NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

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


6 October 2004 (Exceptional place and time: room 403, Wednesday.)

SURVEY OF NONCOMMUTATIVE GEOMETRY

The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the noncommutative torus which plays a role in M-theory compactification, and finally the space of Q-lattices which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic.

On the other hand the stretching of geometric thinking imposed by passing to noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. This has been achieved a long time ago by operator algebraists as far as measure theory is concerned. The theory of nonabelian von-Neumann algebras is indeed a far reaching extension of measure theory, whose main surprise is that such an algebra inherits from its noncommutativity a god-given time evolution.

The development of the topological ideas was prompted by the Novikov conjecture on homotopy invariance of higher signatures of ordinary manifolds as well as by the Atiyah-Singer Index Theorem. It has led to the recognition that not only the Atiyah-Hirzebruch K-theory but more importantly the dual K-homology admit Noncommutative Geometry as their natural framework. The cycles in K-homology are given by Fredholm representations of the C*-algebra A of continuous functions. A basic example is the group ring of a discrete group and restricting oneself to commutative algebras is an obviously undesirable assumption.

The development of differential geometric ideas, including de Rham homology, connections and curvature of vector bundles, took place during the eighties thanks to cyclic cohomology which led for instance to the proof of the Novikov conjecture for hyperbolic groups but got many other applications. Basically, by extending the Chern-Weil characteristic classes to the general framework it allows us for many concrete computations on noncommutative spaces.

The very notion of Noncommutative Geometry comes from the identification of the two basic concepts in Riemann's formulation of Geometry, namely those of manifold and of infinitesimal line element. It was recognized at an early stage that the formalism of quantum mechanics gives a natural place both to infinitesimals (the compact operators in Hilbert space) and to the integral (the logarithmic divergence in an operator trace). It was also recognized long ago by geometers that the main quality of the homotopy type of a manifold, (besides being defined by a cooking recipee) is to satisfy Poincare duality not only in ordinary homology but in K-homology.

In the general framework of Noncommutative Geometry the confluence of the two notions of metric and fundamental class for a manifold led very naturally to the equality ds=1/D which expresses the infinitesimal line element ds as the inverse of the Dirac operator D, hence under suitable boundary conditions as a propagator. The significance of D is two-fold. On the one hand it defines the metric by the above equation, on the other hand its homotopy class represents the K-homology fundamental class of the space under consideration.

We shall discuss three of the recent developments of Noncommutative Geometry. The first is the understanding of the noncommutative nature of spacetime from the symmetries of the Lagrangian of gravity coupled with matter. The starting point is that the natural symmetry group G of this Lagrangian is isomorphic to the group of diffeomorphisms of a space X, provided one stretches one's geometrical notions to allow slightly noncommutative spaces. The spectral action principle allows to recover the Lagrangian of gravity coupled with matter from the spectrum of the line element ds.

The second has to do with various appearances of Hopf algebras relevant to Quantum Field Theory which originated from my joint work with D.Kreimer and led recently in joint work with M.Marcolli to the discovery of the relation between renormalization and one of the most elaborate forms of Galois theory given in the Riemann-Hilbert correspondence and the theory of motives. A tantalising unexplained bare fact is the appearance in the universal singular frame eliminating the divergence of QFT of the same numerical coefficients as in the local index formula. The latter is the corner stone of the definition of curvature in noncommutative geometry.

The third is the spectral interpretation of the zeros of the Riemann zeta function from the action of the idele class group on the space of Q-lattices and of the explicit formulas of number theory as a trace formula of Lefschetz type.

ALAIN CONNES (IHES, Bures-sur-Yvette, France)


11 October 2004

BIALGEBRA CYCLIC (CO)HOMOLOGY AFTER A. KAYGUN

In a recent series of papers, A.Kaygun has introduced a variant of cyclic homology and cohomology theories for a bialgebra B with coefficients in arbitrary stable module/comodule X over this algebra. He has shown that the theory he had introduced coincides with the one proposed by Hajac, Khalkhali, Rangipour and Sommerhaeuser, if B is a Hopf algebra, and X is a stable anti-Yetter-Drinfeld module over B. The purpose of this talk is to give an exposition of the principal ideas and results obtained by A.Kaygun.

GEORGIY SHARYGIN (ITEP, Moscow, Russia)


25 October 2004

BIALGEBRA CYCLIC (CO)HOMOLOGY AFTER A. KAYGUN, PART 2

This talk is a continuation of the previous one. For a bialgebra B, we shall recall briefly the definition of the cyclic theory of B-module coalgebras with coefficients in a stable module/comodule over B. We shall also give the dual construction of cyclic homology of B-module algebras over B and discuss the examples of computations of the cohomology groups of this type related to the foliation Hopf algebras of Connes and Moscovici and to the quantum universal enveloping algebras.

GEORGIY SHARYGIN (ITEP, Moscow, Russia)


8 November 2004

CLASSICAL LOCAL INDEX FORMULAS

This talk is a classical introduction to the noncommutative equivariant local index formula expressed by means of K-theory and cyclic cohomology. Classical Lefschetz, Atiyah and Bott theorems will be discussed in a unified manner. The future perspective, culminating in theorems of Connes-Moscovici and Neshveyev-Tuset, will be sketched. The latter work includes equivariant index computations for the Dabrowski-Sitarz Dirac operator on the standard Podles quantum sphere.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)


15 November 2004

NON-CLASSICAL LOCAL INDEX FORMULAS

This talk is a continuation of the classical introduction to the noncommutative local index formula of Connes-Moscovici. Fredholm modules and spectral triples will be defined. The global index formula of Connes and the pairing of cyclic periodic cohomology with K-theory will be explained. The global index formula and the Diximier trace will be used to obtain the local index formula.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)


22 November 2004

MIRROR-TYPE QUANTUM TWO-SPHERES

We present a new topological quantum two-sphere. Much as the generic Podles quantum sphere, it is obtained by a gluing of quantum discs. This gluing is given by the fibre product of Toeplitz algebras over the circle via a non-trivial boundary identification. We show that the resulting C*-algebra is not Morita equivalent to the C*-algebra of the generic Podles quantum sphere. This is so despite the fact that they have identical K-groups and primitive ideal spaces. We also discuss the relationship of the aforementioned mirror-type and Podles spheres with the Heegaard-type noncommutative 3-sphere. (Based on the joint paper with T.Brzezinski and R.Matthes. Its preliminary version is available at PMH's home page.)

PIOTR M. HAJAC (Instytut Matematyczny PAN / Katedra Metod Matematycznych Fizyki UW), WOJCIECH SZYMANSKI (The University of Newcastle, Australia)


29 November 2004

EXOTIC TRACES

As we know, the ordinary trace is not the only operator trace on a (separable) Hilbert space. There are many "exotic" traces of which those constructed by Dixmier have already acquired a classical status. The talk will describe all positive traces which result from renormalization of the finite dimensional trace. (Based on the article "Vestigia Investiganda" available at the author's home page.)

MARIUSZ WODZICKI (University of California at Berkeley, USA)


6 December 2004 (Exceptional time: 10:30 - 11:30 and 12:00 -13:00, respectively.)

RENORMALIZATION AND MOTIVIC GALOIS THEORY

In this joint work with Alain Connes we investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain "motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, it is a semi-direct product by the multiplicative group of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over the field of rational numbers, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of acategory of mixed Tate motives.

QUANTUM STATISTICAL MECHANICS OF Q-LATTICES

In this joint work with Alain Connes we study the space of commensurability classes of Q-lattices and the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system. We give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field.

MATILDE MARCOLLI (MPIM, Bonn, Germany)


7 December 2004 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5810, 12:00 Tuesday.)

TOWARDS THE FRACTIONAL QUANTUM HALL EFFECT: A NONCOMMUTATIVE GEOMETRY PERSPECTIVE

In this joint work with Varghese Mathai we propose an approach to the fractional Quantum Hall Effect within the framework of noncommutative geometry, using hyperbolic geometry to simulate electron-electron interactions. By computing the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, we determine the range of values of the Connes-Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in our case is that the Connes-Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. The set of possible fractions obtained in this model can be compared with recently available experimental data.

MATILDE MARCOLLI (MPIM, Bonn, Germany)


13 December 2004

SPECTRAL TRIPLES ON QUANTUM THREE AND TWO SPHERES

All known spectral triples (A,H,D), where A is the coordinate algebra of SUq(2), or of the standard or equatorial Podles quantum sphere, H is a Hilbert space and D is a Dirac operator, will be presented.

LUDWIK DABROWSKI (SISSA, Trieste, Wlochy)


20 December 2004

SPECTRAL TRIPLES ON QUANTUM THREE AND TWO SPHERES, PART 2

This talk will be foused on spectral triples on the standard and equatorial Podles quantum sphere.

LUDWIK DABROWSKI (SISSA, Trieste, Wlochy)


3 January 2005

TWISTED HOMOLOGY OF SLq(2) (full contents in pdf)

In this talk we present the results of a joint paper with Hadfield in which we compute the twisted cyclic theory for the simplest quantum group SLq(2) with respect to arbitrary twisting automorphisms acting diagonally on the standard generators of SLq(2). The standard cyclic theory was computed by Masuda, Nakagami and Watanabe in 1990. More general computations by Feng and Tsygan done for all the standard quantum groups at once identified the result with the homology of a quotient space of the classical SL(2). In particular, the Hochschild dimension of SLq(2) is 1. In the twisted setting, we found a distuinguished family of automorphisms which avoids this "dimension drop". Remarkably, the simplest member of this family is the modular automorphism of the Haar functional on SLq(2).

One of Connes' approaches to cyclic cohomology is based on the correspondence between differential calculi equipped with closed graded traces and cyclic cocycles. This directly leads to the interpretation of cyclic cohomology as a generalization of de Rham homology. Recently, Kustermans, Murphy and Tuset showed that this correspondence can be extended to functionals on differential calculi whose tracial properties are of the more general form h(ab) = +-h(s(b)a), where s is an automorphism of the calculus. The resulting cocycles are elements of a variant of cyclic cohomology in which the usual definitions are modified by an automorphism of the algebra under consideration. The resulting theory was called twisted cyclic cohomology and is a special case of Hopf-cyclic cohomology that arose from the work of Connes and Moscovici on the local index formula.

A main motivation for the above generalization is the theory of covariant differential calculi over quantum groups as developed by Woronowicz. These calculi are naturally equipped with such "twisted" graded traces, and it was demonstrated by Kustermans, Murphy and Tuset that one can construct interesting covariant calculi via the new approach.

ULRICH KRAEHMER (Humboldt Universitaet, Berlin, Germany)


10 January 2005

LOCAL QUANTUM PHYSICS OVER SPECTRAL GEOMETRIES

We propose a generalization of the axioms of (algebraic) local quantum field theory which allows for a noncommutatvity of spacetime, and which is in accordance with the equivalence principle that underlies general relativity. Hence it does not presuppose a fixed background geometry. This notion of "spectral local quantum field theory" is entirely formulated in the language of Connes' noncommutative geometry. However, in order to describe the causal structure, it has been necessary to generalize the notion of spectral triples to (noncommutative) "ghysts", i.e. globally hyperbolic spectral triples. After a brief introduction into the physical motivation of our proposal for spectral local QFT, we shall mainly describe this notion of ghysts in the talk.

MARIO PASCHKE (Max-Planck-Institut, Leipzig, Germany)


17 January 2005

THE K-GROUPS OF THE QUANTUM REAL PROJECTIVE SPACE

In analogy with the commutative situation, the quantum real projective space RP2q is defined as a quotient by the Z/2Z-antipodal action on the equatorial Podles sphere. The representations of the *-algebra of RP2q are classified and the corresponding C*-algebra is defined. Using the six-term exact sequence, a derivation of the K-groups of RP2q is given in detail.

RAINER MATTHES (Katedra Metod Matematycznych Fizyki UW)


14 February 2005

QUANTUM BOHR COMPACTIFICATION

I will describe the construction of a universal compactification for quantum groups. This compactification is a quantum group version of the Bohr compactification of topological groups. In case of discrete quantum groups there is a more explicit description of this compactification and it yields new examples of compact quantum groups.

PIOTR M. SOLTAN(Katedra Metod Matematycznych Fizyki UW)


21 February 2005

THE MATSUMOTO GLUING THEOREM

Taking an advantage of the concept of a factor state and the Weak Stone Weierstrass Theorem, we prove a general criterion on the fiber product of *-algebras to be dense in the fiber product of their C*-completions.

RAINER MATTHES (Katedra Metod Matematycznych Fizyki UW)


28 February 2005

SPECTRAL GEOMETRY A LA CONNES: WHAT NEXT?

The program of construction of noncommutative geometries based on the axioms of spectral triples was boosted last year by the success of examples linked to quantum deformations. We shall briefly outline these examples (2 and 3-dimensional quantum spheres) and present the next step: examples of different spin structures on noncommutative tori as well as extension of some results to compact noncommutative manifolds with Lorentz signature.

ANDRZEJ SITARZ (Instytut Fizyki UJ)


7 March 2005

ON A PAIRING BETWEEN SUPER LIE AND PERIODIC HOMOLOGY

For any action of a Lie (super)algebra g on a (super)algebra A by derivations we define a pairing between the homology of g with coefficients in traces on A and periodic homology of A. This pairing generalizes the index formula for summable Fredholm modules and the Connes-Kubo formula for the Hall conductivity. The analogy with the characteristic map for the Hopf-cyclic cohomology will be discussed.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)


14 March 2005

EQUIVARIANT CHERN CHARACTER FOR COMPACT LIE GROUPS

Constructions of equivariant Chern character for G-CW complexes, where G is a compact Lie group, are in contrast to the case with G finite. It is still not known what should be the target of such homomorphism from equivariant K-theory. I will describe few constructions starting from the paper of P. Baum, J-L. Brylinski and R. MacPherson in which they described such Chern character in case of S^1. Next, I will introduce ideas from Brylinski's papers in which he describes the Chern character homomorphism to equivariant periodic cyclic homology of the algebra of smooth functions on a compact manifold. At the end I will give a survey of current state of art in the subject, which involves papers of J. Block and E. Getzler, I. Rosu and others.

PAWEL WITKOWSKI (Instytut Matematyki UW)


21 March 2005

WEIGHT FILTRATION IN COHOMOLOGY OF ALGEBRAIC VARIETIES AND GROUP ACTIONS

In many cases important examples of group actions appearing in topology come from complex geometry. Therefore it is reasonable not to forget the structures which are preserved by such actions. In particular algebraic actions preserve the mixed Hodge structure defined by Deligne in rational cohomology. I will describe the weight filtration in cohomology and equivariant cohomology for actions of a connected algebraic linear group. It makes cohomology very rigid, in many cases spectral sequences such as Eilenberg-Moore spectral sequence degenerates. It allows to prove that $H^*(X)=Tor_H^*(BG)(H^*_G(X),Q)$. I will illustrate this phenomenon for singular toric varieties also for integer coefficients.

ANDRZEJ WEBER (Instytut Matematyki UW)


4 April 2005

TRANSGRESSION OF THE CHERN CHARACTER FOR FAMILIES AND CYCLIC COHOMOLOGY

The aim of this talk is to establish a link between the Chern character in K-homology and the version of the standard Chern character and of its transgression developed by D.Quillen and J.-M.Bismut. We show that the Chern character in K-homology, which associates to every finitely summable Fredholm module over an involutive algebra a cyclic cohomology class, is equal via the Loday-Quillen isomorphism to the transgressed Chern character of the family of conjugates of the Fredholm operator by the unitary group of the algebra. (Based on the joint work with Alain Connes.)

HENRI MOSCOVICI (Ohio State University, USA)


11 April 2005

TRANSVERSE GEOMETRY AND MODULAR FORMS (full contents in pdf)

This talk will present aspects of our joint work with Alain Connes relating the geometry of codimension 1 foliations to modular forms acted upon by Hecke operators. We will discuss the metamorphosis in the modular context of the characteristic classes of the Hopf algebra that governs the transverse geometry of such foliations. The analogue of the Godbillon-Vey class takes the form of a rational Euler cocycle on PSL(2,Q) and acquires arithmetic significance. A related Hopf-cyclic cocycle performs the function of the Schwarzian and leads to a general algebraic definition of a projective structure. Finally, the Hopf-cyclic cocycle representing the transverse fundamental class gives the first term of a universal deformation formula based on the extension of the Rankin-Cohen brackets.

HENRI MOSCOVICI (Ohio State University, USA)


18 April 2005

THE RANGE OF K-INVARIANTS FOR C*-ALGEBRAS OF INFINITE GRAPHS

Countable directed graph algebras (a generalisation of Cuntz-Krieger ones) are examples of nontrivial C*-algebras with readily computable K-theory: K0 is countable abelian, K1 is free abelian. Natural question arises if all possible pairs of such groups can be obtained this way. I will show that, indeed, any countable abelian and any free abelian group can be realised at the same time as K0 and K1, respectively, of a purely infinite simple stable graph C*-algebra. (Based on the work of Wojciech Szymanski, 2002.)

WOJCIECH KAMINSKI (Instytut Fizyki Teoretycznej UW)


25 April 2005

EQUIVARIANT CYCLIC HOMOLOGY AND QUANTUM GROUPS (full contents in pdf)

In the first part of this talk we survey the definition of equivariant cyclic homology for locally compact groups. This theory generalizes a number of constructions including equivariant Bredon homology and chamber homology for totally disconnected groups, as well as equivariant de Rham cohomology for compact Lie groups. As an application we sketch the construction of an equivariant Chern character for the equivariant K-homology of totally disconnected groups. In the second part we outline how the theory can be generalized to an appropriate class of quantum groups. This class contains algebraic quantum groups in the sense of van Daele as well as smooth convolution algebras of Lie groups and their duals. The concept of a modular pair is discussed in this context and used to relate covariant modules (also known as Anti-Yetter-Drinfeld modules) and Yetter-Drinfeld-modules. This is an important conceptual ingredient in the construction of an analogue of the Baaj-Skandalis duality isomorphism which relates equivariant cyclic homology of a quantum group with the equivariant cyclic theory of the dual quantum group. We finally discuss some applications of this duality theorem.

CHRISTIAN VOIGT (Universitaet Muenster, Germany)


9 May 2005

CLEFT EXTENSIONS AND QUANTUM FUNCTION ALGEBRAS AT ROOTS OF 1

The Frobenius homomorphism of quantum function algebras at roots of 1 introduced by Lusztig is a finite central ring extension. It was shown by De Concini and Lyubashenko that this extension is projective, and recently by Brown, Gordon and Stafford, that it is free. Following a suggestion of Masuoka, I will prove that this extension is even H-cleft where H is the quotient Hopf algebra. The proof is based on the theory of cleft extensions and on results by Lusztig on representations of quantum enveloping algebras at roots of 1.

HANS-JUERGEN SCHNEIDER (Universitaet Muenchen, Germany)


23 May 2005 (exceptional place: room 403)

A HOMOTOPY THEORY FOR QUANTUM PRINCIPAL BUNDLES

Hopf Galois extensions are noncommutative analogues of principal fibre bundles with structural group replaced by a Hopf algebra. I'll discuss a concept of homotopy for Hopf Galois extensions and show how it allows a certain classification of such extensions. In particular, I'll show how the Hopf Galois extensions over a Drinfeld-Jimbo quantum enveloping algebra can be completely classified up to homotopy (the latter is joint work with Hans-Juergen Schneider).

References: - C. Kassel, Quantum principal bundles up to homotopy equivalence, in The Legacy of Niels Henrik Abel, The Abel Bicentennial, Oslo, 2002, O. A. Laudal, R. Piene (eds.), Springer-Verlag 2004, 737--748. - C. Kassel and H.-J. Schneider, Homotopy theory of Hopf Galois extensions, math.QA/0402034, to appear in Ann. Inst. Fourier (Grenoble) 55 (2005).

CHRISTIAN KASSEL (CNRS - Universite Louis Pasteur, Strasbourg, France)


27 May 2005 (exceptional time and place: Friday at 16:15, room 403)

NONCOMMUTATIVE METHODS IN ALGEBRAIC TOPOLOGY (full contents in pdf)

abstract in pdf

MAX KAROUBI (Universite Paris 7, France)