structures

Topological Methods in Nonlinear Analysis, 2005

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced... more 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 over arbitrary CW-spaces using quasi-connections [3]. Connes, Gromov and Moscovici [2] showed that for any almost flat bundle α over the manifold M, the index of the signature operator with values in α is a homotopy equivalence invariant of M. From here it follows that a certain integer multiple n of the bundle α comes from the classifying space Bπ 1 (M). The geometric arguments discussed in this paper allow us to show that the bundle α itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace Y ⊂ Bπ 1 (M) trough the classifying mapping.

Research paper thumbnail of Description of coupling in the category of transitive Lie algebroids

In our previous paper ([4]) we have given a sufficient and necessary condition when the coupling ... more In our previous paper ([4]) 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 ([5], Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on the group Aut (g) of all automorphisms of the Lie algebra g, say Aut (g) δ , and show that tangent bundle T M can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology. But the question how many couplings exist under these conditions still remains open. Here we make the result more accurate and prove that there is a one-to-one correspondence between the family Coup(L) of all coupling of the Lie algebra bundle L with fixed finite dimensional Lie algebra g as the fiber and the structural group Aut (g) of all automorphisms of Lie algebra g and the tangent bundle T M and the family LAB δ (L) of equivalent classes of local trivial structures with structural group Aut (g) endowed with new topology Aut (g) δ. This result gives a way for geometric construction of the classifying space for transitive Lie algebroids with fixed structural finite dimensiaonal Lie algebra g. Hence we can clarify a categorical description of the characteristic classes for transitive Lie algebroids and a comparison with that by J. Kubarski ([2],[3])

Research paper thumbnail of C*-algebras and Elliptic Theory

Trends in Mathematics, 2006

The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devo... more The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devoted to the index theory on open manifolds. In the first part of the paper, a short review of index theory on open manifolds is given. In the second part, a general relative index theorem admitting compact topological perturbations and Sobolev perturbations of all other ingredients is established. V. Nazaikinskii and B. Sternin in the paper "Lefschetz Theory on Manifolds with Singularities" extend the Lefschetz formula to the case of elliptic operators on the manifolds with singularities using the semiclassical asymptotic method. In the paper "Pseudodifferential Subspaces and Their Applications in Elliptic Theory" by A. Savin and B. Sternin the method of so called pseudodifferential projectors in the theory of elliptic operators is studied. It is very useful for the study of boundary value problems, computation of the fractional part of the spectral AtiyahPatodiSinger eta invariant and analytic realization of topological K-groups with finite coefficients in terms of elliptic operators. In the paper "Residues and Index for Bisingular Operators" F. Nicola and L. Rodino consider an algebra of pseudo-differential operators on the product of two manifolds, which contains, in particular, tensor products of usual pseudo-differential operators. For this algebra the existence of trace functionals like Wodzickis residue is discussed and a homological index formula for the elliptic elements is proved. B. Bojarski and A. Weber in their paper "Correspondences and Index" define a certain class of correspondences of polarized representations of C *-algebras. These correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms between two manifolds. In this situation an index is defined. The additivity of this index is studied in the paper. Noncommutative aspects of Morse theory: In the paper "New L2-invariants of Chain Complexes and Applications" by V.V. Sharko homotopy invariants of free cochain complexes and Hilbert complex are studied. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds. A. Connes and T. Fack in their paper "Morse Inequalities for Foliations" outline an analytical proof of Morse inequalities for measured foliations obtained by them previously and give some applications. The proof is based on the use of a twisted Laplacian. Riemannian aspects: The paper "A Riemannian Invariant, Euler Structures and Some Topological Applications" by D. Burghelea and S. Haller discusses a numerical invariant associated with a Riemannian metric, a vector field with isolated zeros, and a closed one form which is defined by a geometrically regularized integral. This invariant extends the ChernSimons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. They discuss a generalization of Turaevs Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincare dual concept of co-Euler structures. The duality is provided by a geometrically regularized integral and involves the invariant mentioned above. Euler structures have been introduced because they permit to remove the ambiguities in the definition of the Reidemeister torsion. Similarly, co-Euler structures can be used to eliminate the metric dependence of

Research paper thumbnail of Lie algebroids: spectral sequences and signature

Sbornik: Mathematics, 2003

It is proved that for any transitive Lie algebroid L on a compact oriented connected manifold wit... more 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 the isotropy Lie algebra is equal to its inner automorphism group, or such that the adjoint Lie algebra bundle g induces a trivial homology bundle H * (g) in the category of flat bundles. Bibliography: 27 titles.

Research paper thumbnail of Construction of Fredholm representations and a modification of the Higson-Roe corona

Russian Journal of Mathematical Physics, 2009

The Fredholm representation theory is well adapted to construction of homotopy invariants of non ... more The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula [σ(M)] = L(M)chAf * ξ, [M ] ∈ K 0 A (pt) ⊗ Q,

Research paper thumbnail of Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Journal of Functional Analysis, 2011

We investigate orthonormality-preserving, C *-conformal and conformal module mappings on full Hil... more We investigate orthonormality-preserving, C *-conformal and conformal module mappings on full Hilbert C *-modules to obtain their general structure. Orthogonalitypreserving 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 as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfils the equality T (x), T (y) = |λ| 2 x, y for any elements x, y of the Hilbert C *-module. At the contrary, C *-conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators. 1991 Mathematics Subject Classification. Primary 46L08; Secondary 42C15, 42C40. Key words and phrases. C *-algebras, Hilbert C *-modules, orthogonality preserving mappings, conformal mappings, isometries.

Research paper thumbnail of Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

Russian Journal of Mathematical Physics, 2009

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signatur... more 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 operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

Research paper thumbnail of On analytic torsion over C*-algebras

Banach Center Publications, 1999

In this paper, we present an analytic definition for the relative torsion for flat C∗-algebra bun... more In this paper, 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 L2 torsion as defined in [7], [5].

Research paper thumbnail of The existence of coupling in the category of transitive lie algebroid

The coupling of the tangent bundle T M with the Lie algebra bundle L ([5], Definition 7.2.2) play... more The coupling of the tangent bundle T M with the Lie algebra bundle L ([5], 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 g as a fiber of L. Here we give a necessary and sufficient condition for the existence of such a coupling. Namely we define a new topology on the group Aut (g) of all automorphisms of Lie algebra g and show that tangent bundle T M can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology.

Research paper thumbnail of Critical Analysis of Amino Acids and Polypeptides Geometry

Studies in Systems, Decision and Control, 2015

Research paper thumbnail of Adequacy of Mathematical Models in Control Theory, Physics, and Environmental Science

Technological Concepts and Mathematical Models in the Evolution of Modern Engineering Systems, 2004

In contemporary Russian, the word “model” is often associated with a showing of fashionable cloth... more 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 express the real essence of a phenomenon and the causes of the processes in question. It must be clearly perceived that all natural phenomena are interrelated. The modest abilities of the human intellect do not allow taking all relations into account. But, luckily, some relations are strong, and others are vanishing. So we have a possibility to reveal the main acting forces, discarding all that are of secondary importance. This is the common basic paradigm of all natural sciences, if we exclude their purely descriptive aspects. This paradigm preordains the leading role of mathematics in the process of designing and exploring models.

Research paper thumbnail of K-theory over C * -algebras

Geometry and Topology of Manifolds, 2007

Research paper thumbnail of K-theory from the point of view of C *-algebras and Fredholm representations

Central European Journal of Mathematics, 2005

These notes represent the subject of five lectures which were delivered as a minicourse during the

Research paper thumbnail of Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

Central European Journal of Mathematics, 2004

The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied i... more 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 cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid coho...

Research paper thumbnail of Relations between asymptotic and Fredholm representations

Research paper thumbnail of Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерности

Matematicheskie Zametki, 2015

Общероссийский математический портал А. А. Арутюнов, Редукция нелокальных псевдодифференциальных ... more Общероссийский математический портал А. А. Арутюнов, Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерно

Research paper thumbnail of In memory of Yurii Petrovich Solovyev

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2008

was born on 8 October 1944 in Krasnoyarsk, a city in south central Siberia, on the Yenisei river.... more was born on 8 October 1944 in Krasnoyarsk, a city in south central Siberia, on the Yenisei river. In 1948, his family moved to Bryansk, a city in Russia which is 369 kilometers southwest of Moscow. He did his primary and secondary school education there. In 1963, he graduated with distinction from Bryansk Technical College of Mechanical Engineering. After three years of service in the Soviet Army, he entered the Mechanics and Mathematics Department of the Moscow State University in 1968. From that point on, his life was closely tied to the university. In 1973, he graduated from the Department of Mechanics and Mathematics with distinction. His diploma thesis [1] was published two years later while he was already working on his Candidate of Sciences thesis. In 1976, he defended this thesis and received his Candidate of Science degree, which in the Soviet Union was the equivalent of a Ph.D. degree. After that he became an assistant professor in the Department and from 1983 till 1992 he was a senior researcher. In 1988, he defended his Doctoral thesis and received his Doctor of Science degree. In the Soviet Union this degree was much higher than the Candidate of Science degree and relatively few mathematicians received it. In 1992, he was appointed Professor at the Department of Mechanics and Mathematics.

Research paper thumbnail of Hypergraph partitions

ArXiv, 2019

We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous op... more We suggest a reduction of the combinatorial problem of hypergraph partitioning to a continuous optimization problem.

Research paper thumbnail of Geometric description of the Hochschild cohomology of group algebras

Topology, Geometry, and Dynamics, 2021

There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eile... more There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space B G BG (or the Eilenberg–MacLane complex K ( G , 1 ) K(G,1) ). Note that the space B G BG can be interpreted as a classifying space of the groupoid of the trivial action of the group G G . The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra R [ G ] R[G] and their derivative functors Ext ⁡ ( R [ G ] , M ) \operatorname {Ext}(R[G],M) , for which no geometric interpretation has been known so far. The key point for calculating the Hochschild cohomology H H ∗ ( R [ G ] ) HH^*(R[G]) is the new groupoid G r Gr associated with the adjoint action of the group G G . For this groupoid, the classical cohomology of the corresponding classification space B G r BGr with the finiteness condition for t...

Research paper thumbnail of Comparison of categorical characteristic classes of a transitive Lie algebroid with the Chern-Weil homomorphism

Russian Journal of Mathematical Physics, 2015

Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algeb... more 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 bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. Due to to K.Mackenzie ([1]) the construction can be managed as a homotopy functor T LAg from category of smooth manifolds to the transitive Lie algebroids. The functor T LAg associates with each smooth manifold M the set T LAg(M) of all transitive algebroids with fixed structural finite dimensional Lie algebra g. Hence one can construct ([4],[5]) a classifying space Bg such that the family of all transitive Lie algebroids with fixed Lie algebra g over the manifold M has one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. It allows to describe characteristic classes of transitive Lie algebroids from the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with that derived from the Chern-Weil homomorphism by J.Kubarski([3]). As a matter of fact we show that the Chern-Weil homomorphism does not cover all characteristic classes from categorical point of view.

Research paper thumbnail of Almost flat bundles and almost flat structures

Topological Methods in Nonlinear Analysis, 2005

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced... more 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 over arbitrary CW-spaces using quasi-connections [3]. Connes, Gromov and Moscovici [2] showed that for any almost flat bundle α over the manifold M, the index of the signature operator with values in α is a homotopy equivalence invariant of M. From here it follows that a certain integer multiple n of the bundle α comes from the classifying space Bπ 1 (M). The geometric arguments discussed in this paper allow us to show that the bundle α itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace Y ⊂ Bπ 1 (M) trough the classifying mapping.

Research paper thumbnail of Description of coupling in the category of transitive Lie algebroids

In our previous paper ([4]) we have given a sufficient and necessary condition when the coupling ... more In our previous paper ([4]) 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 ([5], Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on the group Aut (g) of all automorphisms of the Lie algebra g, say Aut (g) δ , and show that tangent bundle T M can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology. But the question how many couplings exist under these conditions still remains open. Here we make the result more accurate and prove that there is a one-to-one correspondence between the family Coup(L) of all coupling of the Lie algebra bundle L with fixed finite dimensional Lie algebra g as the fiber and the structural group Aut (g) of all automorphisms of Lie algebra g and the tangent bundle T M and the family LAB δ (L) of equivalent classes of local trivial structures with structural group Aut (g) endowed with new topology Aut (g) δ. This result gives a way for geometric construction of the classifying space for transitive Lie algebroids with fixed structural finite dimensiaonal Lie algebra g. Hence we can clarify a categorical description of the characteristic classes for transitive Lie algebroids and a comparison with that by J. Kubarski ([2],[3])

Research paper thumbnail of C*-algebras and Elliptic Theory

Trends in Mathematics, 2006

The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devo... more The paper "Index Theory for Generalized Dirac Operators on Open Manifolds" by J. Eichhorn is devoted to the index theory on open manifolds. In the first part of the paper, a short review of index theory on open manifolds is given. In the second part, a general relative index theorem admitting compact topological perturbations and Sobolev perturbations of all other ingredients is established. V. Nazaikinskii and B. Sternin in the paper "Lefschetz Theory on Manifolds with Singularities" extend the Lefschetz formula to the case of elliptic operators on the manifolds with singularities using the semiclassical asymptotic method. In the paper "Pseudodifferential Subspaces and Their Applications in Elliptic Theory" by A. Savin and B. Sternin the method of so called pseudodifferential projectors in the theory of elliptic operators is studied. It is very useful for the study of boundary value problems, computation of the fractional part of the spectral AtiyahPatodiSinger eta invariant and analytic realization of topological K-groups with finite coefficients in terms of elliptic operators. In the paper "Residues and Index for Bisingular Operators" F. Nicola and L. Rodino consider an algebra of pseudo-differential operators on the product of two manifolds, which contains, in particular, tensor products of usual pseudo-differential operators. For this algebra the existence of trace functionals like Wodzickis residue is discussed and a homological index formula for the elliptic elements is proved. B. Bojarski and A. Weber in their paper "Correspondences and Index" define a certain class of correspondences of polarized representations of C *-algebras. These correspondences are modeled on the spaces of boundary values of elliptic operators on bordisms between two manifolds. In this situation an index is defined. The additivity of this index is studied in the paper. Noncommutative aspects of Morse theory: In the paper "New L2-invariants of Chain Complexes and Applications" by V.V. Sharko homotopy invariants of free cochain complexes and Hilbert complex are studied. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds. A. Connes and T. Fack in their paper "Morse Inequalities for Foliations" outline an analytical proof of Morse inequalities for measured foliations obtained by them previously and give some applications. The proof is based on the use of a twisted Laplacian. Riemannian aspects: The paper "A Riemannian Invariant, Euler Structures and Some Topological Applications" by D. Burghelea and S. Haller discusses a numerical invariant associated with a Riemannian metric, a vector field with isolated zeros, and a closed one form which is defined by a geometrically regularized integral. This invariant extends the ChernSimons class from a pair of two Riemannian metrics to a pair of a Riemannian metric and a smooth triangulation. They discuss a generalization of Turaevs Euler structures to manifolds with non-vanishing Euler characteristics and introduce the Poincare dual concept of co-Euler structures. The duality is provided by a geometrically regularized integral and involves the invariant mentioned above. Euler structures have been introduced because they permit to remove the ambiguities in the definition of the Reidemeister torsion. Similarly, co-Euler structures can be used to eliminate the metric dependence of

Research paper thumbnail of Lie algebroids: spectral sequences and signature

Sbornik: Mathematics, 2003

It is proved that for any transitive Lie algebroid L on a compact oriented connected manifold wit... more 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 the isotropy Lie algebra is equal to its inner automorphism group, or such that the adjoint Lie algebra bundle g induces a trivial homology bundle H * (g) in the category of flat bundles. Bibliography: 27 titles.

Research paper thumbnail of Construction of Fredholm representations and a modification of the Higson-Roe corona

Russian Journal of Mathematical Physics, 2009

The Fredholm representation theory is well adapted to construction of homotopy invariants of non ... more The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula [σ(M)] = L(M)chAf * ξ, [M ] ∈ K 0 A (pt) ⊗ Q,

Research paper thumbnail of Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules

Journal of Functional Analysis, 2011

We investigate orthonormality-preserving, C *-conformal and conformal module mappings on full Hil... more We investigate orthonormality-preserving, C *-conformal and conformal module mappings on full Hilbert C *-modules to obtain their general structure. Orthogonalitypreserving 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 as long as some polar decomposition conditions for the specific element λ are fulfilled inside that multiplier algebra. Generally, T always fulfils the equality T (x), T (y) = |λ| 2 x, y for any elements x, y of the Hilbert C *-module. At the contrary, C *-conformal and conformal bounded module maps are shown to be only the positive real multiples of isometric module operators. 1991 Mathematics Subject Classification. Primary 46L08; Secondary 42C15, 42C40. Key words and phrases. C *-algebras, Hilbert C *-modules, orthogonality preserving mappings, conformal mappings, isometries.

Research paper thumbnail of Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

Russian Journal of Mathematical Physics, 2009

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signatur... more 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 operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

Research paper thumbnail of On analytic torsion over C*-algebras

Banach Center Publications, 1999

In this paper, we present an analytic definition for the relative torsion for flat C∗-algebra bun... more In this paper, 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 L2 torsion as defined in [7], [5].

Research paper thumbnail of The existence of coupling in the category of transitive lie algebroid

The coupling of the tangent bundle T M with the Lie algebra bundle L ([5], Definition 7.2.2) play... more The coupling of the tangent bundle T M with the Lie algebra bundle L ([5], 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 g as a fiber of L. Here we give a necessary and sufficient condition for the existence of such a coupling. Namely we define a new topology on the group Aut (g) of all automorphisms of Lie algebra g and show that tangent bundle T M can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology.

Research paper thumbnail of Critical Analysis of Amino Acids and Polypeptides Geometry

Studies in Systems, Decision and Control, 2015