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Abstracts of the Students' Seminar

C*-bundles and their K-theory, Prahald Vaidyanathan

My plan for the talk on Wednesday is as follows : 1. Introduce continuous C* bundles 2. Introduce a K-Theory sheaf used to analyze them. 3. Explain an approximation result for bundles over [0,1]

Z/k manifolds and K-homology with Z/k-coefficients, Robin Deeley

A Z/k manifold is a singular space formed from a manifold which has boundary that decomposes into k disjoint diffeomorphic pieces (the singular space is created by identifying the "k-parts" of the boundary). They were introduced by Sullivan in the 1960s to study geometric topology. Later (in the 1980s) Freed and Melrose proved an index theorem for operators acting on such spaces (where the index takes values in Z/k). Also in the 1980s, Baum and Douglas defined a geometric model for K-homology using spin^c manifolds. We will consider a geometric model for K-homology with Z/k-coefficients where we replace the spin^c manifold theory used in the Baum-Douglas model with the corresponding theory for Z/k manifolds. The Freed-Melrose index theorem then plays the role that the Atiyah-Singer index theorem plays in the Baum-Douglas model.

C*-algebras generated by C*-correspondences and applications to quantum spheres, Dave Robertson

The C*-algebras generated by C*-correspondences are a class of C*-algebras introduced by Katsura in 2003 simultaneously generalising both Cuntz-Pimsner algebras and graph algebras. I will begin by defining a C*-correspondence and then showing how we may associate a universal C*-algebra to it. I will then present a result regarding pullbacks of such algebras, and show how this can be used in the construction of new examples of 'glued' quantum spaces with a particular focus on quantum spheres. This talk is based on work with my Ph.D supervisor Dr. Wojciech Szymanski.

KK-Theory for Cuntz-Pimsner Algebras, Christina Cerny

We follow the 2nd chapter of Pimsner's article to gain two six-term exact sequences for the KK-Theory of O_X

Poisson Geometry aspects of Non-Commutative Geometry, Hanh Duc Do

I talk about some obstruction to quantize non-integrable Poisson Manifold, and how to deal with that in the category of stack. Finally I talk about moduli stack of quantum algebras, with noncommutative torus bundle as an example.

Kadison's Similarity Problem for Z-stable C*-algebras, Miroslava Johanesova

We will show that unital separable Z-stable C*-algebras verify Kadison's Similarity Problem.

Introduction to Atiyah Conjecture, Łukasz Grabowski

Atiyah conjecture is a certain statement of the group theory. Its most basic (and slightly imprecise) formulation is "The group ring of a torsion free group can be embedded into a skew field". In the first part of my talk I will try to show why this conjecture is interesting and point its connections to other well-known conjectures about various group algebras. In the second part I will show a proof of the Atiyah conjecture for the free group (after Peter Linnel). This proof is very elegant, in the sense that it is easy to point the two key ideas in it. Finally, I will talk briefly about my Ph.D. project - proving Atiyah conjecture for new classes of groups.

On almost representations of groups with property (T), Chao You

Kazhdan's property (T) for groups is equivalent to the alternative that a representation of such a group either has an invariant vector or does not have even almost invariant vectors. It is shown that, under a somewhat stronger condition due to Zuk, a similar alternative holds for almost representations.

Property RD for Hecke pairs, Vahid Shirbisheh

We study Property RD (Rapid Decay) for Hecke pairs. We adapt Jolissant’s works to this situation and show that when a Hecke pair (G, H) has Property RD, the algebra of rapidly decreasing functions is closed under holomorphic functional calculus of the associated (reduced) Hecke C∗ -algebra, and consequently, they have the same K0 -groups.

Coarse geometry and the Baum-Connes conjecture, Rufus Willett

I will introduce the coarse assembly map of Higson-Roe, and the coarse co-assembly map of Emerson-Meyer, via the example of the real line. I will then comment on the relationship these have with the Baum-Connes conjecture and Dirac-dual-Dirac method, and discuss some techniques for computing the K-theory groups of the Roe algebra and stable Higson corona (which arise respectively as the right hand side of the coarse assembly, and left hand side of the coarse co-assembly, maps).

Morphims of spectral triples and an algebraic formula for he Kasparov product, Bram Mesland

We show how unbounded KK-cycles equipped with a universal connection can, under a suitable smoothness hypothesis, be interpreted as morphisms of spectral triples. Moreover, the composition, defined on the level of cycles with connection, gives rise to an algebraic formula for the Kasparov product. This is done by interpreting the bounded transform as a functor.

Gap-labelling of the pinwheel tiling, Haija Moustafa

I will begin by introducing the definition of pinwheel tiling and more generally of aperiodic tilings of the plane. I will then present how we can associate a topological space (the continuous hull) to such a tiling which encodes the combinatorial properties of the tiling into topological and dynamical properties of the continuous hull. This space is endowed with an action of the rigid motions and thus we can consider the associated crossed product C*-algebra wich is in fact endowed with a trace (associated to a measure on the continuous hull). This trace induces a linear map on the K-theory of the crossed product and then, the aim of the gap labelling is to compute the image under this trace of the K-theory of the C∗-algebra. I will then present how we can compute this image using the index theorem on foliations of A. Connes and show how this can be used to do explicit computations.

Hilbert modules and intertwining operators for the principal series, Pierre Clare

We first provide a description of the principal series representations of a semisimple Lie group in terms of Hilbert C*-modules. The construction is adapted from Rieffel's work in a way that takes some special features of these representations into account, and the modules can be realised in different pictures, related to the structure of the group. We then construct analogues of Knapp-Stein intertwining operators at this level, the point being that no meromorphic extension techniques are needed to do so. The singularities of these operators nevertheless manifest through some unboundedness properties, which can somehow be measured. Then we explain the principle of normalisation for these operators, which we can perform in special cases. An irreducibility result can also be proved, inspired from Bruhat's work.

The Elliott conjecture and dimension theories of C*-algebras, Hannes Thiel

The Elliott conjecture predicts that nuclear (simple) C*-algebras are classified by their Elliott invariant (K-theory, states, and the pairing between both). The conjecture is known to hold for many classes of C*-algebras. It is therefore natural to ask, how properties (like dimension) of a C*-algebra are reflected in its Elliott invariant.

While for commutative spaces there is only one (natural) concept of dimension, there are various dimension theories for C*-algebras (e.g. real and stable rank). We consider the "ASH-dimension", which is \leq n if the C*-algebra is a limit of n-dimensional subhomogeneous algebras. The famous AF-agebras are exactly the algebras with "ASH-dimension" = 0. We show how the ASH-dimension is detected in the Elliott invariant.

Twists of Isometries, Moritz Weber

The irrational rotation algebra, also known as the non-commutative torus, is the universal C*-Algebra generated by two unitaries, that commute up to a scalar. What happens, if you weaken the condition of being unitaries to the case, when two isometries commute up to a scalar?

This algebra turns out to be too general to be a nice object, but anyhow, it is the starting point of my way of looking at twists of isometries. Therefore I will say a few words about this algebra and how the property of nuclearity gets lost on the way from the (unitary) rotation algebra to the isometry rotation algebra.

The second part is about a twist of two copies of the Cuntz algebra O_2 - hence again a twist of isometries, but with additional structure. This part of the talk refers to a paper written by J. Cuntz in 1981 (K-theory for certain C*-algebras. II.). I will introduce the original definition of Prof. Cuntz, then give a slightly more modern one, explain the connection (or possible connection) between them and finally say something about representing this structure on a Hilbert space.

The Baum-Connes conjecture for p-adic SL2, Tyrone Crisp

I'll discuss the Baum-Connes Conjecture for the p-adic special linear group. I want to compare the p-adic case with the real case (which we heard about, for instance, in the lectures of Pierre Julg). This will involve a construction of the so-called "affine Bruhat-Tits building" (actually a tree) for SL2, and of certain associated equivariant homology groups. I will also briefly mention the general construction, due to Kasparov and Skandalis, of Dirac and dual Dirac elements for groups acting on affine buildings.

The "conjecture" is in fact known to be true for such groups, and if time permits I'll suggest how one might hope to use the Baum-Connes isomorphism to say something interesting about representation theory.

This last part of the talk will necessarily be fairly sketchy, but the discussion of buildings and trees will be appropriately concrete and down-to-earth.

Twisted K-theory and the fractional index theorem, Elkaioum Moutuou

I will introduce the twisted K-theory of a topological space, and show how to express geometrically its elements in the case when the twist is of finite dimension. Then I will talk about pseudodifferential operators acting of twisted vector bundles, and maybe (depends on the time) I will give the analog of the Atiyah-Singer index theorem in the twisted case, following recent works by Mathai, Melrose and I. Singer (2004).

Heat kernel and index theorem, Hang Wang

The heat kernel approach to index theory is an analytical way to work out index for Dirac-type operators. It have been used to prove various index theorems (on cohomology level) including compact manifold (with boundary), families of Dirac operators, L^2 index theorem of homogenous space... I will present a classical heat kernel proof of a simple case: index theory for Dirac operator on even dimensional compact manifold.