# Events - Page 4

Michael Whittaker from University of Glasgow will give a talk with title: New directions in self-similar group theory

Abstract: A self-similar group (G,X) consists of a group G acting faithfully on a homogeneous rooted tree such that the action satisfies a self-similar condition. In this talk I will generalise the above definition to faithful groupoid actions on the path space of more general graphs. This new definition allows us to work out the structure of the KMS state space of associated Toeplitz and Cuntz-Pimsner algebras. This is joint work with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge.

Rasmus Bryder (University of Copenhagen) will give a talk with title: Twisted crossed products over C*-simple groups

Abstract: A twisted C*-dynamical system consists of a C*-algebra, a discrete group and a "twisted" action of the group on the C*-algebra, i.e., the group acts by automorphisms on the C*-algebra in a manner determined by a 2-cocycle of the group into the unitary group of the C*-algebra. Whenever the 2-cocycle (or twist) is trivial, the action is given by a group homomorphism of the group into the automorphism group of the C*-algebra. We consider twisted C*-dynamical systems over C*-simple groups (i.e.,groups whose reduced group C*-algebra is simple) and how C*-simplicity affects the ideal structure of reduced crossed products over such dynamical systems.

Andreas Andersson (UiO): An introduction to duality for compact groups in algebraic quantum field theory

Bartosz Kwasniewski (Odense) will give a talk with title: Paradoxicality and pure infiniteness of C*-algebras associated to Fell bundles

Abstract: Abstract: In this talk we present conditions implying pure infiniteness of the reduced cross-sectional $C^*$-algebra $C^*_r(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ over a discrete group $G$. We introduce notions of aperiodicity, $\mathcal{B}$-paradoxicality and residual $\mathcal{B}$-infiniteness. We discuss their relationship with similar conditions studied, in the context of crossed products, by the following duos: Laca, Spielberg; Jolissaint, Robertson; Sierakowski, R{\o}rdam; Giordano, Sierakowski and Kirchberg, Sierakowski. (based on joint work with Wojciech Szyma{\'n}ski)

Abstract: Exploring connections between subfactors and conformal field theories, Vaughan Jones recently observed that planar algebras give rise to unitary representations of the Thompson group F, and more generally, to unitary representations of the group of fractions of certain categories. Remarkably, this procedure applies to oriented link invariants. In particular, a suitably normalized HOMFLYPT polynomial is a positive definite function on the oriented Thompson group. (Based on joint work with V. Aiello and V. Jones.)

In this talk I will present a paper by D. Bisch, R. Nicoara and S. Popa where continuous families of irreducible subfactors of the hyperfinite II_1 factor which are non-isomorphic, but have all the same standard invariant are constructed. In particular, they obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II_1 factor with Jones index 6, which have all the same standard invariant with property (T).

Abstract: This talk addresses some of the fundamental barriers in the theory of computations. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations such as computing spectra of operators, solutions to inverse problems, roots of polynomials using rational maps, solutions to convex optimization problems, imaging problems etc. such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. The SCI phenomenon is independent of the axiomatic setup and hence any theory aiming at establishing the foundations of computational mathematics will have to include the so called SCI Hierarchy. We will specifically discuss the vast amount of classification problems in this non-collapsing complexity/computability hierarchy that occur in inverse problems, compressed sensing problems, l1 and TV optimization problems, spectral problems, PDEs and computational mathematics in general.

The conference "Quantum groups: geometry, representations, and beyond" will take place on May 9-14, 2016.

Abstract: In a recent work with R. Conti (La Sapienza Univ., Rome), we have introduced a notion of positive definiteness for certain functions associated to a (unital, discrete) C*-dynamical system. We will sketch the proof of a Gelfand-Raikov type theorem for such functions and use it to construct complete positive maps on the full and the reduced C*-crossed products of the system. We will also explain how a natural definition of amenability for C*-dynamical systems emerges from our work.

Abstract: The talk will be on positive linear maps of the n x n matrices into itself, a topic which has become quite popular in quantum information theory. The maps closest to physics are the completely positive ones. I´ll discuss an approximation by a completely positive map to a positive map via the trace , called the “structural physical approximation”, the SPA of the map. Much of the talk will circle around a counter example to a conjecture on the SPA.

Abstract: In the classification program for C*-algebras some of the usual assumptions put on the algebras are that they are simple or have at most have finitely many ideals. We often also want algebras that have real rank 0. In this talk we will discuss how to classify certain graph algebras with uncountably many ideals and without real rank 0. There will be examples and applications. Joint work with S. Eilers, G. Restorff, and E. Ruiz

Abstract: There are many interesting examples of groups acting on trees, arising in various fields (e.g. combinatorial group theory, number theory, geometry). When a group acts on a tree, it necessarily also acts on the boundary of the tree, a (totally disconnected) compact Hausdorff space. The C*-algebras obtained from the crossed product construction include many fundamental examples. I will describe methods for analyzing such crossed products, developed in joint work with Nathan Brownlowe, Alex Mundey, David Pask and Anne Thomas.

Abstract: In this follow-up talk, I shall outline how the boundary quotient diagram may be useful for K-theoretic considerations. We start with the diagram within the context of integral dynamics, and then speculate about potentially promising directions of generalizations.

Abstract: In this follow-up talk, we shall review the results on the structure of KMS states from the case studies of - the ax+b semigroup over the natural numbers (Laca-Raeburn and Brownlowe-an Huef-Laca-Raeburn), - integer dilation matrices (Laca-Raeburn-Ramagge), - self-similar actions (Laca-Raeburn-Ramagge-Whittaker), and - Baumslag-Solitair monoids (Clark-an Huef-Raeburn) from the perspective of the boundary quotient diagram for the respective right LCM semigroups. We will also discuss (to some extent) similarities and differences of the proofs among these cases.

Abstract: For the ax+b semigroup over the natural numbers, which is known to be part of a quasi-lattice ordered group, Laca and Raeburn considered its Nica-Toeplitz algebra and its Cuntz-Nica-Pimsner algebra, with a special appeal to nice presentations by generators and relations as well as the structure of KMS states for a natural dynamics. Shortly thereafter, Brownlowe-an Huef-Laca-Raeburn showed that there are two intermediate quotients between the Nica-Toeplitz algebra and the Cuntz-Nica-Pimsner algebra that exhibit interesting structural properties, especially with regards to KMS states. Since then, analogous quotients have been considered (partly in disguise) in a growing list of case studies on the KMS state structure, e.g. for dilation matrices, self-similar actions, and Baumslag-Solitair monoids. Somewhat surprisingly, all these case studies can be viewed from the perspective of semigroup C*-algebras of right LCM semigroups, and in this talk, I shall describe a unifying perspective on such boundary quotient diagrams. Thereby several questions concerning the general structure of right LCM semigroups are raised.

Abstract: As has been observed by many authors, the Drinfeld double of the q-deformation of a compact Lie group can be regarded as a quantization of the complexification of the original Lie group. Using this point of view, I will discuss irreducible unitary representations of these Drinfeld doubles.

Abstract: We discuss a way of constructing noncommutative projective manifolds as inductive and projective limits, generalizing the so-called Berezin quantization for ordinary compact Kähler manifolds. We first review the physical motivation for Berezin quantization and then discuss how the restriction to commutative manifolds limits the use of this quantization. We will also outline how our more general construction appears naturally in the study of the long-time limit of open quantum systems.

Bas Jordans will continue his talk from last week.

Bas Jordans will give a talk with title " Random walks on discrete quantum groups: convergence to the boundary"

Abstract:

For classical random walks there exist two boundaries: the Poisson boundary and the Martin boundary. The relation between these two boundaries is described by the so-called "convergence to the boundary". For random walks on discrete quantum groups both the Poisson boundary and Martin boundary are defined and a non-commutative analogue of convergence to the boundary can be formulated. However, no proof is known for a such a theorem. In the first part of the talk we will discuss the classical and quantum version of convergence to the boundary, explain how these are related and give an overview of what is known in general for the quantum case. In the second part we will discuss the boundary convergence for SUq(2) and for monoidally equivalent quantum groups.

Piotr Soltan (University of Warsaw) will give a talk with title: Subgroups of quantum groups, the center and inner automorphisms

Arnaud Brothier, Vanderbilt University (USA) will give a talk with title: Analytic properties for subfactors

Abstract:

I will discuss analytic properties for groups and their generalizations to subfactors, standard invariants, and certain tensor categories.

I will present a class of subfactor planar algebras that are constructed with a group acting on a bipartite graph.

I will show that if the group satisfies a given approximation property (such as amenability, Haagerup property, or weak amenability), then the subfactor planar algebra satisfies it as well.

I will exhibit an infinite family of subfactor planar algebras with non-integer index that are non-amenable, have the Haagerup property, and have the complete metric approximation property.

Alfons van Daele (Leuven) will give a talk with title: The Haar measure on quantum groups

Abstract: At this moment, there is no theory of locally compact quantum groups with axioms from which one can prove the existence of the Haar weights. In general, the existence is part of the axioms. There are however a few cases where the existence can be proven. This is true for compact quantum groups and for discrete quantum groups. We will discuss some aspects of these proofs and see which of them are useful in the general case.

Judith Packer, University of Colorado (Boulder), USA, will give a talk with title: Wavelets associated to representations of higher-rank graph C*-algebras

Abstract: Let $\Lambda$ denote a finite $k$-graph in the sense of A. Kumjian and D. Pask that is strongly connected, and let $\Lambda^{\infty}$ denote its infinite path space. I discuss some recent joint work with C. Farsi, E. Gillaspy, and S. Kang, where we construct a system of functions that we call ``wavelets" on a Hilbert space of square-integrable functions on $\Lambda^{\infty}.$ In so doing, we generalize work of M. Marcolli and A. Paolucci for finite directed graphs to the higher rank case. The key tool is the construction of a representation of the graph $C^*$-algebra $C^{\ast}(\Lambda)$ on $L^2(\Lambda^{\infty},M)$ for the appropriate measure $M.$ When the finite $k$-graph $\Lambda$ in question is strongly connected and aperiodic, the representation of $C^{\ast}(\Lambda)$ that we obtain is faithful.

Antoine Julien, NTNU, will give a talk with title: Links between cut-and-project tilings and Diophantine approximation

Abstract: Cut-and-project tilings are obtained by cutting a slice of a higher dimensional lattice and projecting it on a lower dimensional space. The result is a point set which is regular enough (since it originates from a lattice), but is not periodic, provided the direction of the slice is irrational in a suitable sense. In one dimension, typical examples of this construction are Sturmian subshifts. It is known that some of their dynamical properties depend on the arithmetic properties of a certain parameter. In this talk, I will recall some known results by Hedlund and Morse on Sturmian subshifts. Then, I will describe how, even in higher dimensions, the repetition properties of some cut-and-project sets can be linked to problems of simultaneous Diophantine approximation. This is joint work with A. Haynes, H. Koivusalo and J. Walton.