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

1 October 2007

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.

8 October 2007

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.

15 October 2007

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.

22 October 2007

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.

29 October 2007

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).

5 November 2007

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.

12 November 2007

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.

19 November 2007

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.

26 November 2007

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.

3 December 2007

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.)

10 December 2007

Connection-like objects, termed

17 December 2007

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.

7 January 2008

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.

14 January 2008

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.

21 January 2008

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.

18 February 2008

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.

25 February 2008

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.

3 March 2008

10 March 2008

17 March 2008

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.

31 March 2008

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.

7 April 2008

14 April 2008

21 April 2008

28 April 2008

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.)

5 May 2008

12 May 2008

19 May 2008

26 May 2008

2 June 2008