Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/bedlewo2009/abstracts2.pdf Äàòà èçìåíåíèÿ: Wed Jan 14 22:15:02 2009 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:06:06 2016 Êîäèðîâêà:

Stefan Banach International Mathematical Center International Conference C*-algebras and elliptic theory. III

Bdlewo, January 26­31, 2009

Organizing Committee:
Alexander Mishchenko (chairman), Bogdan Bo jarski, Jan Kubarski, Alexander Pavlov, Evgenij Troitsky

Scientific Committee:
P. F. Baum, J. Bro dzki, D. Burghelea, P. M. Ha jac, V. M. Manuilov, R. Melrose, A. S. Mishchenko, V. Nistor, N. Teleman, E. V. Troitsky


The conference will b e the next in the series of conferences starting from · the International Conference C -algebras and el liptic theory, February 23­28, 2004, Banach Center, Bdlewo, Poland, · the International Conference Topology, analysis and applications in mathematical physics Moscow, Russia, February 14­19, 2005, dedicated to the memory of Professor Yu. P. Solovjev, · the International Conference C -algebras and el liptic theory. II, January 23­28, 2006, Banach Center, Bdlewo, Poland, · the International Conference Operator algebras and topology, Moscow, Russia, January 29­February 03, 2007, which were organized by Moscow mathematicians.

Main topics:
· · · · · · · C -algebras and their applications, K-theory of op erator algebras, index theory, geometric group theory and group C -algebras, noncommutative geometry and top ology, pseudo differential op erators on singular manifolds, deformation quantization.

2


C*-algebras and elliptic theory -- I I I Schedule
Monday, 26.01.2009 8:45­ 9:00 OPENING CEREMONY 9:00­ 9:45 Vladimir Sharko: L2 -invariants and Morse numbers of manifolds 10:00­10:45 Vladimir Manuilov: On some new tensor product functors for C*-algebras COFFEE BREAK 11:15­12:00 Sara Azzali: Large time asymptotic for L2 invariants for families 12:15­13:00 Alexander Pavlov: Multipliers of C*-algebras and Hilbert C*-modules 13:15­14:00 Alexander Helemskii: Semi-Ruan one-sided modules, the extreme flatness and Arveson-Wittstock type theorems LUNCH 15:00­15:45 Chao You: On almost representations of groups with property (T ) 16:00­16:45 Tatiana Shulman: On lifting problems in C*-algebras COFFEE BREAK 17:15­18:00 Alexey Pirkovskii: Homological dimensions of the algebras of smooth and holomorphic functions on the quantum torus 18:15­19:00 DINNER Tuesday, 27.01.2009 9:00­ 9:45 Jean-Paul Brasselet: The Hochschild-Kostant-Rosenberg Theorem for singular varieties 10:00­10:45 Yuri Kordyukov: Transverse Dirac operators on foliated manifolds and some applications COFFEE BREAK 11:15­12:00 Edwin Beggs: Noncommutative sheaves and complex structures 12:15­13:00 James Heitsch: Leafwise homotopy invariance of the twisted higher harmonic signature for foliations 13:15­14:00 LUNCH ´ 15:00­15:45 Jesus Antonio Alvarez LÑp ez: Transversality and Lefschetz numbers for fo´ liation maps 16:00­16:45 Jacek Lech: On the group of C r,s -diffeomorphisms on a foliated manifold COFFEE BREAK 17:15­18:00 18:15­19:00 CONFERENCE DINNER Wednesday, 28.01.2009 EXCURSIONS DINNER
3


9:00­ 9:45 10:00­10:45 11:15­12:00 12:15­13:00 13:15­14:00

15:00­15:45 16:00­16:45

17:15­18:00 18:15­19:00

Thursday, 29.01.2009 Ulrich Bunke: Classification results for differentiable extensions of generalized cohomology theories Piotr M. Ha jac: The Peter-Weyl-Galois theory of compact principal bund les COFFEE BREAK Levan Alania: Moduli space of flat bund les and cohomology with local system Dan Burghelea: What can we do with differential forms besides familiar manipulations? String cohomology Andrei Ershov: Topological obstructions to embedding a matrix algebra bund le into a trivial one LUNCH Jan Kubarski: The signature of Lipschitz manifolds from more general point of view Alexander Mishchenko: Bordisms of manifolds with proper action of discrete group COFFEE BREAK Quitzeh Morales Melendez: Equivariant vector bund les for semifree proper action Nicolae Teleman: TBA DINNER Friday, 30.01.2009 Alexander Fel'shtyn: Twisted Burnside-Frobenius theorem for Metabelian groups Anton Savin: Noncommutative el liptic theory over C*-algebras COFFEE BREAK Elmar Schrohe: Index theory for boundary value problems via continuous fields of C*-algebras Severino Toscano do Rego Melo: K-Theory of DOs with periodic symbols on a cylinder

9:00­ 9:45 10:00­10:45 11:15­12:00 12:15­13:00 13:15­14:00

LUNCH 15:00­15:45 Boguslaw Ha jduk: On some problems related to geometry of moduli spaces of pseudoholomorphic spheres 16:00­16:45 Alexei Tuzhilin: Infinite subsets of metric spaces which can be spanned by finite length trees COFFEE BREAK 17:15­18:00 Seytek Tabaldyev: On homological dimensions of algebras of continuous functions 18:15­19:00 DINNER Saturday, 31.01.2009 9:00­10:45 INFORMAL DISCUSSIONS COFFEE BREAK

4


Abstracts
Mo duli space of flat bundles and cohomology with lo cal system Levan ALANIA
Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: alania@mech.math.msu.su

Let M be a closed, smooth manifold with nontrivial fundamental group. Cohomology with local system for abelian representation of fundamental group can be calculated be means of spectral sequence with Massay type differentials, as Novikov showed. We generalized this spectral sequence for nonabelian case and studied some conditions of convergence. Transversality and Lefschetz numb ers for foliation maps ´ Jesus Antonio ALVAREZ LñPEZ ´
aticas, Facultade de Matem´ Universidade de Santiago de Comp ostela, Santiago de Comp ostela, Spain E-mail address: jalvarez@usc.es

Let F be a smooth foliation on a closed Riemannian manifold M , and let be a transverse invariant measure of F . Suppose that is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the -Lefschetz number of any leaf preserving diffeomorphism (M , F ) (M , F ) is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological -Lefschetz number is equal to the analytic -Lefschetz number defined by Heitsch and Lazarov, which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to F .


Large time asymptotic for L2 invariants for families Sara AZZALI (Joint work with Sebastian GOETTE and Thomas SCHICK)
Mathematisches Institut, Georg-August-Universit¨ at, G¨ ottingen, Germany E-mail address: azzali@uni-math.gwdg.de

A new approach to large time asymptotic for the heat operator of the Bismut-Lott superconnection in a L2-setting is explained. As an application, we show that the L2 index theorem for the family of signature operators holds, without assumptions on the fibrewise Novikov­Shubin invariants. Under the same hypothesis, the L2-rho form is well defined. Noncommutative sheaves and complex structures Edwin BEGGS
Department of Mathematics, Swansea University, Wales, UK E-mail address: E.J.Beggs@swansea.ac.uk

Joint work with T. Brzezinski and P. Smith. Classically sheaves are used as a general coefficient system for cohomology theories. I will describe how to view sheaves in the context of noncommutative geometry, using modules with connections over a differential graded algebra. (Essentially this uses a flat connection to replace the local homeomorphism part of the classical definition). It will turn out that we can define cohomology with coefficients in modules with flat connection, and several classical results for sheaf cohomology transfer to the noncommutative case, including the Serre spectral sequence. I will consider complex structures on noncommutative differential calculi, and the associated integrability condition. This can be combined with the idea of a sheaf in noncommutative geometry to give a holomorphic sheaf, and a resulting cohomology theory. The eventual aim would be to form a noncommutative version of the bridge between algebraic geometry and complex differential geometry (classically using K¨ ahler manifolds). I will say something of the many problems still to be considered in building this bridge -- the simplest being the existence of critical points.
6


The Ho chschild-Kostant-Rosenb erg Theorem for singular varieties Jean-Paul BRASSELET (joint with Andr´ LEGRAND) e
IML - CNRS, Case 907 - Luminy 13288, Marseille, cedex 9, France E-mail address: jpb@iml.univ-mrs.fr

We generalize the Hochschild-Kostant-Rosenberg Theorem in the case of singular varieties. The tool we use is the Teleman localisation process. We deal with manifolds with boundary and with varieties with isolated singularities. In each case, we define suitable algebras of functions and study the localization of the corresponding Hochschild homology. In the case of isolated singularities, we prove that the closed Hochschild homology corresponds to the complex of intersection differential forms.

Classification results for differentiable extensions of generalized cohomology theories Ulrich BUNKE (report on joint work with Thomas SCHICK)
Regensburg University, Regensburg, Germany E-mail address: ulrich.bunke@mathematik.uni-regensburg.de

A differentiable generalized cohomology theory is a geometric refinement of a generalized cohomology on smooth manifolds. It is designed to capture primary and secondary topological invariants on equal footing and also formalizes the construction of invariants of geometric structures. Typical examples are characteristic classes of vector bundles, their Chern-Weyl representatives and Chern-Simons invariants. In the talk I will present an axiomatic approach to differentiable extensions of generalized cohomology theories. The main results concern existence, uniqueness of the differentiable extensions themselves and of additional structures like products or integration maps. The most interesting example is complex K-theory. The comparison of geometric, homotopy theoretic and analytic models leads to interesting insights into the structure of primary and secondary invariants of local index theory.

7


What can we do with differential forms b esides familiar manipulations? String cohomology Dan BURGHELEA
Department of Mathematics, Ohio State University, Columbus, USA E-mail address: burghele@math.ohio-state.edu

Less familiar manipulation with invariant differential forms on a (finite or infinite dimensional) smooth S 1 -manifold leads to a mild modification of equivariant cohomology in finite dimensional case of some interest. However, when applied to the free loop space of a manifold, regarded as an infinite dimensional manifold, this leads to an interesting homotopy functor (string cohomology of the manifold) which: a) unifies Atiyah-Hirzebruch and Waldhausen algebraic K-theory at least for 1-connected space, b) provides a convenient homological interpretation of expressions e on the free loop space of interest in string theory. The functor is computable, and can be actually defined on the category of connected commutative differential graded algebras; however we do not know HOW.

8


Top ological obstructions to emb edding a matrix algebra bundle into a trivial one Andrei ERSHOV
Chair of Geometry, Saratov State University, Saratov, Russia E-mail address: ershov.andrei@gmail.com

The starting point of our work was the following question. Let X be (say) a compact manifold, Ak X a locally trivial bundle with fibre a complex matrix algebra Mk (C) considered as C*-algebra (so its "natural" structure group is Aut(Mk (C)) PUk (C)). Then is Ak a subbund le of a (finite = dimensional) trivial bund le X â Mkl (C), i.e. is there a fibrewise map (in fact embedding) A
µ kA AA AA AA A /

X â Mkl (C) (1)

X such that x X its restriction µ |x embeds the fibre (Ak )x into Mkl (C) as a unital subalgebra? It turns out that the answer is positive if we do not impose any constraint on l. But if we require, say, l to be relatively prime to k , then stable obstructions arise. The obstructions can be described explicitly by reducing the embedding problem (1) to the lifting problem for a suitable fibre bundle because after that we can apply the usual methods of the topological obstruction theory. For example, the first obstruction is the class (Ak ) H 2 (X, Z/k Z) which is the obstruction for reduction of the structure group of A k from PU(k ) to SU(k ). If (Ak ) = 0, then Ak = End(k ) for some vector Ck -bundle k with the structure group SU(k ). Then the second obstruction is c 2 ( ) mod k H 4 (X, Z/k Z), where c2 is the second Chern class. It turns out that the fibre bundle related to our problem has the structure of a principal groupoid bundle for some groupoid related to subalgebras in a fixed matrix algebra. This structure sheds some light on relations between matrix algebra bundles with additional structures (such as an embedding into a trivial bundle). References
[1] A. V. Ershov, Topological obstructions to embedding a matrix algebra bund le into a trivial one, // arXiv:0807.3544v4 [math.KT]

r rrr rrr r xrrr

9


Twisted Burnside-Frob enius theorem for Metab elian groups Alexander FEL'SHTYN
University of Szczecin, Szczecin, Poland E-mail address: felshtyn@gmail.com

It is proved for a finitely generated Metabelian groups that the Reidemeister number of a automorphism is equal to the number of finite-dimensional fixed points of induced map on the unitary dual space, if the Reidemeister number is finite.This theorem is a natural generalization to Metabelian groups of the classical Burnside-Frobenius theorem. On the other hands it implies the congruences for Reidemeister numbers of iterations of automorphism.This congruences give necessary conditions for the realization problem of Reidemeister numbers of iterations. For Polycyclic-by-finite groups twisted Burnside-Frobenius theorem was proven in: Fel'shtyn A., Troitsky, E., Twisted Burnside-Frobenius theory for discrete groups. J. Reine Angew. Math. (Crelle Journal), 613 (2007), 193­210. The Peter-Weyl-Galois theory of compact principal bundles Piotr M. HAJAC
IMPAN, Warsaw, Poland E-mail address: pmh@impan.gov.pl

We define a functor from the category of unital C*-algebras with compact quantum group actions to the category of comodule algebras by extending the notion of the algebra of regular functions (spanned by the matrix coefficients of the irreducible unitary corepresentations) from compact quantum groups to unital C*-algebras on which they act. We call it the Peter-Weyl functor. Combined with the Gelfand transform, it translates compact group actions on compact Hausdorff spaces into a general algebraic framework. On the other hand, the Galois condition for finite field extensions is also translated into this comodule-algebraic setting, and is the founding stone of noncommutative Hopf-Galois theory. Our main result is the equivalence of the freeness of a classical compact group action on a compact Hausdorff space and the Galois condition for its Peter-Weyl comodule algebra. This parallels the well-known equivalence of Galois coverings and discrete group principal bundles. (Joint work with Paul F. Baum.)
10


On some problems related to geometry of mo duli spaces of pseudoholomorphic spheres Boguslaw HAJDUK
Institute of Mathematics, Wro claw University, Wro claw, Poland E-mail address: hajduk@math.uni.wroc.pl

We discuss the problem of singular points for the evaluation map M â S 2 X, where M is the moduli of pseudoholomorphic spheres for an almost complex structure on a symplectic manifold X. We give some positive answers in the case X = S 2 â T 2n. Leafwise homotopy invariance of the twisted higher harmonic signature for foliations James L. HEITSCH
University of Illinois (Chicago), and Northwestern University, USA E-mail address: heitsch@math.uic.edu

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact oriented Riemannian manifold M with coefficients in a leafwise flat complex bundle is a leafwise homotopy invariant.

11


Semi-Ruan one-sided mo dules, the extreme flatness and ArvesonWittsto ck typ e theorems Alexander HELEMSKI I
Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: helemskii@rambler.ru

We show that a certain class of normed modules over the algebra of all bounded operators on a Hilbert space possesses a certain functional-analytic version of the standard homological property of flatness. This version is formulated in terms of the preservation, under the pro jective tensor product of modules, of the property of a given morphism to be isometric. To give the exact definition, at first we fix an space L and denote, for brevity, B (L) by B . tive right B -module Y a semi-Ruan module, pro jections P, Q B and arbitrary x, y Y , 1 ( x · P 2 + y · Q 2) 2 . infinite-dimensional Hilbert We call a normed contracif for arbitrary orthogonal we have x · P + y · Q

We call a normed contractive left B -module X extremely flat, if for every isometric morphism : Y Z of right semi-Ruan B -modules the operator 1X : Y X Z X is also isometric.
B B B

Theorem. Let H be a Hilbert space, S (H, L) the left Banach B -module of Schmidt operators from H into L with the outer multiplication defined as the operator composition. Then S (H, L) is an extremely flat module. As an application, we obtain several theorem on the norm-preserving extension of morphisms for different types of (bi)modules, called Arveson­ Wittstock type theorems. These, in their turn, have, as a straight corollary, the `genuine' Arveson-Wittstock Theorem on the extension of completely bounded operators (in the frame-work of the non-matricial presentation of operator space theory; cf. [3]). Concerning the importance of the latter theorem and its history see the textbooks [2,4] and the original papers [5,1].
References [1] W. Arveson, Acta Math. 123 (1969) 141­224. [2] E. G. Effros, Z.-J. Ruan, Operator spaces. Clarendon Press, Oxford, 2000. [3] A. Ya. Helemskii. In: Topological Algebras and Applications, (Eds. A. Mallios and M. Haralampidou) AMS, Prov., 2007, 199­224. [4] V. I. Paulsen. Completely bounded maps and operator algebras. Cambridge Univ. Press, Cambridge, 2002. [5] G. Wittsto ck, J. Funct. Anal. 40 (1981) 127­150.

12


Transverse Dirac op erators on foliated manifolds and some applications Yuri KORDYUKOV
Institute of Mathematics, Russian Academy of Sciences, Ufa, Russia E-mail address: ykordyukov@yahoo.com

We introduce transverse Dirac operators on Riemannian foliated manifolds and discuss basic properties of these operators. In particular, we state Lichnerowicz and Bochner type formulas. Our main result is an analogue of the Kodaira vanishing theorem for the transverse Spinc Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation. We will also discuss some applications of our results in transverse index theory and geometric quantization of foliations. The signature of Lipschitz manifolds from more general p oint of view Jan KUBARSKI
Institute of Mathematics, Technical University of LÑ d, LÑ d, Poland E-mail address: kubarski@p.lodz.pl

13


On the group of C r,s-diffeomorphisms on a foliated manifold Jacek LECH
AGH University of Science and Technology, KrakÑw, Poland E-mail address: lechjace@galaxy.agh.edu.pl

Let M be an n-dimensional manifold and let Diff r (M )0 be the identity c r component of the group of C -diffeomorphisms with compact supports. Mather and Epstein showed that this group is perfect and simple, whenever r = n + 1 or r = . Now let us take a foliated manifold (M , F ), dim F = k . We consider the perfectness of the group Diff r (M , F )0 of C r c diffeomorphisms acting along leaves. For rk we introduce the notion of C r,s -mappings which are of class C r along leaves and of class C s transversally. Then we may prove, by modifications of Mather-Epstein's method, that the group Diff r,s (M , F )0 of C r,s-diffeomorphisms acting along leaves is c perfect for r - s > k + 1 or r = . Notice that the perfectness of Diff r (M )0 c r and Diff c (M , F )0 are related as each diffeomorphism from the first group may be decomposed into diffeomorphisms preserving leaves. On some new tensor pro duct functors for C*-algebras Vladimir MANUILOV
Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: manuilov@mech.math.msu.su

On the category of C*-algebras, we define new tensor product functors, which are related to asymptotic representations. In order to calculate these tensor norms, we discuss asymptotic representations of some C*-algebras. In particular, we construct an asymptotic representation of the reduced group C*-algebra of a free group, which is not equivalent to any genuine representation. We also show that symmetric tensor product functors can be non-associative.

14


K-Theory of DOs with p erio dic symb ols on a cylinder Severino T. MELO
Instituto de Matematica e Estatistica, Universidade de S~ Paulo, ao S~ Paulo, Brazil ao E-mail address: toscano@ime.usp.br

Let B denote a compact Riemannian manifold and let A denote the smallest norm-closed adjoint-invariant sub-algebra of the algebra of all bounded operators on L2 (B â R) containing: (i ) all operators of multiplication by functions that extend continuously to the compactification B â [-, +] and by continuous functions which are 2 -periodic on the second variable, (ii ) = (1 - )-1/2, where denotes the laplacian on B â R, (iii ) t , where t denotes the derivative with respect to t R and (iv ) L, for every smooth vector field L on B . This algebra contains the classical zeroorder pseudodifferential operators on B â R with 2 -periodic symbols. Two C*-algebra homomorphisms are defined on A: the usual principal symbol and an operator valued symbol . A given A A is a Fredholm operator if and only if both (A) and (A) are invertible (Cordes - M., 1988, 1990). In this talk, I plan to report on joint work with Patr´ Hess, in which we icia use those old results to study the K-theory of A/K, where K denotes the compact ideal. We have reduced the problem of computing the K-theory index-mapping associated to to the problem of computing the Fredholm index of elliptic operators on the compact manifold S 1 â B . In the case B = S 1 we could completely compute that index-mapping, showing that K0(A/K) has no torsion. Also in this case, we have shown that the image of is the crossed product of a smaller algebra by an automorphism and, using Pimsner and Voiculescu's sequence, computed its K-theory. Equivariant vector bundles for semifree prop er action Quitzeh MORALES MELENDEZ
Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: balamquitzeh@yahoo.com

We give a description, up to homotopy, of equivariant vector bundles with action of a discrete group G when the action on the base space of the bundle happens to be semifree.
15


Bordisms of manifolds with prop er action of discrete group Alexander MISHCHENKO
Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: asmish@mech.math.msu.su

There is a combinatorial construction of non-commutative signature of the manifolds with proper action of a discrete group. On the other hand one can defined a proper bordism relation between oriented manifolds with proper action. The non-commutative signature turns out to be homotopy invariant and invariant of oriented bordism. As a consequence the bordism group of proper action is not trivial. The Hirzebruch type formula will be presented. Multipliers of C*-algebras and Hilb ert C*-mo dules Alexander PAVLOV (report on joint work with Michael FRANK)
Dipartimento di Matematica e Informatica, Universit` di Trieste, a Trieste, Italy E-mail address: axpavlov@gmail.com

We are going to discuss how to apply the categorial approach to multipliers for constructing the multiplier theory of Hilbert C*-modules. There will be emphasized some properties of multipliers and, in particular, their property of maximality among all strictly essential extensions of Hilbert C*-modules as well as relations between left essential and left strictly essential extensions in different contexts. Also the topological outlook on multipliers in terms of strict topologies will be considered. In conclusion we are planning to consider briefly some possible applications.

16


Homological dimensions of the algebras of smo oth and holomorphic functions on the quantum torus Alexey PIRKOVSKI I
Peoples' Friendship University of Russia, Moscow, Russia E-mail address: pirkosha@sci.pfu.edu.ru, pirkosha@online.ru

Let A be a Fr´ het algebra, and let dg A (resp. db A) denote the globec al dimension (resp. the bidimension) of A. The corresponding weak (i.e., flat) dimensions will be denoted by w.dg A and w.db A, respectively. It is known that dg A db A and w.dg A w.db A for every A. However, it is an open problem whether there exists a Fr´ het algebra A for which any ec of the above inequalities is strict. We show that w.dg A = w.db A provided that A is nuclear and satisfies w.db A < . If, in addition, A has a "nice" pro jective bimodule resolution, then we show that dg A db A dg A + 1. Next we specialize to nuclear Fr´ het algebras satisfying Van den Bergh's ec condition VdB(n), which can be viewed as a kind of Poincar´ duality bee tween Hochschild homology and cohomology. We give a number of examples of such algebras and show that for each such algebra A we have dg A = db A = w.dg A = w.db A = n. As an application, we show that for the algebras of smooth and holomorphic functions on the quantum n-torus all the above dimensions are equal to n. This contrasts with the result of McConnell and Pettit (1988) who proved that the global dimension of the algebra of polynomial functions on the generic quantum n-torus equals 1. Non-commutative residue for pseudo-differential op erators in R Luigi RODINO
Dipartimento di Matematica, University of Torino, Torino, Italy E-mail address: luigi.rodino@unito.it
n

The non-commutative residue of Wodzicki and Connes was initially defined for pseudo-differential operators on compact manifolds. The definition was then extended in different contexts, in particular Boggiatto and Nicola (2003) introduced a non-commutative residue for a class of pseudodifferential operators in Rn , including the Quantum Harmonic Oscillator and its real powers. In the same frame, here we study Laurent expansions for regularized integrals of holomorphic symbols, recapturing the notion of non-commutative residue in Rn . We also establish connections between the residue and Einstein metrics on non-compact manifolds.
17


Noncommutative elliptic theory over C*-algebras Anton SAVIN
Indep endent University of Moscow, Moscow, Russia E-mail address: antonsavin@mail.ru

Boris STERNIN
Peoples' Friendship University of Russia, Moscow, Russia E-mail address: sternin@mail.ru

In a number of problems related to noncommutative geometry, there arise naturally elliptic operators, whose symbols are elements of essentially noncommutative algebras (e.g. elliptic theory on noncommutative torus, noncommutative spheres, etc.) In the talk, we shall describe elliptic theory for operators of this type, which act in modules over C*-algebras (in the spirit of the classical MishchenkoFomenko theory). The main result is an index formula in this situation. Following applications of this theory are considered: we use it to obtain Bott periodicity for some discrete groups and construct new homotopy invariants of noncommutative elliptic operators. Index theory for b oundary value problems via continuous fields of C*-algebras Elmar SCHROHE
¨ Institut fur Analysis, Leibniz Universit¨ Hannover, at Hannover, Germany E-mail address: schrohe@math.uni-hannover.de

For a smooth manifold X with boundary we construct a semigroupoid T - X and a continuous field Cr (T - X ) of C*-algebras which extend Connes' construction of the tangent groupoid. We show the asymptotic multiplicativity of semiclassical pseudodifferential boundary value problems with smoothing symbols and compute the Ktheory of the associated symbol algebra. We then use these results to derive a deformation theoretic index theorem for boundary value problems in Boutet de Monvel's calculus using ideas of Elliott-Natsume-Nest.
18


L2-invariants and Morse numb er of manifold Vladimir SHARKO
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine E-mail address: sharko@imath.kiev.ua

The Morse number M(W n) of a manifold W n is the minimum of the total number of critical points over all Morse functions on W n . The following equality holds for the Morse number of W n . Theorem. Let (W n, V0n-1, V1n-1) (n 6) be a compact smooth manifold with boundary W n = V0n-1 V1n-1 and = 1 (W n) be the fundamental group of the manifold W n. Suppose that (Vin-1) - 1 (W n) is isomorphism, W h( ) = 0, where W h( ) Whitehead group of , and D0 (W n ) = D r for all i. Then M(W n) = 2
n-2 i=1 n- n- i (S(2) (W n) + S(2) 1(W n) + i=01(dimN [] (H i((2) (W n, Z )))) + n 2µ(H n(W n , Z [ ])) - dimN [] (H(2) (W n, Z )). n-1 l

(W n) = D0 (W n ) = D r

n-2 l

(W n) = Di (W n ) = 0

On lifting problems in C*-algebras Tatiana SHULMAN
University of Cop enhagen, Cop enhagen, Denmark E-mail address: shulman@math.ku.dk

There is a well known theorem of Titze-Urysohn about extensions of continuous functions. We are going to discuss noncommutative analogs of problems of this kind called lifting problems. In particular there will be considered a question of Loring and Pedersen about lifting of nilpotent contractions. This will be discussed in the contexts of semipro jectivity and pro jectivity questions in C*-algebras. On homological dimensions of algebras of continuous functions Seytek TABALDYEV
Bauman Moscow State Technical University, Department of Higher Mathematics, FN­1, Moscow, Russia

19


Infinite subsets of metric spaces which can b e spanned by finite length trees Alexey TUZHILIN Department of Mechanics and Mathematics, Moscow State Lomonosov University, Moscow, Russia E-mail address: tuz@mech.math.msu.su Being motivated by some gaps in Du-Hwang proof of Gilbert-Pollak conjecture on the Steiner ratio of Euclidean plane, and of the experimental fact that the Steiner ratio of Euclidean 3-space can not be achieved on a special set of the space, we decided to generalize the theory of Minimal Spanning Trees (MST) and Steiner Minimal Trees (SMT) to the case of infinite subsets of metric spaces. The first step is to describe the subsets for which the lengths of MST and SMT are finite. The later is equivalent to the fact that the subset can be spanned by a finite length tree. We have obtained the corresponding criterion, and described some properties of such subsets in Banach spaces. On almost representations of groups with prop erty (T) Chao YOU
Harbin Institute of Technology, Harbin, China E-mail address: hityou1982@gmail.com

Kazhdan's property (T) for groups is equivalent to the alternative that a representation of such a group either has an invariant vector or does not have even almost invariant vectors. It is shown that, under a somewhat stronger condition due to Zuk, a similar alternative holds for almost representations.

20


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.