NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Sniadeckich 8, room 408, Mondays, 10:15-12:00


1999/2001 2002/2003 2003/2004 2004/2005 2005/2006 2006/2007


1 October 2007

PROJECTIVE MODULES AND QUANTUM GROMOV-HAUSDORFF DISTANCE

Earlier I have shown how one can give precise meaning to statements in the literature of high-energy physics of the kind "Matrix algebras converge to the sphere". I did this by introducing and applying the concept of a "compact quantum metric space", and a corresponding "quantum Gromov-Hausdorff distance". But the physics literature continues with discussion of superstructure in this situation, such as vector bundles, connections, Dirac operators, etc. For example, certain projective modules over matrix algebras are asserted to be the monopole bundles corresponding to the usual monopole bundles on the sphere. This suggests that one needs to make precise the idea that for two compact metric spaces that are close together for Gromov-Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles over the other space. In a recent paper I showed how to do this for ordinary metric spaces. I will describe my strategy, and report on my progress in extending my results to the case of quantum metric spaces. This is work in progress, i.e. I do not yet have a final answer.

MARC A. RIEFFEL (University of California at Berkeley, USA)


8 October 2007

(CO)CYCLIC (CO)HOMOLOGY OF BIALGEBROIDS: AN APPROACH VIA (CO)MONADS

In a recent joint paper with Dragos Stefan we proposed a universal approach to Hopf (co)cyclic (co)homology, via (co)monads. In the resulting framework not only earlier examples can be recovered, but also one can go beyond: describe para-(co)cyclic objects corresponding to bialgebroids. The existence of a truly (co)cyclic quotient in these para-(co)cyclic objects is governed by stable anti Yetter-Drinfel'd modules, as in the Hopf algebra case. Our abstract results are applied to obtain explicit formulae for Hochschild and cyclic homologies of a groupoid with finitely many objects.

GABRIELLA BÖHM (Research Institute for Particle and Nuclear Physics, Budapest, Hungary)


15 October 2007

GEOMETRIC TOPOLOGY AND FIELD THEORY

During the past two decades a surprising number of new structures have appeared in the geometric topology of low-dimensional manifolds influenced by physical theories. Their precise mathematical fomulation is often obtained by using non-commutative objects. For example, WRT invariants use the framework of quantum groups. We will discuss some topics where this interaction has led to new results, including gauge theory to string theory correspondence and conformal field theory and Khovanov homology.

KISHORE MARATHE (Department of Mathematics, City University of New York, USA)


22 October 2007

MORITA CONTEXTS IN GALOIS THEORY

One of the main concerns in (Hopf) Galois theory is to prove an equivalence between the category of generalised descent data (provided by comodules of a coring) and the category of modules of an (invariant) algebra. If the coring in question is finitely generated and projective, then this problem reduces to an equivalence between two module categories. This explains the role Morita theory plays therein. In this talk, we will show how Morita contexts originally did appear in Hopf Galois theory, sketch the process of generalisation to arbitrary comodules of corings, and discuss applications to the generalised descent problem. The original results in the talk have been obtained in collaboration with Joost Vercruysse.

GABRIELLA BÖHM (Research Institute for Particle and Nuclear Physics, Budapest, Hungary)


29 October 2007

INVARIANT SUBSPACE CONJECTURE

In its simplest form, Invariant Subspace Conjecture states that every bounded operator on a Hilbert space has a non-trivial, closed, invariant subspace. During the talk, I will concentrate on the relative version of the conjecture requiring the additional condition that the projection onto the invariant subspace must belong to the von Neumann algebra generated by the considered operator. (This algebra is always assumed to be a II_1 factor.) In the last decade, the ideas from random matrix theory and Voiculesu's free probability theory influenced the research in the field of Invariant Subspace Conjecture resulting in the study of exotic candidates for counterexamples (Dykema, Haagerup, Speicher, Śniady) and construction of invariant subspaces for operators with non-trivial Brown spectrum (Haagerup, Schultz).

PIOTR SNIADY (Instytut Matematyczny, Uniwersytet Wroclawski, Poland)


5 November 2007

NONCOMMUTATIVE CORRESPONDENCES AND CYCLIC HOMOLOGY

In classical differential geometry, de Rham cohomology is a (contravariant) functor with respect to smooth maps between manifolds. The aim of the talk is to present a construction in terms of monoidal categories of cyclic homology with coefficients that is functorial with respect to a wide class of correspondences regarded as generalized regular morphisms between spaces. It will be shown that the case of finite flat correspondences of schemes is a particular example of this construction.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


12 November 2007

TOPOLOGICAL DESCENT THEORY

Grothendieck Descent Theory is a wide generalization of methods involving localization in topology and geometry applicable in many parts of mathematics. A map E ---> B is said to be an effective descent map if it allows solving certain problems on B using their solutions for E. We will discuss precise mathematical definitions of "allows solving" and indicate how to characterize effective descent morphisms in topological categories.

GEORGE JANELIDZE (University of Cape Town, South Africa)


19 November 2007

GLOBALIZING HOPF-GALOIS EXTENSIONS

Hopf-Galois extensions are noncommutative generalizations of G-principal bundles where both the total space and the base are affine, and the group is replaced by a Hopf algebra. In noncommutative algebraic geometry, we may need the case when the total or base space are not necessarily affine. I will explain how to make sense of gluing together charts which are Hopf-Galois extensions, into noncommutative bundles which may be viewed as a global generalizaton of Hopf-Galois extensions. One of the prerequisites is to explain what the Hopf algebra action on a category of "quasicoherent sheaves" on nonaffine noncommutative scheme is, and what the corresponding "equivariant sheaves" are. The distributive laws for actions of monoidal categories play a major role in generalizations of this picture, like the entwining structures did in affine case.

ZORAN SKODA (Institut Ruder Boskovic, Zagreb, Croatia)


26 November 2007

TWISTING COCHAINS AND LOCAL FORMULAS FOR THE CHERN CLASSES OF A PRINCIPAL BUNDLE

It is well-known that principal bundles (over a manifold) are uniquely (up to an isomorpfism) determined by their "gluing functions" -- the noncommutative Cech cocycle with values in the structure group of the bundle. One can look for the formulas that would express the Chern classes of this bundle in terms of this cocycle. In the talk, we shall describe a pretty simple way to obtain all such formulas. Our approach is based on the notion of the "twisting cochain" -- an object, well-known to algebraic topologists, but rarely appearing in Geometry.

GEORGY SHARYGIN (Institute for Theoretical and Experimental Physics, Moscow, Russia)


3 December 2007

PULLBACK DIAGRAMS OF PRINCIPAL COMODULE ALGEBRAS

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra. (We view such objects as noncommutative compact principal bundles.) I will show that the fibre product (pullback) of principal comodule algebras given by morphisms of which at least one is surjective is again a principal comodule algebra. Then I will explain how to derive from this result the following corollary: If F is a flabby sheaf of H-comodule algebras over any topological space with a finite open cover {Ui}i such that F(Ui) is principal for any i, then F(U) is principal for any open set U. This demonstrates the piecewise (local) nature of the principality of comodule algebras. (This is a joint work with U.Krähmer, R.Matthes, E.Wagner and B.Zielinski.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)


10 December 2007

NON-COMMUTATIVE CONNECTIONS OF THE SECOND KIND

Connection-like objects, termed hom-connections, are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature of a hom-connection is defined, and it is shown that flat hom-connections give rise to homology theory.

TOMASZ BRZEZINSKI (Swansea University, G. Britain)


17 December 2007

ON THE CLASSIFICATION OF GALOIS OBJECTS OF Oq(SL(2))

In this talk, we will remind how Galois objects can be interpreted as fiber functors and use this interpretation to classify Galois objects over the quantum groups of a nondegenerate bilinear form, including the quantum groups Oq(SL(2)). We will show in details the classification of these objects up to isomorphism. We will also consider the homotopy relation on Galois objects and give results for the classification problem of Galois object of Oq(SL(2)) up to homotopy equivalence.

THOMAS AUBRIOT (Université de Bourgogne, Dijon, France)


7 January 2008

GEOMETRY OF THE STANDARD MODEL AND NEUTRINO MASS TERMS

The apparently random collection of particles in the standard model of particle physics has a beautiful explanation in terms of the non-commutative geometry of a certain internal "space", using the framework developed over a long period by Alain Connes. I will explain this geometrical framework, focussing on the relatively recent discovery, made independently by both Alain and myself, that the internal space has dimension 6 mod 8. I will also mention some of the physical predictions, such as the mass of the Higgs field and the structure of the neutrino mass terms, which are implied by the framework.

JOHN BARRETT (University of Nottingham, G. Britain)


14 January 2008

COACTIONS OF COSEPARABLE COALGEBRAS AND COVERINGS OF ALGEBRAS AS SOURCES OF GALOIS STRUCTURES

In non-commutative geometry Galois structures such as Galois type extensions and Galois corings play the role of free group actions, principal fibre bundles and also covers of non-commutative spaces. This lecture is devoted to two aspects of Galois structures. In the first part, we give an explicit formula for a strong connection form in a principal extension by a coseparable coalgebra (or a non-commutative principal bundle). In the second part, we show that Galois corings provide an effective way to cover non-commutative algebras by ideals.

TOMASZ BRZEZINSKI (Swansea University, G. Britain)


21 January 2008

SCHNEIDER TYPE THEOREMS AND RELATIVE SEPARABLE FUNCTORS

Hopf Galois extensions are defined by means of bijectivity of a certain canonical map. One of the most important tools in the theory is Schneider's theorem providing criteria for the bijectivity of this map. The aim of this talk is to show how can one prove (crucial steps of) Schneider's theorem by analysing properties of a forgetful functor. The relevant notion of `relative separability' of a functor will be introduced and behaviour of such functors will be studied. To stress the power of the approach, other `Schneider type' theorems in the literature, based (implicitly) on the existence of a relative separable forgetful functor, we be listed. The results in the talk were obtained in collaboration with Alessandro Ardizzoni and Claudia Menini.

GABRIELLA BÖHM (Research Institute for Particle and Nuclear Physics, Budapest, Hungary)


18 February 2008

NONCOMMUTATIVE CORRESPONDENCES AND GALOIS-TANNAKA RECONSTRUCTION

According to Grothendieck-Galois theory, there is a close relation between splittings of commutative rings by an appropriate base change and (groupoid) actions. The reconstruction of the action from a given splitting is called the Galois reconstruction. According to Grothendieck-Deligne-Saavedra Rivano-Tannaka theory, there is another close relation between representations of a given groupoid and the groupoid itself. The reconstruction of the groupoid from its representations is called the Tannaka reconstruction. We show that both reconstructions are particular cases of our theorem about splittings of flat covers in the bicategory of monoidal categories.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


25 February 2008

CONTINUOUSLY SQUARE-INTEGRABLE QUANTUM GROUP REPRESENTATIONS ON HILBERT MODULES

I will report on joint work with Alcides Buss. The first part of the lecture deals with continuous spectral decompositions of actions of Abelian locally compact groups on C*-algebras. The continuity of the decomposition is expressed in terms of Fell bundles. Fell bundles over non-Abelian groups can be interpreted similarly, but they are related to coactions of the underlying group. The relevant analysis is related to square-integrability of actions of locally compact quantum groups on Hilbert modules. In the second part, I discuss this notion and the equivariant analogue of the Kasparov Stabilisation Theorem for locally compact quantum groups.

RALF MEYER (Universität Göttingen, Germany)


3 March 2008

ON SPLITTING POLYNOMIALS WITH NONCOMMUTATIVE COEFFICIENTS

For every splitting of a polynomial with noncommutative coefficients into linear factors (X - Ai) with Ai's commuting with coefficients, we will show that any cyclic permutation of linear factors gives the same result and all Ai's are roots of that polynomial. It implies that, although Ai's appearing in a splitting of a polynomial with commutative coefficients in some noncommutative extension do not form a set (in general), they do form a cyclic order consisting of roots. We will give examples of this phenomenon.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


10 March 2008

THE SPECTRAL GEOMETRY OF THE QUANTUM PROJECTIVE PLANE CPq(2)

This talk is based on a joint work with F.D'Andrea and G.Landi. It will be focused on the description of a Dirac operator D which gives a Uq(su(3))-equivariant 0-summable triple. The spectrum of D is computed by relating the square of D to a Casimir element of Uq(su(3)) for which a left and a right action on spinors coincide. An interesting aspect is that classically CP(2) is a spin-c but not a spin manifold. A possibility of extending this and previous examples to q-deformations of higher dimensional groups and homogeneous spaces will be briefly discussed.

LUDWIK DABROWSKI (SISSA, Trieste, Wlochy)


17 March 2008

SQUARE-INTEGRABLE QUANTUM GROUP REPRESENTATIONS ON HILBERT MODULES

I will report on joint work with Alcides Buss. We use the theory of unbounded weights on C*-algebras to define integrable actions of quantum groups on C*-algebras and square-integrable actions of quantum groups on Hilbert modules. The latter allows us to prove an equivariant analogue of the Kasparov Stabilisation Theorem for locally compact quantum groups.

RALF MEYER (Universität Göttingen, Germany)


31 March 2008

A NONCOMMUTATIVE FAMILY OF INSTANTONS

We describe the construction of a noncommutative family of instantons on an isospectral deformation of the 4-sphere. The family, obtained by coacting on a basic instanton, is parametrized by a noncommutative space which turns out to be a deformation of the moduli space of charge one instantons on the classical 4-sphere. This talk based on a joint work with Giovanni Landi, Cesare Reina and Walter van Suijlekom.

CHIARA PAGANI (Uniwersytet Warszawski)


7 April 2008

K-THEORETIC FIBRATIONS

We study non-commutative analogues of Serre-fibrations in algebraic topology for C*-algebra bundles. We shall present several examples of such fibrations and give applications for the computation of the K-theory of C*-algebras.

SIEGFRIED ECHTERHOFF (Universität Münster, Germany)


14 April 2008

SPECTRAL TRIPLES FOR MANIFOLDS AND ORBIFOLDS

After reviewing recent work on the reconstruction of manifolds from spectral triples with commutative coordinate algebras, we consider how orbifolds are disallowed by this procedure. In the simple case of the orbit space of a compact oriented manifold under the action of a finite group, the obstruction comes from the vanishing of any putative volume form on the singular locus.

JOSEPH C. VARILLY (University of Costa Rica, Costa Rica)


21 April 2008

K-THEORY OF A REAL BANACH ALGEBRA AND ITS COMPLEXIFICATION

Real K-theory (periodicity 8) seems more complicated than complex K-theory (periodicity 2). In this talk, we show how to relate these two theories using three ingredients: the real K-theory of Atiyah, Clifford algebras and the theory of homotopy fixed point sets. One philosophy which emerges from these considerations is the following: any "general" theorem on complex Banach algebras extends to real ones. During the lecture, we make this statement more precise and consider its applications to the Lichtenbaum-Quillen and real Baum-Connes conjectures.

MAX KAROUBI (Paris, France)


28 April 2008

MORITA EQUIVALENCE REVISITED

Let k be the co-ordinate algebra of a complex affine variety. A k-algebra is an algebra A over the complex numbers with a given k-module structure. The k-module structure is required to be compatible (in an evident way) with the algebraic operations of A. This talk introduces an equivalence relation on k-algebras called "geometric equivalence". This equivalence allows a tearing apart of strata in the primitive ideal space, while Morita equivalence gives a homeomorphism of primitive ideal spaces. The talk is intended for non-specialists. All the basic definitions will be precisely stated. Examples will be given to show how the new equivalence relation works. An application to the representation theory of reductive p-adic groups will be briefly indicated. There should be an analogous equivalence relation (with analogous properties) for an appropriate class of C* algebras.

PAUL F. BAUM (Pennsylvania State University, State College, USA)


29 April 2008 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5870, 12:00 Tuesday.)

TWISTED K-THEORY OLD AND NEW

The purpose of this lecture is to give a historical view of the subject, starting from the work of Atiyah, Bott and Shapiro on Clifford modules. There will be also an exposition of some recent results, essentially in the equivariant case, related to algebraic K-theory and operator algebras. Finally, some applications in various mathematical subjects will be given.

MAX KAROUBI (Paris, France)


5 May 2008

FLABBY SHEAVES OF ALGEBRAS AND FINITE FREE DISTRIBUTIVE LATTICES

An ordered N-covering of an algebra is an ordered family of N algebra surjections whose ideals intersect to zero and generate a distributive lattice. We prove that the category of ordered N-coverings is equivalent to the category of flabby sheaves of algebras over the projective space P^(N-1)(Z/2Z) with topology generated by the covering of affine spaces. A key step in the proof is to show that all non-empty open subsets of this projective space form a free distributive lattice. (Determining the number of elements in this lattice is the celebrated Dedekind problem open since 1897.) As a corollary, we obtain an equivalence of the category of flabby sheaves of commutative unital C*-algebras and the opposite category of closed coverings of compact Hausdorff spaces. To consider all finite coverings at the same time, we take sheaves over the infinite Z/2-projective space. (This talk is based on joint work with P.M.Hajac, U.Krähmer, R.Matthes and C.Pagani.)

BARTOSZ ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


12 May 2008

C*-ALGEBRAS OF SUq(2)-RELATIONS

We will prove that universal C*-algebras for SUq(2)-relations, as considered by Dabrowski and Landi, are isomorphic for all complex values of q with absolute value strictly less than 1. This extends Woronowicz's theorem for real values of q. For unitary values of q, these algebras are the Matsumoto C*-algebras exemplifying isospectral deformations of Connes and Landi.

PIOTR M. SOŁTAN (Uniwersytet Warszawski)


19 May 2008

ON THE HOCHSCHILD AND CYCLIC HOMOLOGY OF QUANTUM HOMOGENEOUS SPACES

In this talk, I discuss the structure of the Hochschild and cyclic homology of quantum homogeneous spaces, in particular of the Podles spheres, with special emphasis on dualities arising from the product structures in Hochschild (co)homology.

ULRICH KRÄHMER (University of Glasgow, Scotland)


26 May 2008

MONOIDAL EQUIVALENCE FOR LOCALLY COMPACT QUANTUM GROUPS

Recently, monoidal equivalence has been studied for compact quantum groups by Bichon, De Rijdt and Vaes, to provide examples of compact quantum group actions with large quantum multiplicity. It has also been applied succesfully by Vaes and Vander Vennet to determine Poisson boundaries of some (non-amenable) discrete quantum groups. In this talk, I will discuss a theory of monoidal equivalence for locally compact quantum groups (on the level of von Neumann algebras). As in the purely algebraic setup, the main notion is that of a Galois object which links both quantum groups. I show how the Galois object and one of the locally compact quantum groups can be used to reconstruct the other locally compact quantum group, the main problem being the construction of the invariant weights. On the other hand, the Galois object and both locally compact quantum groups can also be combined into a single object, namely a measured quantum groupoid (in the sense of Lesieur and Enock). This basic picture seems the most convenient one to treat the 'monoidality' of the strong Morita equivalence between the C*-algebraic duals of the two quantum groups.

KENNY DE COMMER (Katholieke Universiteit Leuven, Belgium)


2 June 2008

CONTRACTIONS OF SEMISIMPLE GROUPS AND THE MACKEY ANALOGY

Suppose that G is a connected Lie group and that K is a maximal compact subgroup of G. There is a smooth family of Lie groups G_t, t a real number, such that G_t = G when t is not 0, and such that G_0 is the semidirect product group associated to the adjoint action of K on the quotient of the Lie algebra of G by the Lie algebra of K. The group G_0 is called a contraction of G, and in a 1975 paper Mackey proposed that, when G is semisimple, the unitary representation theories of G and G_0 ought to be analogous to one another. Mackey's proposed analogy is very closely related to the Connes-Kasparov conjecture in C*-algebra K-theory. I shall briefly review this fact, and then examine the analogy from the related, but different, point of view of Harish-Chandra modules and Hecke algebras.

NIGEL HIGSON (Pennsylvania State University, State College, USA)