OPERATOR ALGEBRAS AND TOPOLOGY

A trace formula for foliated flows via adiabatic limits
Jesu´s Antonio A´LVAREZ LO´PEZ
Departamento de Xeometr´ıa e Topolox´ıa,
Facultade de Matema´ticas,
Campus Universitario Sur,
Universidade de Santiago de Compostela,
15782 Santiago de Compostela, Spain
E-mail address: jalvarez@usc.es
Let F be a C∞ foliation of codimension one on a closed manifold M , and let φt be a
foliated flow on (M,F). The subbundle of vectors tangent to the leaves will be denoted
by TF ⊂ TM . Suppose that each fixed point x of φt is non-degenerate in the sense
that the map 1 − Txφt : TxM → TxM is an isomorphism. Thus the fixed point set of
φt is finite. For each fixed point x, there is some real number κx 6= 0 such that the map
TxM/TxF → TxM/TxF induced by Txφt is multiplication by eκxt, and the sign of the
determinant of 1−Txφt : TxF → TxF will be denoted by �x. For simplicity, suppose that
φt has no periodic orbits different from fixed points.
Now, pick any riemannian metric g on M , which may not be bundle-like (F may not be
given by riemannian submersions). Consider the decomposition g = g⊥ ⊕ gF according
to the decomposition TM = TF⊥ ⊕ TF . We can also consider the family of metrics
gh = h−2g⊥ ⊕ gF for h > 0, whose limit as h → 0 is called the adiabatic limit. Observe
that the local distance between the leaves blows up at the adiabatic limit.
Let (Ω, d) denote the de Rham complex of M , let δgh be the gh-adjoint of d, and ∆gh =
dδgh + δghd the gh-laplacian. To work in a fixed Hilbert space for all h > 0, a rescaling
Θh is introduced by multiplying hu each differential form of “transverse degree” u. Then
δh = ΘhδghΘ
−1
h is adjoint of dh = ΘhdΘ
−1
h with respect to the inner product defined by g,
and we get the positive essentially self-adjoint operator ∆h = dhδh + δhdh = Θh∆ghΘ
−1
h .
There are limits as h → 0 of dh, δh and ∆h, which are respectively denoted by d0, δ0
and ∆0; these are differential operators which involve only leafwise derivatives, and ∆0 is
leafwise elliptic. We will also use φt∗h = Θhφ
t∗Θ−1h , whose limit as h → 0 is denoted by
φt∗0 .
Let pi0,· denote the orthogonal projection of Ω onto its subspace of differential forms with
“transverse degree” 0, and let pi0,v denote the orthogonal projection of Ω onto its subspace
with “transverse degree” 0 and “tangential degree” v. Then pi0,·d0pi0,· can be identified to
the de Rham derivative of the leaves acting on leafwise differential forms smooth on M
(the leafwise complex). Therefore the limit
Trv(φt,F) = lim
r→∞
lim
h→0
Tr(pi0,vφ
t∗
h e
−r∆hpi0,v) ,
1

2if it exists, can be considered as a version of the trace of the action of φt∗0 on the leafwise
reduced cohomology, and we have a corresponding “leafwise Lefstchetz number”
L(φt,F) =
∑
v
(−1)v Trv(φt,F) .
It will be shown that each Trv(φt,F) is well defined, and moreover
L(φt,F) =
∑
x
�x |1− e
κxt|−1 ,
where x runs in the set of fixed points.
Non-commutative algebraic geometry and the representation the-
ory of p-adic groups
Paul Frank BAUM
312 McAllister Bldg.,
Dept. of Math., Penn State University,
University Park, PA 16802, USA
E-mail address: baum@math.psu.edu
Non-commutative topology begins with Gelfand’s theorem asserting that commutative C∗
algebras and locally compact Hausdorff topological spaces are the same thing. Another
classical theorem states that unital commutative finitely-generated nilpotent-free algebras
(over the complex numbers) is the same thing as complex affine algebraic varieties. This
can be taken as the starting point for non-commutative algebraic geometry. Based on
this point of view, the talk states a conjecture within the representation theory of p-adic
groups. The idea of the conjecture is that a simple geometric structure underlies many
delicate and intricate results in this representation theory.
The above is joint work with Anne-Marie Aubert and Roger Plymen.
Complex Ray Singer torsion
Dan BURGHELEA (joint with Stefan Haller)
Department of mathematics,
Ohio State University,
231 West 18-th Avenue,
Columbus, OH 43210, USA
E-mail address: burghele@math.ohio-state.edu
This is joint work with Stefan Haller.

3Ray Singer torsion is real valued numerical invariant associated to a closed Riemannian
manifold and a flat connection in a vector bundle equipped with a Hermitian structure. It
is based on regularized determinants of Laplace Beltrami operators associated with these
geometric data, and when properly modified provides a topological invariant for a smooth
compact manifold, a complex representation of its fundamental group and an Euler or
equivalently co-Euler structure.
We define a complex number invariant associated with a closed Riemannian manifold
and flat connection in a complex valued vector bundle equipped with a non degenerate
symmetric bilinear form. This is again based on regularized determinants of Laplace
Beltrami type operators which are however non selfadjoint but have regularized complex
valued determinants without angular anomaly. When modified by the same additional
data one obtains a topological invariant whose absolute value is essentially the square of
the modified Ray-Singer torsion.
Our invariant depends holomorphically on the flat connection. Recently we have shown
that this invariant is equal to the one defined by Milnor and improved by Turaev associated
with a triangulation instead of Riemannian metric.
Our proof is based on a non selfadjoint Witten-Hellfer-Sjo¨strand theory. An alternative
proof was also provided by Su and Zhang.
Local operator spaces and noncommutative rational functions
Anar DOSIEV
Department of Mathematics, Atilim University,
Incek 06836, Ankara, Turkey
E-mail address: adosiev@atilim.edu.tr, dosiev@yahoo.com
We talk about realization of a local operator space (respectively, system) as a noncom-
mutative rational functions in the multinormed C∗-algebra context. Being unbounded
operators, these functions are appeared as quotients of bounded operators on a fixed
Hilbert space.
The Ricci flow on open manifolds
Ju¨rgen EICHHORN
Institut fu¨r Mathematik und Informatik,
Universita¨t Greifswald,
D-17487 Greifswald Germany
E-mail address: eichhorn@uni-greifswald.de

4The Ricci flow is the natural evolution of the geometry on a manifold. We present some
short time existence theorems and prove invariance properties for Sobolev spaces, the ex-
istence of spectral gaps, the continuous and essential spectrum and characteristic numbers
(defined by integration).
On subgroups of the unitary group of a Hilbert space and a gen-
eralization of the twisted K-theory
Andrei V. ERSHOV
Chair of Geometry, Saratov State University
E-mail address: ershov@higeom.math.msu.su
Let H be a separable Hilbert space; B, K its algebras of bounded and compact operators
respectively. Let K+ be the algebra obtained by adjoining a unit to K and Mn(K+) the
algebra of n × n matrices over K+. We study homotopy types of the topological groups
Aut(Mn(K+)), n ∈ N (equipped with the norm topology), and show how they would
provide a more general example of twistings in K-theory than the usual one (corresponding
to the subalgebra K+ ⊂Mn(K+) whose automorphism group is PGL(B)).
Generalized Burnside-Frobenius theorem
Alexander FELSHTYN
Department of Mathematics, Boise State University, 1910
University Drive, Boise, Idaho, 83725-155, USA
E-mail address: felshtyn@diamond.boisestate.edu
It is proved for a residually finite groups that the Reidemeister coincidence number of two
surjective endomorphisms φ and ψ is equal to the number of finite-dimensional coincidence
points of dual maps φ̂ and ψ̂ on the unitary dual, if Reidemeister coincidence number is
finite. This theorem is a natural generalization to infinite groups of the classical Burnside-
Frobenius theorem and immediately follows from the twisted Burnside-Frobenius theorem
for automorphisms of residually finite groups which was proved in the article Alexander
Fel’shtyn, Evgenij Troitsky. Twisted conjugacy separable groups. Math.GR/0606764,
2006. For Abelian groups we prove the Generalized Burnside-Frobenius theorem for any
two endomorphisms φ and ψ. Some counter-examples for groups with extreme properties
are discussed.
KK-theory of C∗-algebras related to Pimsner algebras
Emmanuel GERMAIN

5Institut de mathe´matique de Jussieu
175 rue du Chevaleret
75013 PARIS
FRANCE
E-mail address: germain@math.jussieu.fr
A-valued semi-circular system, introduced by D. Shlyakhtenko, is a common object in
free probability theory. We show that the K-theory of the C*-algebra it generates can be
investigated in relation with Pimsner algebras. We also consider the case of q-Fock spaces
and q-semi-circular systems.
Ihara zeta function for periodic and fractal graphs
Daniele GUIDO (in collaboration with T. Isola and M.L. Lapidus)
Department of Mathematics,
Univ. Roma ”Tor Vergata”
Via della Ricerca Scientifica 1,
I-00133 Roma, Italy
E-mail address: guido@mat.uniroma2.it
Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta
functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and
Sato, to name just a few authors. Then, Bass (1992) and Clair and Mokhtari-Sharghi
(2001) have studied zeta functions for infinite graphs acted upon by a discrete group of
automorphisms. The main formula in all these treatments, the so-called determinant for-
mula, establishes a connection between the zeta function, originally defined as an infinite
product, and the Laplacian of the graph. On the one hand we show an approximation
formula for the zeta function of infine periodic graphs acted upon by an amenable group.
On the other hand, we define and study zeta functions for self-similar fractal graphs. We
prove a determinant formula, functional equations, and a formula which allows approxi-
mation of the zeta function by the zeta functions of finite subgraphs.
On the quantum versions of injective tensor product and duality:
the non-matricial approach
Alexander Ya. HELEMSKII
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia

6E-mail address: alexander@helemskii.mccme.ru, helemskii@comail.ru
We continue the program of the presentation of basic notions and results of quantum
functional analysis in the framework of the non-matricial, or non-coordinate approach.
In this talk we introduce, in this context, the quantum version of the classical injective
tensor product of normed spaces and establish some of its properties. At the end we
shall present, in the same framework, the comparatively easy calculation of quantum dual
spaces to some Hilbertians.
Some examples for the twisted Burnside theory
Fedor INDUKAEV
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: indukaev@mail.ru
Let G be a discrete group and φ its automorphism. The number of classes of the equiva-
lence x ∼ gxφ(g−1), g ∈ G is called the Reidemeister number of φ. Depending on G and
φ this number can be finite or infinite. Fel’shtyn and Troitsky proved that for a large
class of groups the following is true: if the Reidemeister number of an automorphism is
finite then it is equal to the number of fixed points of the induced map φ̂ on the finite-
dimensional part of the unitary dual object. The class in question is called RP and is
known to contain all almost polycyclic groups. The first example of not almost polycyclic
group satisfying this condition will be presented in the talk. Also it will be shown that
the discrete Heisenberg group has an automorphism with Reidemeister number N for any
even number N .
Coverings in noncommutative geometry
Nickolay IVANKOV
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: monstr3d@korolev-net.ru

7An alalogue of unramified coverings [1] of spectral triples [2] has been introduced. This
analogue enables us to denote fundamental group of spectral triples and unoriented spec-
tral triples. It was shown that in commutative case this fundamental group is a profi-
nite completion of fundamental group of corresponding Riemann manifold. Fundamental
group of noncommutative torus has been calculated.
References [1] J.S. Milne, ”Etale cohomology” , Princeton Univ. Press 1980.
[2] J. C. Varilly. An Introduction to Noncommutative Geometry , arXiv:physics/9709045
v 1997.
Uniqueness of Steiner minimal tree for general position boundary
in the plane
A.O. IVANOV, A.A. TUZHILIN
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: tuz@mech.math.msu.su
Steiner minimal tree is the shortest tree among all ones connecting points of a finite subset
M of a metric space X. For a given M there can be more then one Steiner minimal tree.
We prove that in the case when X is the standard Euclidean plane and M is ”in general
position” the corresponding Steiner minimal tree is unique. We’ll give a few consequences
of this main result and state some related problems.
Pre-differential operator on graded algebra
Vyacheslav KALNITSKY
Higher Geometry Dept.
St.-Petersburg State University,
Universitetsky Prosp. 28
St.-Petersburg 198504, Russia
E-mail address: skalnitsky@hotmail.com
Let’s consider the graded Lie algebra L of homogeneous vector fields with respect to
Liouville field L on TM L =
⊕∞
k=0{X|[L,X] = kX}. Due to the complete lift of tensor
fields with respect to any linear connection the algebra L can be described as the Lie
algebra S =
⊕∞
k=0 ST
1
k (M) with symmetrized convolution ◦ as multiplication. The
structure of Lie algebra on it is induced from L and can be described in terms of ◦ and
the inner differential δ. We say that L is δ-lift of S and δ is pre-differential [1].

8The spray S of the connection in turn lies in L. The subalgebra A = KeradS is called
the Jacobi algebra of connection [2]. We show that Jacobi algebras A(R1) and A(R2) of
the affine line and plane have the structure of δ-lifts of some algebras.
Let P [v] be the algebra of polynomials with pre-differential δ ≡ 0, then A(R1) is isomor-
phic to δ-lift of P [v]⊗ R.
Let’s now introduce the complex
C• : 0 → h0
(1)
δ→ h1
(2)
δ→ ...
where hk = {δ2A = 0|A ∈ ST 1k (R
2)}, h0 ∼= aff2(R). The algebra A(R2) is isomorphic to
the δ-lift of the algebra
∞⊕
i=0
(
Ker
(2i+1)
δ ⊕Im
(2i+1)
δ
)
∼=
(
P [v0, v1, v2]/P [v
2
0]
)
⊗ R2,
where vi corresponds the spray invariant functions which are linear on the fibers of TM
and P [v0]⊗ R2 ∼= H•(C•).
References
[1] Kalnitsky V.S., D-lift of convolution algebras //Proc. Jacobi Int. Conf., Albertina univ. June 20-26,
2005
[2] Kalnitsky V.S., Algebra of generalized Jacobi fields // J. Math. Sci. (New-York) 91 (1998), no. 6, pp.
3476–3491.
A K-theoretic index formula for transversally elliptic operators
Gennadi KASPAROV
Dept. of Mathematics,
1326 Stevenson Center,
Vanderbilt University,
Nashville, TN 37240, USA
E-mail address: gennadi.kasparov@vanderbilt.edu
In the index theory of elliptic operators, there is a K-theoretic index formula (essentially
equivalent to the Atiyah-Singer index formula), namely, the index of an elliptic operator in
K-homology is equal to the intersection product of the symbol of this operator (which is an
element of K-cohomology) and a certain K-homology class called the ”Dolbeault element”.
Conjecturally, a similar K-theoretic index formula also exists for transversally elliptic
operators, except that the ”Dolbeault element” for transversally elliptic operators has not
been constructed until now. In this talk, I will explain the construction of the ”Dolbeault
element” for transversally elliptic operators and outline the proof of the resulting K-
theoretic index formula.

9Geodesic flows in transverse geometry of Riemannian foliations
Yuri KORDYUKOV
Ufa Institute of Mathematics,
Russian Academy of Sciences
112 Chernyshevsky str.
450077 Ufa, RUSSIA
E-mail address: ykordyukov@yahoo.com
We will introduce classical and quantum analogues of the geodesic flow on the leaf space
of a Riemannian foliation on a compact manifold as well as the noncommutative geodesic
flow associated with the spectral triple given by a transverse Dirac type operator on the
foliated manifold. We will describe some relationships between these objects, which are
based, in particular, on Egorov’s theorem for matrix valued transversally elliptic operators
on Riemannian foliations. Some related topics will be discussed.
Jan KUBARSKI
Institute of Mathematics, Technical University of Lodz,
ul. Wolczanska 215
PL-93005 Lodz, Poland
E-mail address: kubarski@mail.p.lodz.pl
In the paper J.Kubarski and N.Teleman, ”Linear Direct Connections”, Banach Center
Publications (in printing) to each linear direct connections τ : U → GL (E) in a vector
bundle E [∆ ⊂ U ⊂M×M, τ (x, y) : Ey
∼=→ Ex, τ (x, x) = id ] is associated infinitesimally
an usual linear connection ∇τ in E. Next, it is proved that image of the periodic cyclic
cocycle Φτ2k through the Connes’ isomorphism coincides with the differential form provided
by the classical Chern - Weil theory applied upon the underlying linear connection ∇τ
Ω2k(Φ
τ
2k) =
1
(2k)!
·
1
2k
· Tr Rk,
where R = (∇τ )2 is the curvature of the underlying linear connection ∇τ . We recall
that the Connes’ isomorphism associates with any periodic cyclic cycle f an even/odd
non-homogeneous closed differential form Ω(f) on M . As an application it is obtained a
direct proof of Theorem 6.2 from the paper Teleman N.: Direct Connections and Chern
Character, Proceedings of the International Conference in Honor of Jean-Paul Brasselet,
Luminy, May 2005, which shows how we can extract the Chern-character of a smooth
vector bundle E from any direct linear connection τ in E.
The aim of this talk it is to give a concept of the linear direct connections τ in arbitrary
(transitive) Lie groupoid Φ and to show haw we can extract the Chern-Weil homomor-
phism of Φ from τ.

10
Some quadratic equations in the free group of rank 2
Elena KUDRYAVTSEVA
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: eakudr@mech.math.msu.su
We present here a joint work (2006) with Daciberg Lima Gonc¸alves (Brasil) and Heiner
Zieschang (Germany).
For a given quadratic equation with any number of unknowns in any free group F , with
right-hand side an arbitrary element of F , an algorithm for solving the problem of the ex-
istence of a solution was given by Culler (1981), using a surface method and generalizing
a result of Wicks (1962). Based on another geometric techniques, the problem has been
studied by the authors (2001-02) for parametric families of quadratic equations arising
from continuous maps between closed surfaces, with certain conjugation factors as the
parameters running through the group F . In particular, for a one-parameter family of
quadratic equations in the free group F2 of rank 2, corresponding to maps of absolute
degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence
of faithful solutions has been solved in terms of the value of the self-intersection index
µ : F2 → Z[F2] on the conjugation parameter. The present paper investigates the ex-
istence of faithful, or non-faithful, solutions of similar families of quadratic equations
corresponding to maps of absolute degree 0. The existence results are proved by con-
structing solutions. The non-existence results are based on studying two equations in
Z[pi] and in its quotient Q, respectively, which are derived from the original equation
and are easier to work with, where pi is the fundamental group of the target surface, and
Q is the quotient of the abelian group Z[pi \ {1}] by the system of relations g ∼ −g−1,
g ∈ pi \ {1}. Unknown variables of the first and second derived equations belong to pi,
Z[pi], Q, while the parameters of these equations are the projections of the conjugation
parameter to pi and Q, respectively. In terms of these projections, sufficient conditions for
the existence, or non-existence, of solutions of the quadratic equations in F2 are obtained.
Noncommutative geometry and quantum groups
Giovanni LANDI
Dipartimento di Matematica e Informatica,
Universita’ di Trieste,
via A. Valerio 12/1

11
I-34141 Trieste, Italy
E-mail address: landi@univ.trieste.it
We present some recent constructions of equivariant spectral triples on spaces associated
with quantum groups. These include the manifold of quantum SU(2), two-dimensional
quantum spheres as well as four-dimensional orthogonal quantum spheres. The spectral
triples are isospectral in that the spectrum of the Dirac operator is the same as the one
of the corresponding classical operator. The conditions for a real structures need to be
weakened and are satisfied only up to infinitesimals of arbitrary high order.
Новое условие гомотопической инвариантности дифференциа-
ла в теории возмущений дифференциальных модулей
S.V.LAPIN (С.В.Лапин)
кафедра алгебры и геометрии
Мордовского государственного университета им. Н.П.Огарева
E-mail address: slapin@mail.ru
Напомним, что в теории возмущений дифференциальных модулей под возмущени-
ем дифференциального модуля (X, d) понимается рассмотрение нового дифференци-
ального модуля (X,D), дифференциал D которого, вообще говоря, никак не связан с
дифференциалом d. Отображение t = d−D называется возмущением дифференциала
d. Основной задачей в теории возмущений дифференциальных модулей является ис-
следование условий гомотопической инвариантности дифференциала D, т.е. инвари-
антностиD при гомотопических эквивалентностях вида (η : (X, d) −→←− (Y, d) : ξ , h), где
ηξ = 1Y , dh+hd = ξη−1X , ηh = 0, hξ = 0, hh = 0, называемых SDR-ситуациями диф-
ференциальных модулей. В работе [1] было показано, что если задана SDR-ситуация
( η : X −→←− Y : ξ , h ) дифференциальных модулей с возрастающими фильтрациями
{Xn}, {Y n} и задано возмущение (X,D = d + t) дифференциального модуля (X, d),
удовлетворяющее условию t(Xn) ⊆ Xn−1, n > 0, то дифференциал D гомотопически
инвариантен, т.е. на дифференциальном модуле (Y, d) также появляется дифферен-
циал D˜ = d + t˜ и, кроме того, возникает SDR-ситуация ( η˜ : (X,D) −→←− (Y, D˜) : ξ˜ , h˜ )
дифференциальных модулей с возрастающими фильтрациями, где t˜(Y n) ⊆ Y n−1.
В следующем утверждении дается новое фильтрационное условие, отличное от усло-
вия t(Xn) ⊆ Xn−1 из [1], гарантирующее гомотопическую инвариантность диффе-
ренциала D.
Теорема. Пусть задана SDR-ситуация ( η : X −→←− Y : ξ , h ) дифференциальных моду-
лей с возрастающими фильтрациями {Xn}, {Y n} и задано возмущение (X,D = d+ t)
дифференциального модуля (X, d), удовлетворяющее условию D(Xn) ⊆ Xn−1, n > 0.

12
Тогда дифференциал D гомотопически инвариантен, т.е. на дифференциальном мо-
дуле (Y, d) также имеется дифференциал D˜ = d + t˜ и, кроме того, определена SDR-
ситуация ( η˜ : (X,D) −→←− (Y, D˜) : ξ˜ , h˜ ) дифференциальных модулей с возрастающими
фильтрациями {Xn} и {Y n}, где D˜(Y n) ⊆ Y n−1. �
Теперь отметим, что если рассматривать SDR-ситуации дифференциальных модулей
с дополнительными алгебраическими структурами [2], то следствием гомотопиче-
ской инвариантности дифференциала D, при выполнении указанного выше условия
t(Xn) ⊆ Xn−1 из [1], является гомотопическая инвариантность структур дифферен-
циальных A∞-алгебр [3] и дифференциальных E∞-алгебр [4]. Следствием же гомо-
топической инвариантности дифференциала D, при выполнении условия D(Xn) ⊆
Xn−1 из сформулированной выше теоремы, является гомотопическая инвариант-
ность структур градуированных A∞-алгебр и градуированных E∞-алгебр относи-
тельно дифференциала, с которым данные структуры, вообще говоря, никак не свя-
заны.
Список литературы
[1] Gugenheim V.K.A.M. On a chain complex of a fibration // Illinois J. Math. 1972. V.3. P.398-414.
[2] Gugenheim V.K.A.M., Lambe L.A., Stasheff J.D., Perturbation theory in differential homological
algebra II // Illinois J. Math. 1991. V.35. N 3. P. 357-373.
[3] Gugenheim V.K.A.M., Stasheff J.D., On perturbations and A∞-structures // Bull. Soc. Math. Belg.,
1986. V.38. P. 237-246.
[4] Лапин С.В. D∞-дифференциальные E∞-алгебры и мультипликативные спектральные после-
довательности. // Матем. сб. 2005. Т. 196. 11. С. 75-108.
An index theorem for pseudomanifolds with conical singularities
Jean-Marie LESCURE
Laboratoire de Mathe´matiques
Universite´ Blaise Pascal
Campus Universitaire des Ce´zeaux 63177 Aubie`re cedex France
E-mail address: lescure@math.univ-bpclermont.fr
This is a joint work with Claire Debord and Victor Nistor. Using an appropriate notion
of tangent space of a pseudomanifold with conical singularities, I will explain how to
define analytical and topological indices on a pseudomanifold with conical singularities
in the spirit of M.F. Atiyah and I.M. Singer. Moreover, I will show that groupoids and
bivariant K-theory allow a proof of the Atiyah-Singer index theorem which generalizes
straightforwardly to the case of pseudomanifolds with conical singularities.
The Maslov dequantization, convex geometry and linear opera-
tors

13
G. L. LITVINOV (joint with G. B. SHPIZ)
Independent University of Moscow
E-mail address: islc@dol.ru
1. For functions defined on Cn (or Rn+) the well-known Maslov dequantization [1] generates
a dequantization transform f 7→ fˆ :
f 7→ fˆ(x) = lim
h→+0
h log | f(exp(x/h))|,
where h is a (small) real parameter and x ∈ Rn.
Theorem. If f is a polynomial, then the subdifferential ∂fˆ of fˆ at the origin coincides
with the Newton polytope of f . For the semiring of polynomials with nonnegative coef-
ficients, the transform f 7→ ∂fˆ is a homomorphism of this semiring to the semiring of
convex polytopes with respect to the well-known Minkovski operations.
Using the dequantization transform it is possible to generalize this result to a wide class
of functions and convex sets.
2. Suppose that A is a linear operator in a finite dimensional linear space V over C,
v ∈ V , and v′ : v 7→ 〈v′, v〉 is a linear functional on V . The following upper limit
lim
h→+0
h log | 〈v′, exp(Av/h)〉 |
is called a dequantization of the matrix element 〈v′, Av〉.
Theorem. The set of all dequantizations of all matrix elements of A coincides with the
set of real parts of all eigenvalues of the operator A.
There are many generalizations of this result and applications to the representation theory.
References: [1]. G.L. Litvinov, The Maslov dequantization, idempotent and tropical math-
ematics: a brief introduction // Journal of Mathematical Sciences, 2006; E-print: arXiv:
math.GM/ 0507014, 2005 (http://arXiv.org).
Some recent progress in the theory of C∗-algebra extensions
Vladimir MANUILOV
Moscow State University and Harbin Institute of Technology
E-mail address: manuilov@mech.math.msu.su
Some recent results in the theory of C∗-algebra extensions obtained jointly with K. Thom-
sen will be discussed.

14
First, we discuss non-invertibility of extensions. We show that this deficiency of the BDF
theory (Ext is not always a group) persists on the level of homotopy. We also explain
how non-exact C∗-algebras can be used to construct extensions that are not homotopy
invertible. The construction of such extensions generalizes that of S. Wassermann.
Second, we define a relative BDF theory for a C∗-algebra and its C∗-subalgebra. The
definition is based on the notion of absorbing representation developed by G. Kasparov
and generalized by K. Thomsen. We present a six-term exact sequence for this relative
Ext functor. We also apply this functor to some approximation problems for normal
operators.
Massey products in graded Lie algebra cohomology
Dmitri MILLIONSCHIKOV
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: million@mech.math.msu.su
We discuss Massey products in a N-graded Lie algebra cohomology. One of the main
examples is so-called ”positive part” L1 of the Witt algebra W . Buchstaber conjectured
that H∗(L1) is generated with respect to non-trivial Massey products by H1(L1). Feigin,
Fuchs and Retakh represented H∗(L1) by trivial Massey products and the second part of
the Buchstaber conjecture is still open. We consider an associated graded algebra m0 of
L1 with respect to the filtration by its descending central series and prove that H∗(m0) is
generated with respect to non-trivial Massey products by H1(m0).
Poincare duality and signature for topological manifolds
Alexander MISHCHENKO (joint with Peter POPOV)
Dept. of Mechanics and Mathematics,
Moscow State Lomonosov University
Leninskie Gory, Moscow,
119992, Russia
E-mail address: asmish@mech.math.msu.su
The signature of the Poincare duality of compact topological manifolds with local sys-
tem of coefficients can be described as a natural invariant of nondegenerate symmetric
quadratic forms defined on a category of infinite dimensional linear spaces.

15
The objects of this category are linear spaces of the form W = V ⊕ V ∗ where V is
abstract linear space with countable base. The space W is considered with minimal
natural topology.
The symmetric quadratic form on the space W is generated by the Poincare duality
homomorphism on the abstract cochain group induced by nerves of the system of atlases
of charts on the topological manifold.
Boutet de Monvel Calculus and Groupoids
Bertrand MONTHUBERT
Universite´ P. Sabatier
Address:
32 chemin de la Pe´lude
F-31400 Toulouse, France
E-mail address: bertrand@monthubert.net
Joint work with J. Aastrup, S.T. Melo and E. Schrohe. This paper is part of our effort
to answer the question: Can Boutet de Monvel’s algebra on a compact manifold with
boundary X be obtained as the algebra of pseudodifferential operators Ψ0(G) on some
Lie groupoid G? If it could, the kernel of the principal symbol would have to be isomorphic
to the groupoid C∗-algebra C∗(G). We exhibit a groupoid G such that C∗(G) possesses
an ideal I isomorphic to the algebra of Singular Green Operators G, which is the kernel
of the principal symbol homomorphism on Boutet de Monvel’s algebra. We prove that G
is isomorphic to the algebra of pseudodifferential operators on the boundary Y tensored
by the compact operators, that both I and G are extensions of C(S∗Y )⊗K by K (where
K denotes the compact ideal and S∗Y denotes the co-sphere bundle over the boundary of
X), and that the index mappings in the standard K-theory cyclic exact sequences induced
by these two extensions are the same.
On homotopy classification of elliptic operators on manifolds with
singularities
Vladimir NAZAYKINSKIY (joint with Anton SAVIN and Boris STERNIN)
Institute for Problems in Mechanics, RAS,
Independent University of Moscow
Moscow, Russia
E-mail address: nazaikinskii@yandex.ru, antonsavin@mail.ru, sternin@mail.ru
We compute the group of stable homotopy classes of elliptic operators (Ell-group) on two
classes of manifolds with singularities.