Wolfram Bauer (Uni. Mainz) Mean oscillation and Hankel operators on
the Segal-Bargmann space
For a large symbol-class S of complex valued
functions on $\mathbb{C}^n$ we completely characterize the elements f in S for
which the Hankel operators $H_f$ and $H_{\bar{f}}$ are simultaneously bounded or
compact on the Segal Bargmann space in terms of the mean oscillation of f. The
analogous description holds for commutators $[M_f,P]$ where $M_f$ denotes the
multiplication by f and P is the Toeplitz projection. Finally we characterize
the entire functions $f$ in S and the polynomials p in the complex variables $z$
and $\bar{z}$ for which $H_{\bar{f}}$ and $H_p$ are bounded (resp. compact).
Paul Baum (Penn State University) Dirac Operators and K theory for
discrete groups
In this talk the BC (Baum-Connes) conjecture for
discrete groups is reformulated using Dirac operators. The universal example for
proper actions and Kasparov equivariant K homology are not used. One corollary
of this approach is an explicit Chern character for the left side of BC. This
Chern character is obtained via the Atiyah-Singer index formula for the Dirac
operator.
Brasselet Jean-Paul (IML - CNRS - Marseille) Algebras of functions on
stratified spaces
This is a survey on join works with A. Legrand, N.
Teleman and with M. Pflaum. A. Connes proved that Hochschild homology of the
algebra of smooth functions on a manifold corresponds to the Rham complex and
that periodic cyclic homology (of the algebra) corresponds to the de Rham
cohomology (of the manifold). One defines suitable algebras on singular
(stratified) varieties in order to prove a similar result in the singular case.
Jacek Brodzki (Univ. Southampton) Approximation properties and
exactness for discrete groups
It is known from the work of Kirchberg and
Wassermann that if a C*-algebra admits a completely bounded approximation
property (CBPAP) then it is exact. In the context of discrete we show that the
reduced C*algebra has the CBPAP when the group admits uniformly bounded
representations and so such groups are exact. We illustrate these concepts using
properties of groups acting on trees and Coxeter groups.
Dan Burghelea (OSU, Columbus, Ohio) Euler and co-Euler structures and
a New invariant for non simply connected manifolds
This is joint work
with S. Haller. We revisit an old concept introduced by Turaev (Euler
structure), extend it to all manifolds and use it to remove the geometric
ambiguity of torsion. In the way an interesting new Riemanian invariant defined
by geometric regularization of some divergent integrals appear. As a consequence
we extend the Alexander polynomial (associated to the complement of a knot) to
an invariant defined for any non simply connected manifold. Our new invariant is
a meromorphic function on the space of complex degree N representations of the
fundamental group. When the fundamental group made commutative is the group of
integers and N=1 the space of representation is the complex plane but zero so
our invariant is a meromorphic function of one variable . If the manifold is the
complement of a knot this function is essentially the Alexander polynomial.
Gratiela Cicortas (University of Oradea) Perturbation methods on
Morse theory for continuous functionals
Alexander Helemskii (Moscow State University) Homology in
$C^*$-algebras: "classical" and "quantized" approaches
The aim of this
talk is to say about some recent developments of homology in $C^*$-algebras. In
particular, we have completed the program of the description of projective
Hilbert modules over $C^*$-algebras in all existing homology theories. (We mean
Banach, or 'classical" homology and two versions of "quantized" homology). All
irreducible projective Hilbert modules are characterized in terms of elementary
projections, and they are the same in all three theories. Further, in both
quantized homology theories general projective Hilbert modules are described as
$l_2$-sums of arbitrary families of irreducible modules, whereas in the
classical theory the respective sums must satisfy the additional condition of
the so-called essensial finiteness. As a result of joint efforts with
Yu.Selivaniv and O.Aristov, biprojective $C^*$-algebras also obtained their
description. Here, in the classical homology theory as well as in the quantized
homology, based on the operator projective tensor product, biprojective
$C^*$-algebras turned out to be $c_0$-sums of full matrix $C^*$-algebras. At the
same time, in the quantized homology, based on the Haagerup tensor product,
biprojective $C^*$-algebras turned out to be $c_0$-sums of $C^*$-algebras of
compact operators (i.e. dual $C^*$-algebras in the sense of Kaplansky).
Yuri. A. Kordyukov (Russian Academy of Sciences) Semiclassical
spectral asymptotics for periodic magnetic Schroedinger operators
We
study semiclassical spectral approximations of periodic magnetic Schroedinger
operators on covering spaces of compact manifolds in two cases. In the first
case, the electric field is strong, and, in the second case, the electric field
vanishes, and the magnetic field is strong. Under some Morse type assumptions,
we prove existence of arbitrarily large number of gaps in the spectrum of
Schroedinger operators in these semiclassical limits. We also establish a
vanishing theorem for the higher traces in cyclic cohomology of the spectral
projections of Schroedinger operators in the semiclassical limit of the strong
electric field. The results are partially based on a joint work with V. Mathai
and M. Shubin.
Jan Kubarski (Technical University of Lodz) From Poincare duality to
Evans-Lu-Weinstein pairing (joint with A. Mishchenko)
see PDF file or
the home page of the speaker The cohomology pairing coming from Evans-Lu-Weinstein representation of a
Lie algebroid [E-L-W] is very important in many applications of Lie algebroids
(Poisson geometry [W], [E-L-W], intrinsic characteristic classes [C], [F]). This
pairing generalizes the well known pairings that give Poincare duality for Lie
algebra cohomology and de Rham cohomology of a manifold and real cohomology of
transitive invariantly oriented Lie algebroids [K2], [K3]. For a Poisson
manifold, this pairing agree with the pairing on the Poisson homology. The
authors of [E-L-W] give an example of a nontransitive Lie algebroid for which
the pairing is not necessarily non-degenerate and post the problem of when it is
non-degenerate. This paper gives the positive answer for the case of any
transitive Lie algebroids and prove the capacity of this representation: it is
the one (up to isomorphy) for which the top group of compactly supported
cohomology is nontrivial. In proofs of these theorems for Lie algebroids it is
used the Hochschild-Serre spectral sequence and it is shown the general fact
concerning pairings between graded filtered differential R-vector spaces:
assuming that the second terms live in the finite rectangular, nondegeneration
of the pairing for the second terms (which can be infinite dimensional) implies
the same for cohomology spaces. This theorem generalize the theorem for algebras
[K-M].
[C] M. Crainic, Differentiable and algebroid cohomology, Van Est
isomorphisms, and characteristic classes, preprint, arXiv:math.DG/0008064,
Commentarii Mathematici Helvetici, to appear.
[E-L-W] S.Evens, J-H.Lu, and A.Weinstein, Transverse measures, the modular
class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford 50
(1999), 417-436.
[F] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes,
}preprint DG/007132, Advances in Mathematics 170, (2002) 119-179.
[K1] J.Kubarski, Fibre integral in regular Lie algebroids, New
Developments in Differential Geometry, Budapest 1996, Proceedings of the
Conference on Differential Geometry, Budapest, Hungary, July 27-30, 1996;
Kluwer Academic Publishers 1999.
[K2] J.Kubarski, Poincare duality for transitive unimodular invariantly
oriented Lie algebroids, Topology and Its Applications, Vol 121, 3, June 2002,
333-355.
[K-M] J.Kubarski, A.S.Mishchenko, Lie algebroids: spectral sequences and
signature, Sbornik: Mathematics, Volume 194(2003), Number 7, Pages 1079-1103.
[W] A.Weinstein, The modular automorphism group of a Poisson manifold, }J.
Geom. Phys. 23 (1997), 379-394.
Vladimir Manuilov (Moscow State University) Asymptotic homomorphisms
and extensions of C*-algebras
A C*-extension is a short exact sequence
$0-->B-->E-->A-->0$ of C*-algebras. The BDF theory classifies
C*-extensions by K-homology when $A$ is commutative or, more generally, nuclear.
Unfortunately, not much is known beyond the nuclear case. A construction by
Connes and Higson relates C*-extensions to asymptotic homomorphisms. We describe
some modifications of that construction, which are then used to classify
C*-extensions by homotopy classes of asymptotic homomorphisms.
Severino Toscano do Rego Melo (University of Sao Paulo) Boundary
principal symbols in K-theoretic computations
Let A denote the
C*-algebra of bounded operators on $L^2(R)$ generated by all operators of
multiplication by 2$\pi$-periodic continuous functions and by functions having
limits at plus and minus infinity, and by all Fourier multipliers with symbols
having limits at plus and minus infinity. The Fredholm property for operators in
A is equivalent to the invertibility of two "symbols" on A: the principal symbol
and an operator-valued "boundary principal symbol" (the boundary here are two
copies of the circle, one at minus and another at plus infinity). Moreover, the
quotient of the kernel of the principal symbol by the compact ideal is
isomorphic to a compact-operator-valued algebra of functions, with isomorphism
induced by the boundary principal symbol. Such a Fredholm criterion governed
by two symbols, and such a composition series defined by them, are very common
for C*-algebras generated by pseudodifferential operators; that's the case also
for the C*-algebra B generated by all polyhomogeneous Green operators of order
and class zero in Boutet de Monvel's calculus on a compact manifold with
boundary. Both for A and for B, good descriptions of the image and of the
kernel of the boundary principal symbol are enough to compute their K-groups;
that's the subject of my talk. At first glance, it may look more natural to
use the exact sequence defined the principal symbol. What makes that alternative
approach harder, however, is the fact that one would have to explicitly compute
connecting mappings landing on K-groups of compact-operator valued algebras of
functions. While the quotient of the kernel of the boundary-principal symbol by
the compact ideal is commutative, in both examples considered here.
Bertrand Monthubert (Universit Paul Sabatier) A Atiyah-Singer type
index theorem for manifolds with corners
We obtain an index theorem for
the analytic index on the groupoid of a manifold with corners, using an
embedding technique as in the work of Atiyah and Singer.
Igor Nikonov (Moscow State University) Characteristic classes of
approximately finite algebras
We describe the kernel of Connes-Chern
character for two families of C*-algebras: approximately finite dimensional
C*-algebras and von Neumann algebras. A similar result is obtained for the
reduced Connes-Chern character.
Theodore Popelensky (Moscow State Lomonosov University) Cohomology
with internal symmetries for Hopf algebras
We are going to speak about
dihedral and reflexive cohomology of Hopf algebras. It is a generalization of
Connes-Moscovici cyclic cohomology for Hopf algebras. Constructions and some
properties will be discussed.
Peter S. Popov (Moscow State University) (joint with Alexander
Mishchenko) Infinite dimensional generalization of the signature type
invariants of topological manifolds
We have constructed the signature of
compact topological manifold with local system of coefficients, generated by the
natural representation of the fundamental group $\pi$ in the $C^*-$algebra
$C^*(\pi)$. To prove this conjecture we consider $C^*-$ modules of singular
chains and cochains. These modules are infinite dimensional, nevertheless the
signature is represented as a difference of finite dimensional modules and
represents an element of the K- theory of $C^*-$algebra $C^*(\pi).$ Category
of infinite dimensional modules and their self-adjoint mappings with finite
dimensional signature is constructed. The objects of this category are modules
of the form $W=V\oplus V^*$ where $V$ is free module with minimal natural
topology and $V^*$ is its adjoint module.
Luigi Rodino (Universita' di Torino) Lower bounds for
pseudo-differential operators
We report on some results obtained in
collaboration with F.Nicola, concerning classical self-adjoint
pseudo-differential operators. We assume that the principal symbol is
non-negative, and vanishes exactly to the order k on a smooth characteristic
manifold. In the case when the (even) integer k is larger than 2 , we give a
necessary and sufficient condition for the validity of a lower bound with gain
of k/(k-2) derivatives.
Anton Savin (Independent University of Moscow) Elliptic operators on
manifolds with singularities and K-homology
In 1970's Atiyah showed that
elliptic operators on a smooth closed manifold define cycles in K-theory. The
relationship between elliptic theory and $K$-theory is even more precise: the
group Ell(M) of stable homotopy classes of elliptic pseudodifferential operators
on a manifold M is isomorphic to the even K-homology group of the manifold:
Ell(M)=K_0(M). It turns out that a similar isomorphism holds in many situations,
when the manifold is no longer smooth. In the talk, we discuss an analog of this
isomorphism when M is a manifold with edges.
Elmar Schrohe (Hannover) Boundary Value Problems on Manifolds with
Conical Singularities
We study the closed extensions (realizations) of
differential operators subject to homogeneous boundary conditions on weighted
$L_p$-Sobolev spaces over a manifold with boundary and conical singularities.
Under natural ellipticity conditions we determine the domains of the minimal and
the maximal extension. We show that both are Fredholm operators and give a
formula for the relative index. Moreover we find stronger assumptions which
allow us to describe the resolvent of the minimal extension and to prove the
existence of a $H_\infty $-functional calculus and hence maximal regularity FOR
THE SOLUTIONS of the associated evolution equations.
Bert-Wolfgang Schulze (University of Potsdam) Operators with Symbolic
Hierarchies on Spaces with Higher Corners We consider spaces with corners that
are iteratively defined by local wedges with model cones that have singular
bases of lower singularity orders, together with certain regularity conditions
on the transition maps. Examples can be constructed in terms of degenerate
Riemannian metrics; the associated Laplace-Beltrami operators are (in stretched
coordinators) degenerate in a typical way. More generally, there are natural
classes of "corner degenerate" differential operators. The problem is to
organise a pseudo-differential algebra with a symbolic structure which contains
the typical differential operators together with the parametrices of elliptic
elements. We construct such algebras in a number of interesting cases, where the
symbolic information is encoded by hierarchies of scalar and operator-valued
components. We concentrate on higher edge symbols, given in terms of families of
operators on infinite singular cones. This requires an analysis of operators
when edges have conical exits to infinity. We finally construct edge conditions
(of trace and potential type) and obtain Fredholm operators in weighted edge
Sobolev spaces.
Vladimir Sharko (National Academy of Sciences of
Ukraine) L^2-invariants and their applications
Georgiy Sharygin (Moscow State University, ITEP) Cyclic cohomology of
Hopf algebras: properties and examples
In a recent paper by Hajac,
Khalkhali, Rangipour and Sommerhaeuser math.KT/0306288: "Hopf-cyclic homology
and cohomology with coefficientsthere" was given a generalization of the
Connes-Moscovici's Hopf cyclic cohomology for the case of arbitrary stable
anty-Yetter-Drinfeld module as coefficients. In this talk (joint work with
I.Nikonov) we shall present few examples of calculations of such cohomology and
a generalization of Crainic's construction chracteristic classes, induced by
higher equivariant traces, to this case.
Eugenij Sinaiskij (Moscow State Technical University) Translation
Continuous Functionals\\ on the space CB(G)
Let G be a locally compact group and CB(G) be the Banach space of
continuous, bounded, real functions on the group G. A functional F from CB(G)'
is called a left translation continuous functional (an LTC functional) if the
map t -> tF is weak* continuous (here tF stands for the left shift of the
functional F by the element t). In our report we present several examples of LTC
and not LTC functionals, define classes of functionals of compact and infinite
type and describe values of mean LTC functionals of infinite type on a given
function.
Boris Sternin (Independent University of Moscow & Universitat
Potsdam) Elliptic Theory on Manifolds with nonisolated Singularities
For elliptic operators on manifolds with edges, we compute the
obstruction to the existence of Fredholm edge problems and give an index
formula.
Seytek Tabaldyev (Moscow State Institute of Electronics and
Mathematics) Some Algebras $C(\Omega)$ of homological dimension 2
Klaus Thomsen (Aarhus University) Duality in equivariant KK-theory
Let G be a locally compact second countable group, and A and B two
separable C*-algebras with continuous actions by G. I will decribe how to obtain
a C*-algebra D with an action of G such that the Kasparov KK-groups KK^0(A,B)
and KK^1(A,B) (ignoring the G-actions) are the K-groups of D, while the
equivariant Kasparov KK-groups KK^0_G(A,B) and KK^1_G(A,B) are the K-groups of
the fixed point algebra D^G.
Evgenij Troitsky (Moscow State University) Index theory for
gauge-invariant families and twisted K-theory" (joint research with V.Nistor)
Let $G\to B$ be a bundle of compact Lie groups acting on a fiber bundle
$Y\to B$. We introduce and study gauge-equivariant $K$-theory groups $K^i_G$.
These groups satisfy the usual properties of the equivariant $K$-theory groups,
but also some new phenomena arise due to the topological non-triviality of the
bundle $G\to B$. As an application, we define a gauge-equivariant index for a
family of elliptic operators$(P_b)_{b\in B}$ invariant with respect to the
action of $G\to B$, which, in this approach, is an element of $K^0_G(B)$. We
then give another definition of the gauge-equivariant index as an element of
$K_0(C^*(G))$, the $K$-theory group of the Banach algebra $C^*(G)$. We prove
that $K_0(C^*(G))\cong K^0_G(G)$ and that the two definitions of the
gauge-equivariant index are equivalent. The algebra $C^*(G)$ is the algebra of
continuous sections of a certain field of $C^*$-algebras with non-trivial
Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus
examples of twisted $K$-theory groups, which have recently turned out to be
useful in the study of Ramond-Ramond fields. The second part of results is
devoted to the proof of the Thom isomorphism in this cathegory - the main step
in defining the topological index. The results are based on the joint
research with V.Nistor started in MPIM (Bonn).
The research was also
supported by the Russian Foundation or Basic Research (Grant 02-01-00572), Grant
for support of leading scientific schools (H III - 619.203.1), and Grant
"Universities of Russia".
[1] V. Nistor, E. Troitsky. An index for
gauge-invariant operators and the Dixmier-Douady invariant. Trans. Amer. Math.
Soc. 356 (2004), no. 1, 185--218 [2] V. Nistor, E. Troitsky. The Thom
isomorphism for the gauge-invariant K-theory. Preprint, 2004.
Ezio Vasselli (Universita' La Sapienza di Roma) The Pimsner algebra of
a vector bundle, fields of Cuntz algebras and K-theory
We study the
Pimsner algebra associated with the module of continuous sections of a vector
bundle, and prove that it is a continuous bundle of Cuntz algebras. Furthermore,
we study bundles of Cuntz algebras carrying a global circle action, and assign
them an invariant in the representable KK-group of the zero-grade bundle. Such
invariant is proposed for a classification unless graded stable isomorphism, and
is explicitly computed for the Pimsner algebra of a vector bundle.
Hong You (Harbin Institute of Technology) Generating the Kernel of
K2(R,M)--->K2(R)
Suppose that R is a commutative local ring with the
maximal ideal M, and that R/M=Fq is finite. Eventhough we know the the kernel of
K2(R,M)--->K2(R) is cyclic, as far as we know, no paper has given an explicit
cyclic generator for the kernel and generally speaking it is not easy to find
the cyclic generator for the kernel even for some simple cases. In this note, a
generating set of [q+3/2] elements at most for the kernel of K2(R,M)--->K2(R)
is presented. For some concrete local ring, we may reduce the set of generatorts
of the kernel further. As an application, we present the cyclic generator of the
kernel for the case R=Z/13.