NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Śniadeckich 8, room 322, Mondays



1999/2001 2002/2003 2003/2004 2004/2005 2005/2006 2006/2007 2007/2008 2008/2009 2009/2010 2010/2011 2011/2012 2012/2013



7 October 2013, 10:30-12:00

SPECTRAL TRIPLES ON QUANTUM LENS SPACES

In this talk, we construct real spectral triples on quotients of SUq(2) by actions of finite cyclic groups. The real structure is based on the construction of Dąbrowski, Landi, Sitarz, Van Suijlekom and Varilly of a real spectral triple on SUq(2). This approach leads to an easy calculation of the spectra of Dirac operators on (quantum) lens spaces. We will also describe some aspects of the K-theory of quantum lens space C*-algebras, and the relation between our quantum lens spaces and quantum teardrops after Brzeziński and Fairfax. (This talk is based on joint work with Andrzej Sitarz.)

JAN JITSE VENSELAAR (California Institute of Technology)



14 October 2013, 10:30-12:00

HAAGERUP PROPERTY FOR LOCALLY COMPACT CLASSICAL AND QUANTUM GROUPS (PART 1)

The Haagerup property has recently become one of the most intensively studied properties of locally compact groups. This property exhibits both geometric and analytic characters of such groups. In the first lecture, I will discuss various equivalent characterisations of the property for a given group G in terms of the existence of some particular: a) representations of G; b) positive definite functions on G; c) conditionally negative definite function on G; d) affine actions of G on a Hilbert space. Then I will add a new characterisation of the Haagerup property in terms of a dense subset of the space of representations of G. In the second lecture, I will pass to the context of locally compact quantum groups, outlining respective definitions and explaining subtleties arising in the quantum case. We will see that the Haagerup property for discrete quantum groups is preserved under taking free products. Finally, I will describe certain open problems related to the Haagerup approximation property for von Neumann algebra with a non-tracial state. (Partly based on joint work with Matt Daws, Pierre Fima and Stuart White).

ADAM SKALSKI (IMPAN / Uniwersytet Warszawski)



14 October 2013, 14:15-15:45

HAAGERUP PROPERTY FOR LOCALLY COMPACT CLASSICAL AND QUANTUM GROUPS (PART 2)

See above.

ADAM SKALSKI (IMPAN / Uniwersytet Warszawski)



21 October 2013, 10:30-12:00

UNIQUENESS THEOREMS AND TOPOLOGICAL APERIODICITY FOR PRODUCT SYSTEMS OF C*-CORRESPONDENCES

By a uniqueness theorem we mean a result concerning properties of generators and relations of a universal C*-algebra A guaranteeing that the faithfulness of a representation of A on its generators implies that the representation is an isomorphism. We start by giving a brief survey of such results, indicating their fundamental meaning in various problems. Then we present a uniqueness theorem for Cuntz-Pimsner algebras of product systems over Ore semigroups. To this end, we introduce the Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the aforementioned uniqueness theorem and a certain simplicity criterion. They generalize results for crossed products by groups (R. J. Archbold, J. S. Spielberg) and Exel's crossed products (R. Exel, A. Vershik). They also give interesting conditions for topological higher rank graphs, and apply to the new Cuntz C*-algebra QN arising from the "ax+b"-semigroup over natural numbers. (Based on joint work with Wojciech Szymański.)

BARTOSZ K. KWAŚNIEWSKI (IMPAN / Uniwersytet w Białymstoku)



21 October 2013, 14:15-15:45

CYCLIC EILENBERG-MOORE CONSTRUCTION

We introduce a cyclic version of the monoidal Eilenberg-Moore construction, and use it to obtain a categorification of Hopf-cyclic (co)homology. This categorification is an invariant of (op)monoidal functors under suitable transformations. The corresponding (co)cyclic object consists of natural transformations between appropriate compositions of (op)monoidal functors, tensor products and trace functors. As a side concept, we introduce a notion of cyclic scheme, and show that it gives an equivalent description of the spectrum of a commutative ring. However, this notion still makes sense for arbitrary bialgebroids over a noncommutative base.

TOMASZ MASZCZYK (Uniwersytet Warszawski)



28 October 2013, 10:30-12:00

NONCOMMUTATIVE KÄHLER GEOMETRY OF THE STANDARD PODLEŚ SPHERE

The standard Podleś sphere is one of the most studied examples in noncommutative geometry. It has served as a motivating structure for the theories of differential calculi, quantum principal bundles, and spectral triples. In recent years, it has also served as a basic example for the newly emerging field of noncommutative complex geometry. In this talk, I will explain how one can build upon this work, and present the Podleś sphere as an example of a noncommutative Kähler structure. Discussed will be noncommutative versions of the Lefschetz decomposition, the Kähler identities, complex Hodge decomposition, and the equivalence of the de Rham and Dolbeault cohomologies. Time permitting, I shall also discuss the possible extension of this work to all the irreducible quantum flag manifolds, and relate it to Fröhlich, Grandjean, and Recknagel's theory of complex and Kähler spectral triples.

RÉAMMON Ó BUACHALLA (Univerzita Karlova v Praze)



28 October 2013, 14:15-15:45

K-THEORY WITH A DECOMPOSABLE TWIST

Gerbes and twisted K-theory groups on a manifold will be discussed in the case where the twisting class is a cup product of a 1-cocycle and a 2-cocycle. Geometric realizations for these classes (known as gerbes) will be constructed as bundles of projective Fock spaces. A gerbe can be applied to twist K-theory: associated with a gerbe there is a bundle of Fredholm operators, and elements in a twisted K-theory group arise as homotopy classes of sections of this Fredholm bundle. Twisted K-theory elements in this settings will be constructed explicitly, and their twisted families index problem will be solved using the superconnection techniques.

ANTTI J. HARJU (Centrum Kopernika Badań Interdyscyplinarnych, Kraków)



4 November 2013, 10:30-12:00

PARTIAL ACTIONS OF GROUPS AND HOPF ALGEBRAS

In this talk, we give an overview of the theory of partial actions of groups and Hopf algebras. First, we start with partial actions of groups on sets, and show their relationship with groupoids. Then we present partial actions of groups on algebras and describe partial skew group rings with some examples. Finally, we move towards the theory of partial actions and partial coactions of Hopf algebras presenting non-trivial examples like algebras partially graded by some group. We discuss also the globalization theorem for partial actions.

ELIEZER BATISTA (Universidade Federal de Santa Catarina, Brazil)



4 November 2013, 14:15-15:45

PARTIAL REPRESENTATIONS OF HOPF ALGEBRAS AND PARTIAL HOPF MODULES

In this talk, we present the notion of partial representations of Hopf algebras. Starting with the case of groups, we show how partial actions are related to partial representations. For the case of partial representations of a Hopf algebra H, we construct a Hopf algebroid that plays the role of a universal object for partial representations of H. Even for finite-dimensional Hopf algebras, this universal Hopf algebroid can be infinite dimensional as a vector space. We define the monoidal category of partial H-modules. Here the monoidal structure comes from the associated universal Hopf algebroid. The algebra objects in this category of partial modules coincide with the partial actions of the same Hopf algebra. We also give some explicit examples of categories of partial modules.

ELIEZER BATISTA (Universidade Federal de Santa Catarina, Brazil)



18 November 2013, 14:15-15:45

EXAMPLES OF COMPACT QUANTUM METRIC SPACES COMING FROM LENGTH FUNCTIONS

One of the earliest examples of spectral triples on noncommutative algebras were constructed by Connes for group algebras of finitely generated discrete groups by using the length function. Later it turned out that, for some groups, these spectral triples also give examples of compact quantum metric spaces a la Rieffel. We will show that the techniques of Ozawa-Rieffel and Christensen-Antonescu can be used to give compact quantum metric space structures on duals of a certain class of finitely generated discrete quantum groups. (Joint work in progress with C. Voigt and J. Zacharias.)

JYOTISHMAN BHOWMICK (Universitetet i Oslo, Norway)



25 November 2013, 10:30-12:00

TENSORING AN OPERATOR SPACE WITH A DUAL OPERATOR SPACE

An asymmetric tensor product, for an operator space V and a dual operator space Y, has been introduced. It is injective and contains the spatial tensor product of V with Y, and also the normal spatial tensor product of V with Y when V is a dual operator space too. It has been found to be effective in quantum stochastic analysis where one is interested in marrying the topology of a noncommutative state space, encoded in a C*-algebra A, to the measure-theoretic noise, encoded in a filtration of von Neumann algebras, in order to construct stochastic flows on A. In Part I of this talk, I shall outline the key features of this tensor product, in particular, the facility with which it permits ampliation of maps via its connection to mapping spaces, its restriction to the case where V is also a dual operator space (where symmetry is restored), and the connection to other tensor products and matrix spaces over operator spaces. The latter throws light on Neufang's treatment of ampliations of non-normal completely bounded operators between dual operator spaces which, for the case of functionals, goes back to early work of Tomiyama on Fubini tensor products and slice maps. In Part II, I shall give a flavour of the role this `matrix space tensor product' plays in quantum stochastic analysis; specifically, in addressing a question posed by the late Bill Arveson in the nineteen-eighties. (A section of Part I is joint work with Orawan Tripak; the basis for Part II was laid in joint work with Steve Wills, and has been exploited in joint work with Adam Skalski, as well as in that of Alex Belton.)

J. MARTIN LINDSAY (Lancaster University, England)



25 November 2013, 14:15-15:45

QUANTUM MARKOV SEMIGROUPS AND QUANTUM STOCHASTIC FLOWS

The generators of strongly continuous semigroups of contractions on Banach spaces are characterised by the Hille-Yosida theorem. However, in practice it can be difficult to verify that this theorem's hypotheses are satisfied. In the first part of this talk, it will be shown how to construct certain quantum Markov semigroups (strongly continuous semigroups of contractions on C*-algebras) by realising them as expectation semigroups of quantum stochastic flows, which are a type of non-commutative Markov process. The extra structure possessed by these flows is sufficient to avoid the need to use Hille and Yosida's result. The second part of the talk will focus on the perturbation of quantum stochastic flows, and so their associated Markov semigroups. Such perturbations are constructed as solutions to quantum stochastic differential equations. In the setting of von Neumann algebras, this procedure is well understood and relatively straightforward. The problems which arise in the more general C*-algebraic situation will be discussed and partially resolved. (Joint work with J. Martin Lindsay, with Adam G. Skalski, and with Stephen J. Wills.)

ALEXANDER C. R. BELTON (Lancaster University, England)



16 December 2013, 14:15-15:45

COMPACT QUANTUM SURFACES OF ANY GENUS

I will construct a noncommutative version of all closed compact 2-surfaces, show that the corresponding C*-algebras have the same K-groups as their classical counterparts, and comment on the K-homology of noncommutative orientable surfaces.

ELMAR WAGNER (Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico)



13 January 2014, 10:30-12:00

CLIFFORD ALGEBRAS FOR THE QUANTISED COMPACT HERMITIAN SYMMETRIC SPACES

In this joint work with Matthew Tucker-Simmons, the Dolbeault complex of the quantised compact Hermitian symmetric spaces is identified with the Koszul complexes of the quantised symmetric algebras of Berenstein and Zwicknagl, which then leads to an explicit construction of the relevant quantised Clifford algebras.

ULRICH KRÄHMER (University of Glasgow, Scotland)



13 January 2014, 14:15-15:45

BATALIN-VILKOVISKY ALGEBRA STRUCTURES ON (Co)Tor AND POISSON BIALGEBROIDS

In this talk, we discuss higher algebraic structures on (co)homology theories defined by a left bialgebroid (U,A). For a braided commutative Yetter-Drinfeld algebra N, explicit expressions for the canonical Gerstenhaber algebra structure on ExtU(A,N) are given. Similarly, if (U,A) is a left Hopf algebroid, where A is an anti-Yetter-Drinfeld module over U, it is shown that the cochain complex computing CotorU(A,N) defines a cyclic operad with multiplication, and hence the groups CotorU(A,N) form a Batalin-Vilkovisky algebra. In the second part of the talk, Poisson structures and the Poisson bicomplex for bialgebroids are introduced. They allow us to generalise simultaneously both classical Poisson homology and cyclic homology. In case the bialgebroid U is commutative, a Poisson structure on U leads to a Batalin-Vilkovisky algebra structure on TorU(A,A). As an illustration, we show how this generalises the classical Koszul bracket on differential forms. Finally, we conclude by indicating how classical Lie-Rinehart bialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids.

NIELS KOWALZIG (Università degli Studi di Roma "Tor Vergata", Italy)



20 January 2014, 14:15-16:00

CHARACTERISTIC CLASSES OF EQUIVARIANT K-THEORY WITH COEFFICIENTS

In my talk, I will describe two constructions of characteristic classes for what can be regarded as equivariant K-theory with coefficients. The first construction uses characters on a Hopf algebra and takes values in Hopf-cyclic cohomology. In the second one, we deal with an equivariant sheaf of K-groups of a Lie group, and construct a characteristic map to the Block-Getzler sheaf of equivariant differential forms. As the Block-Getzler sheaf appears to be closely related to Hopf-cyclic theory, there might be a way of comparing these two approaches. (The talk is based on a joint paper with I. Nikonov.)

GEORGY SHARYGIN (ITEP, Moscow, Russia)



3 March 2014, 14:15-16:00

APPROXIMATION PROPERTIES OF GROUPS, QUANTUM GROUPS AND VON NEUMANN ALGEBRAS

We give an overview of recent results on the interaction of approximation properties of groups, quantum groups, von Neumann algebras and Banach algebras. The beginning of the talk is motivated purely by quantum group theory. We show how compactness and amenability of a quantum group can be captured in homological properties of its Fourier algebra. In particular, we show that important results of Z.-J. Ruan cannot be extended beyond Kac type quantum groups. Secondly, we will see how the Haagerup property can be defined for an arbitrary von Neumann algebra. In particular, this defines a suitable Haagerup property for recent examples in discrete quantum group theory by De Commer, Freslon and Yamashita. We shall also show that the Haagerup property satisfies several stability properties and that there exists a non-discrete quantum group that has the Haagerup property. If time permits, in the remainder of the talk, we will look at approximation properties of non-commutative Lp-spaces of group von Neumann algebras. The talk is based on several writings, including joint work with Hun Hee Lee, Éric Ricard, Mikael de la Salle and Adam Skalski.

MARTIJN CASPERS (Universität Münster, Germany)



10 March 2014, 14:15-16:00

QUANTUM EXIT TIME ASYMPTOTICS AND ANALYTIC THEORY OF LÉVY PROCESSES ON COMPACT QUANTUM GROUPS

The aim of this talk is twofold. In the first half, we will give an analytic construction of Lévy processes on compact quantum groups of reduced type. The construction gives an analytic footing to the algebraic theory of Lévy processes on bialgebras à la Schürmann, in particular in the context of compact quantum group algebras. Our results complement the analytic theory of Levy processes on compact quantum groups of universal type obtained by Lindsay and Skalski. For instance, we will show that every Lévy process on a reduced compact quantum group is implemented by a family of unitary cocycles consisting of corepresentations of the dual quantum group. In the second half, we will formulate a quantum analogue of exit time in classical probability. It is well known that many geometric objects related to a Riemannian manifold, like the dimension of the manifold and scalar curvature, can be obtained through a systematic study of exit time of the standard Brownian motion on the manifold. In the same spirit, we exemplify results connecting our theory of Lévy processes and noncommutative geometry.

BISWARUP DAS (IMPAN)



31 March 2014, 14:15-16:00

QUANTUM PILLOW

Quotients of noncommutative tori by actions, of finite cyclic groups were introduced as quantum topological spheres. After a brief review of their history and basic properties, I will answer the question whether the quantum pillow is a noncommutative manifold or a noncommutative orbifold. (Based on joint work with Tomasz Brzeziński.)

ANDRZEJ SITARZ (Uniwersytet Jagielloński / IMPAN)



7 April 2014, 12:15-13:45

DIRAC OPERATORS ON NONCOMMUTATIVE PRINCIPAL TORUS BUNDLES

I will present recently proposed noncommutative spectral triple extensions of Riemannian spin geometry of principal circle and torus bundles. Our constructions require several suitable and mutually matching choices for various structures on the total space, the base space and the structure group of a bundle. This includes differential calculi and principal connections, all formulated in terms of Hilbert space operators. The key idea is to relate the noncommutative geometry of the total space of a bundle with the geometry of its base space, and to use strong connections to build new Dirac operators. As a particular case, I will discuss noncommutative tori and theta-deformed manifolds. (Based on joint work with Andrzej Sitarz and Alessandro Zucca.)

LUDWIK DĄBROWSKI (SISSA, Trieste, Italy)



7 April 2014, 14:30-16:00

BRAIDED NONCOMMUTATIVE JOIN CONSTRUCTION

The goal of this talk is to explain a join construction of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the diagonal coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and a certain finite-dimensional noncosemisimple Hopf algebra. The former yields a noncommutative deformation of a nontrivial torus bundle, and the latter a finite quantum covering. (Based on joint work with Ludwik Dąbrowski, Tom Hadfield and Piotr M. Hajac.)

ELMAR WAGNER (Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico)



28 April 2014, 10:15-12:00

RUDIMENTS OF ASYMPTOTIC AND EXACT EXPANSIONS OF HEAT TRACES

The heat trace expansion is one of the most powerful tools in the study of geometry via spectral properties of (pseudo)differential operators on manifolds. With the development of noncommutative geometry, the need came to investigate heat traces associated with general operators of Dirac type in the framework of spectral triples. No general results are available in this setting, and the existence of an asymptotic expansion of the heat trace has been proven rigorously only for a few specific examples. In this talk, we will exploit the Mellin transform to explain the interplay of the heat trace of an unbounded positive operator with its spectral zeta function. We will establish general conditions under which an asymptotic expansion of the heat trace exists. Moreover, we shall discuss the issue of convergence of the expansion.

ARTUR ZAJĄC (Uniwersytet Jagielloński)



28 April 2014, 14:15-16:00

EXAMPLES OF ASYMPTOTIC AND EXACT EXPANSIONS OF HEAT TRACES

In this talk, we will illustrate the general theory of heat traces with several geometrically motivated examples. At first, we will establish the existence of an asymptotic expansion for an important class of operators with polynomial eigenvalues and polynomial multiplicities. We shall also provide explicit examples for convergent expansions leading to (almost) exact representations of heat traces. Next, we will discuss the case of operators with exponential eigenvalues, which appear in the framework of fractals or quantum groups. We shall end with a counterexample exhibiting the limit of applicability of the Mellin transform method.

MICHAŁ ECKSTEIN (Uniwersytet Jagielloński)



12 May 2014, 10:15-12:00

CROSSED PRODUCTS BY ENDOMORPHISMS, TRANSFER OPERATORS AND COMPLETELY POSITIVE MAPS

There are essentially two different constructions of crossed products of C*-algebras by endomorphisms. One of them was initiated by Cuntz, and involves the so-called Stacey crossed product. Another one was introduced by Exel. In this talk, we re-examine these two approaches and show that they can be naturally unified in the realm of crossed products by completely positive maps. Along the way, we plan to discuss the problem of existence of noncommutative Markov shifts on infinite graph C*-algebras, and the method of reversible extensions. In the commutative case, the latter leads to topologically complicated objects of dynamical origin.

BARTOSZ K. KWAŚNIEWSKI (IMPAN / Uniwersytet w Białymstoku)



12 May 2014, 14:15-16:00

UNIVERSAL QUANTUM GROUPOIDS

By a result of T. Banica, any compact quantum group with the same fusion rules as SU(2) is isomorphic to a free orthogonal quantum group. In this talk, we consider quantum groupoids having these fusion rules. Our formalism for quantum groupoids is that of Hayashi's compact Hopf face algebras, slightly generalised to accommodate infinite bases. Our free orthogonal quantum groupoids are obtained by the Tannaka-Krein reconstruction applied to a fiber functor from a Temperley-Lieb category into a category of bigraded Hilbert spaces. As such, these quantum groupoids are closely linked with ergodic actions of the quantum SU(2) (recent joint work with M. Yamashita). As an example, we show that the dynamical quantum SU(2) groups can be obtained from the natural action of the quantum SU(2) on a generic Podleś sphere. (This is work in progress joint with T. Timmermann.)

KENNY DE COMMER (Vrije Universiteit Brussel, Belgium)



19 May 2014, 10:15-12:00

EXPANDERS AND AND MORITA-COMPATIBLE EXACT CROSSED-PRODUCTS

An expander or expander family is a sequence of finite graphs X1, X2, X3,... which is efficiently connected. A discrete group G which "contains" an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. Some care must be taken with the definition of "contains". M. Gromov outlined a method for constructing such a group. G. Arjantseva and T. Delzant completed the construction. Any group so obtained is known as a Gromov group or Gromov monster group, and these are the only known examples of a non-exact groups. The left side of BC with coefficients "sees" any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also "sees" any group as if the group were exact. This corrected form of BC with coefficients uses the unique minimal exact and Morita compatible intermediate crossed-product. For exact groups (i.e. all groups except the Gromov groups) there is no change in BC with coefficients. In the corrected form of BC with coefficients, a Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (May, 2014) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Guentner and R. Willett. This work is based on - and inspired by - a result of R. Willett and G. Yu, and is very closely connected to results in the thesis of M. Finn-Sell.

PAUL F. BAUM (Pennsylvania State University / IMPAN)



19 May 2014, 14:00-15:45

THE EBERLEIN COMPACTIFICATION OF QUANTUM GROUPS

In 1949, Eberlein initiated the theory of weakly almost periodic functions on abelian groups in order to gain understanding of the Fourier-Stieltjes transforms of measures on the dual group. Following his work, in the recent years, there has been a comprehensive study of the notion of "weakly almost periodic functions" on locally compact groups. The weakly almost periodic functions on a locally compact group G form a commutative C*-algebra wap(G), whose character space Gwap gives the largest "semitopological" semigroup compactification of G, while the largest "group" compactification of G (otherwise known as the Bohr compactification) is given by the character space of the algebra of almost periodic functions on G. It is natural to seek a generalisation of these concepts within the framework of quantum groups. A recent work of Sołtan provides the Bohr compactification of quantum groups. The resulting object that he obtains is a compact quantum group. This perfectly agrees with the classical picture: the Bohr compactification of a locally compact group is a compact group. In general, it is difficult to define weakly almost periodic functions on quantum groups due to the absence of a notion of a "quantum semitopological semigroup", i.e. a quantum semigroup with separately continuous products. In this talk, we will partially address this issue by proposing a notion of the Eberlein compactification of quantum groups. The Eberlein compactification (of locally compact groups) is a special type of semigroup compactification studied extensively in the recent years by Spronk and Stokke. We will explain some aspects of our theory, and connect our work with that of Sołtan. (Based on joint work with Matthew Daws.)

BISWARUP DAS (IMPAN)



2 June 2014, 14:00-15:45

CYCLIC COHOMOLOGY AND LOCAL INDEX THEORY FOR LIE GROUPOIDS

In this talk, we will consider smooth actions of Lie groupoids on manifolds. Using cyclic-cohomological methods, we will establish local higher index formulas for certain pseudo-differential-type operators.

DENIS PERROT (Université Lyon 1, France)