Past Events

• Alex Belton: Entrywise preservers for classes of matrices with positivity and negativity constraints

We all know how to multiply matrices, but it is also possible to treat them simply as arrays of numbers and perform algebraic operations entry by entry. For multiplication, this is called the Hadamard product. A notable feature of the Hadamard product is that it preserves the collection of matrices that are positive semidefinite: real symmetric matrices with non-negative eigenvalues. It follows immediately that applying any absolutely monotonic function entrywise also preserves this form of positivity. (A function is absolutely monotonic if its Maclaurin series has non-negative coefficients.) Rather more work is required to show that the converse is true: a function that preserves positive semidefiniteness when applied entrywise to matrices of arbitrary size is necessarily absolutely monotonic.

The situation is more complex for matrices of a fixed size, or when the class of matrices under study has some other form of positivity or negativity constraint, or possesses additional structure, such as Hankel or Toeplitz matrices. This talk will discuss results for some of these situations.  

This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

• Valerio Bianchi: Equivariant higher Dixmier-Douady theory for strongly self-absorbing C*-algebras

A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with the compact operators as fibres by methods from homotopy theory. Dadarlat and Pennig have shown that this can be generalised: precisely, they found out that bundles whose fibres are isomorphic to stabilised strongly self-absorbing C*-algebras are classified by a certain cohomology theory, namely the units of K-theory. Evans and Pennig have also shown that part of the theory carries over equivariantly for circle actions on UHF-algebras. In joint work with Ulrich Pennig we show that by further restricting to a Z/pZ-action on such algebras we get a complete generalisation of the non-equivariant results.

• Claire Debord: Groupoids and index theory

The aim of this lecture is to explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory. The first par of the talk will be a review on Lie groupoids, their C*-algebras and associate pseudodifferential calculus. Then, I will describe the deformation to the normal cone construction and the blowup construction in the groupoid setting and explain how those can be used in order define order 0 pseudodifferential operators in  a (pseudo)differential operator free way, construct new pseudodifferential calculi, go to various generalize index problems. 

This talk, inspired by ideas of A. Connes, will be based on joint works with G. Skandalis.

• Shanshan Hua: Classification of approximately finite-dimensional *-homomorphisms

A C*-algebra is quasi-diagonal if there is a unital embedding into the ultraproduct of matrices. The Tikusis-White-Winter theorem shows that any stably finite, simple, separable, nuclear, unital C*-algebra satisfying the UCT will be quasidiagonal. Thus an interesting question is to understand how unique are such embeddings given the existence.

We are able to answer this question for simple C*-algebras, by obtaining a KK-uniqueness theorem. The major difficulty appears since the codomain of the map of interest does not separably absorb the Jiang-Su algebra. We will explain how to tackle the difficulty by looking at K-theoretical properties of a special C*-algebra, the Paschke Dual algebra associated to the map.

• Julian Kranz: Amenable actions on simple C*-algebras

Amenable actions of groups on C*-algebras are actions that behave like actions of amenable groups. There are many examples of amenable actions of non-amenable groups on commutative C*-algebras and more recently, examples on simple C*-algebras were obtained. It remains open whether a non-amenable group can act amenably on a unital, simple, stably finite C*-algebra. I will try to shed some light on the difficulty of this problem and discuss related regularity properties for actions of non-amenable groups on classifiable C*-algebras. The results are based on joint work together with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro.

• Shinataro Nishikawa: Equivariant K-theory of approximately inner flips

Dominic Enders, André Schemaitat, and Aaron Tikuisis identified the classifiable C*-algebras with approximately inner flip. I will give an answer to the following question: are these flip actions by Z/2 trivial in equivariant KK-theory? To answer this, we apply the equivariant UCT theorem developed by Manuel Köhler, which has been further studied by Ralf Meyer. I will use this opportunity to describe how to use Köhler’s UCT for practical examples for those who have not seen such an application. This talk is based mostly on joint work with Julian Kranz.

• Aaron Tikuisis: Central sequences and topological dynamics

Central sequences play a fundamental role in operator algebras (property Gamma, the MacDuff property, Z-stability for example). Given the deep link between operator algebras and dynamics, it is natural to look for dynamical analogues of central sequence properties. This turns out to be subtle. I will discuss a body of work related to this idea. This is joint work with Grigorios Kopsacheilis, Hung-Chang Liao, and Andrea Vaccaro.

• Francesca Tripaldi: Multicomplexes on Carnot groups

Carnot groups are the easiest examples of subRiemannian manifolds. We show that, on Carnot groups, the de Rham complex has the structure of a multicomplex. The extra algebraic structure given by the multicomplex has several applications, such as extracting sharper subcomplexes that are homotopic to the de Rham complex.

• Nick Wright: K-theory for (Exceptional) Extended Affine Weyl Groups

Affine Weyl groups are a class of groups for which the group C*-algebra can in principle be understood directly. I will give an overview of how Langlands duality gives a Poincare duality for these algebras, and discuss the problem of calculating the K-theory for the extended affine Weyl groups for groups of type A_n and for the exceptional Lie group E_6.

• Adrian Ioana (UCSD): Wreath-like product groups and rigidity of their von Neumann algebras.
  Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T).
  In this talk, I will present several rigidity results for von Neumann algebras of wreath-like product groups. We show that any group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group von Neumann algebra L(G) remembers the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T). For a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups.  As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an icc property (T) group G. These results were obtained in joint works with Ionut Chifan, Denis Osin and Bin Sun.
  I will also mention an additional application of wreath-like products obtained in joint work with Ionut Chifan and Daniel Drimbe, and showing that any separable II_1 factor is contained in one with property (T). This provides an operator algebraic counterpart of the group theoretic fact that every countable group is contained in one with property (T).

• Cornelia Drutu (Oxford): Actions of groups on Banach spaces
  This talk will discuss various versions of fixed point properties, generalizing property (T), and of proper actions on Banach spaces, generalizing a-T-menability. In particular, I will describe a notion of spectrum providing an optimal way to measure ``the strength'' of the property (T) that an infinite group may have, and what can be said about this spectrum, in particular for hyperbolic groups. I will also describe weak versions of a-T-menability for (acylindrically) hyperbolic groups and for mapping class groups. This is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.

• Shanshan Hua (Oxford): Nonstable K-theory for Z-stable C*-algebras
  In Jiang's unpublished paper (1997), it is shown that any Z-stable C*-algebra A is K1-injective and K1-surjective, which means that K1(A) can be calculated by looking at homotopy equivalence classes of U(A), without matrix amplifications. Moreover, for such A, higher homotopy groups of U(A) are isomorphic to K0(A) or K1(A), depending on the dimension of the higher homotopy group. In this talk, I will present Jiang's result for Z-stable C*-algebras. Moreover, I will explain briefly our new strategies to reprove his theorems using an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.

• Sam Richardson (Cardiff): Natural Transformations of Cohomology Theories Induced by Exponential Functors
  Exponential functors are monoidal functors that transform the direct sum of vector spaces into the tensor product of two slightly different vector spaces. The source and target categories of an exponential functor are both strict symmetric monoidal categories and so we may use them to construct cohomology theories.
  Of particular note for C*-algebras is the map induced in the 1st degree. We take the geometric realisation of the nerve of our exponential functor to achieve a map of spaces and it just so happens that the target space is homotopy equivalent to a certain C*-algebra also derived from the exponential functor.
  To better understand the exotic cohomology theories we achieve we investigate the class of the Weyl map W: SU(n)/T x T -> SU(n) which will prove very useful thanks to it’s particular product structure and the suspension-loop adjunction. Since the map we are studying factors through the space U we will be able to investigate using K theory first and then hopefully it will be as easy as applying our exponential functor to the relevant vector bundles!

• Maryam Hosseini (QMUL): Dimension Groups and Topological Factoring of Bratteli-Vershik Systems
  Bratteli diagrams were initially introduced in classifications of Af-algebras. In 1992, R. H. Herman I. Putnam and C. Skau had shown the close connections between /Cantor minimal dynamical systems/, /Bratteli diagrams/ and /Dimension groups/ which led to many studies on Cantor dynamics. Knowing topological factors of a dynamical systems is one of the interests of people in the field. Topological factors of a dynamical system can realize topological or sometimes all measurable /spectrum/ of the system and may give information about the complexity of the system. In 2019, M. Amini, G. Elliott and N. Golestani proved that existence of topological factoring between two Cantor minimal systems is equivalent to the existence of some /morphism/ between their Bratteli-Vershik realizations. In this talk, I will talk about some applications of this equivalence in the study of Cantor minimal systems and by time permission will discuss whether existence of such morphism will make implications toward existence of factoring between two systems from a larger family of Cantor systems. The talk is mostly based on my previous and a current joint work with N. Golestani.

• Steven Flynn (Bath): Inhomogeneous pseudodifferential calculus with noncommutative symbols
  I will summarize various developments of inhomogeneous pseudodifferential calcului adapted to the study of hypoelliptic operators. This includes the constructions by Van Erp/Yuncken and Choi/Ponge of the tangent groupoid for filtered manifolds. I will then explain how operator-valued symbols arise naturally, and how such symbols are used to characterize the asymptotic behavior of high energy solutions to the Schrödinger equation with a hypoelliptic Hamiltonian.

• Francesca Arici (Universiteit Leiden): SU(2)-symmetries of C*-algebras, subproduct systems, and K-theory

This talk is motivated by multivariate operator theory and by the study of symmetries in the context of C*-algebras. We shall consider SU(2)-equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through K(K)- theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an SU(2)-representation is equivariantly KK-equivalent to the algebra of complex numbers so that the (K)K- theory groups of the Cuntz–Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class. Finally, we will discuss why and how C*-algebras in this class satisfy Poincaré duality.

Based on joint work with Jens Kaad (SDU Ondense), Yufan Ge (Leiden), and Dimitris Gerontogiannis (Leiden).

• Christian Bönicke (Newcastle University): Homology and K-theory of totally disconnected dynamical systems

Groupoids form a convenient framework to study various kinds of (topological) dynamical systems. They also play an increasingly important role in the abstract theory of C*-algebras and their classification.  This talk will focus on homological and K-theoretical invariants of groupoids. In particular, I will give an overview on some recent progress on a conjecture posed by Matui that relates the K-theory of a groupoid C*-algebra to the homology of the underlying groupoid itself.

• Samantha Pilgrim (University of Glasgow): Crossed Product C*-algebras and Finite Approximations of Topological Dynamical Systems

We will discuss how finite dimensional approximations of crossed product C^*-algebras can arise from approximations of topological dynamical systems.  In particular, we will give an exposition of residual finiteness and quasidiagonality of group actions in the sense of Kerr and Nowak; and present theorems on finite approximations of isometric actions.  As corollaries, we obtain economical proofs of quasidiagonality for crossed products of isometric actions by amenable groups as well as many new examples of crossed products by non-amenable groups with the MF property.  Interactions with semiprojectivity properties of C*-algebras may also be discussed, time permitting.  

• Ian Charlesworth (Cardiff University): Regularity in Free Probability and Free Stein Dimension

The study of regularity in free probability boils down to the question of how much information about a *-algebra can be gleaned from probabilistic properties of its generators. Some of the first results in this theme come from the theory of Voiculescu's free entropy: generators satisfying certain entropic assumptions generate von Neumann algebras which are non-Gamma, or prime, or do not admit Cartan subalgebras. Free Stein dimension -- a quantity I introduced with Nelson -- is a more recent quantity in a similar vein, which is robust under polynomial transformations and not trivial for variables which do not embeddable in R^\omega.

After giving a brief introduction to free probability and free entropy, I will speak on some recent improvements in related to free Stein dimension. We are now able to compute the free Stein dimension of direct sums, and of tensor products with finite dimensional algebras, which allow us to compute it in a large number of new examples. We are also able to give bounds on Stein dimension based on the presence of algebraic relations among generators. This is joint work with Brent Nelson.

• Kevin Boucher (University of Southampton): Uniformly bounded representations of hyperbolic groups

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.

This is joint work with Jan Spakula.