Past seminars

Title: One-cross bundles  

Abstract: We present a framework for constructing a noncommutative geometry on a quantum homogeneous space. The resulting differential calculus—or quantum de Rham complex—exhibits a remarkably rich structure, encompassing a noncommutative Riemannian metric, a Levi-Civita connection, a complex structure, and associated holomorphic structures. As motivating examples, we discuss the irreducible quantum flag manifolds, and, time permitting, explore the analytic properties of their corresponding Dolbeault–Dirac operators.

Title: A finite-dimensional approach to K-homology

Abstract: Let A be a suitably nice C*-algebra (for example, quasidiagonal with countable K-homology).  I will explain how the K-homology of A can be realized by almost multiplicative ucp maps to matrices.  I’ll sketch some applications to K-homology of spaces, and to approximate representation theory of groups.

Title: Noncommutative Choquet theory

Abstract: In this talk, I will introduce noncommutative convexity and present a brief overview of noncommutative Choquet theory, which provides a structural foundation for the study of noncommutative convex sets. Since the category of compact noncommutative convex sets is dual to the category of operator systems, which are unital self-adjoint subspaces of operators, noncommutative Choquet theory can potentially shed new light on problems of an operator-algebraic nature. I will discuss two recent examples of this. The first relates to the notion of self testing in quantum information theory, and the second provides a new characterization of operator systems satisfying an important approximation-theoretic property called hyperrigidity. This talk will feature joint work with Ken Davidson and Eli Shamovich.

Title: Spectral localizers in KK-theory

Abstract: In this talk we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.

Title: Computing Quantum Automorphism Groups of Graphs

Abstract: The quantum automorphism group of a graph is a natural generalization of the notion of automorphism group of a graph in the framework of compact quantum groups. What makes these objects particularly interesting is their interplay with the Graph Isomorphism nonlocal game from quantum information theory. However, explicit computation of quantum automorphism groups of graphs is quite difficult and few general results were known a few years ago. In this talk, we shall explore algebraic and combinatorial tools which enable us to compute quantum automorphism groups of larger graphs from those of simpler ones. In particular, we give explicit algorithms for computing quantum automorphism groups of large families of graphs such as forests, outerplanar graphs, and block graphs, and study quantum automorphism groups of lexicographic products of graphs.

Title: Stable rank one in nonnuclear crossed products

Abstract: I’ll give an overview of stable rank one and the C*-algebras known to have it, then describe a new approach to studying stable rank one in possibly nonnuclear crossed products. As an application, we show that stable rank one is generic for natural classes of minimal actions of free groups on the Cantor set. This is joint work with Shirly Geffen and David Kerr.

Title: The unitary group of a II_1 factor is SOT-contractible

Abstract:  I show that the unitary group of any SOT-separable II_1 factor M, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type I_n direct summands (n < infinity). The proof for the II_1-factor case uses regularization via free convolution and Popa's theorem on the existence of approximately free Haar unitaries in II_1 factors.  I will also explain some of the bigger picture of the free probability ingredients.

More information at https://sites.google.com/view/scaoaw/overview 

Title: The (Local) Lifting Problem for C*-Algebras

Abstract: A C*-algebra is said to have the Lifting Property (LP) if every completely positive map into a quotient C*-algebra has a completely positive lift. Through the years, the LP and its local version, the LLP, have been getting a lot of traction both because of their usefulness, but also their relevance in connecting various major problems such as Kirchberg's QWEP-conjecture, Connes' Embedding Problem and Tsirelson's Problem. I will introduce these properties and study the problem of finding groups whose full group C*-algebras fail the LLP. I will present a class of examples due to Ioana, Spaas and Wiersma which in particular include SL3(Z).

The seminar is based on my master's thesis: “Lifting Properties of Group C*-Algebras and Operator Spaces.”

Title: Noncommutative coloured entropy

Abstract: Building on the classical noncommutative entropy for automorphisms of nuclear C*-algebras, I will introduce a formally different notion of entropy which uses the more refined cpc approximations given by finite nuclear dimension or finite decomposition rank. In the sequel, I will explore the typical values of this entropy. This is joint work with Bhishan Jacelon.

Title: Classifying group actions on Kirchberg algebras

Abstract: A few years ago, Szabó and I classified all outer amenable discrete group actions on Kirchberg algebras. The classifying invariant is equivariant KK-theory, so at its core, the result is a rigidity theorem. I will talk about how one can strengthen the classification in certain cases and obtain classification with more computable K-theoretic invariants.

Title: Etale groupoids without the Felix H property

Abstract: There are many natural examples of etale groupoids arising from e.g. group actions or self-similar groups that fail to be Hausdorff, and this poses a problem for understanding their (C*)-algebras. In particular, there is a singular ideal and it is not clear when this vanishes, or when the reduced groupoid (C*)-algebra is simple. I will discuss recent joint work with Julian Gonzales, Jeremy Hume, and Xin Li, where we make progress on this question, especially in the algebraic case.

Title: Quantum metrics from quantum groups

Abstract: In Rieffel’s theory of compact quantum metric spaces, it is often quite difficult to verify if a given candidate is indeed a quantum metric. In my talk, I will describe a framework which allows one to reduce the problem to a subalgebra and show how this can be applied to obtain new quantum metrics from length functions on quantum groups, and to recover existing results for q-deformations.

Title: Generalized *-homomorphisms and KK with extra structure

Abstract: In the 1980's, Cuntz developed a new approach to KK-theory by considering *-homomorphisms out of a certain universal algebra qA. For purposes of (in particular) classification of nuclear C*-algebras, one often considers different variations of KK-theory, such as Skandalis' nuclear KK-theory, Kirchberg's ideal-related KK-theory, or equivariant KK-theory. I will explain how this additional structure in KK-theory can be encoded in the qA-formalism of KK-theory. This is joint work with Joachim Cuntz.

Title: The nuclear dimension of C*-algebras of groupoids, with applications to C*-algebras of directed graphs

Abstract: Guentner, Willett and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra. To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite. I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.

This is joint work with Dana Williams.

Title: Property (SI) and approximations using pure states 

Abstract: Property (SI) for C*-algebras was firstly introduced by Matui and Sato in 2012, where they made a breakthrough on the remaining implication of the Toms-Winter conjecture. The property has since then become an important tool to access Z-stability from strict comparison. It was generalized to maps, where a large class of maps are shown to have the property, thus ob-taining a unital copy of the Jiang-Su algebra in the relative commutant. Moreover, an equivariant version of property (SI) is defined by Szabo to show Z-stability for crossed products. 

One of the key steps in these results is to obtain a refined approximation for nuclear C*-algebras using pure states. In the simple case, the approximation is obtained by applying the Voiculescu’s theorem. For non-simple C*-algebras, the approximation was only established under extra tech-nical conditions on the representation theory. We show that this refined approximation property holds for all nuclear C*-algebras. As a consequence, we can remove the technical condition and obtain property (SI) results for much more general maps.

Title: Pure C*-algebras

Abstract: Pureness is a regularity property for C*-algebras that was introduced by Winter in his seminal investigation into Z-stability and finite nuclear dimension of simple, nuclear C*-algebras. We show that a separable C*-algebra is Z-stable if and only if its central sequence algebra is pure, which justifies to view pureness as a non-central version of Z-stability.

We demonstrate that pure C*-algebras form a robust class by proving that pureness follows from very weak comparison and divisibility properties. Using this, we verify pureness for several classes of C*-algebras. In particular, every C*-algebra with the Global Glimm Property (i.e. sufficiently non-elementary) and with finite nuclear dimension is pure, which leads to the verification of the non-simple Toms-Winter conjecture for a large class of C*-algebras.

Title: Simplicity of groupoid algebras associated to Z_2-multispinal groups

Abstract: I will present an approach to showing simplicity of groupoid *-algebras associated to multispinal groups over Z_2 extending the method by Clark-Exel-Pardo-Starling-Sims for the case of the Grigorchuk group (the first example inspiring the constructions of multispinal groups). Along the way we come across a class of full-rank matrices of a kind appearing in the theory of 2-designs. This is joint work with Carla Farsi, Judith Packer and Nathaniel Thiem at UC Boulder, Colorado, USA.

Title: Noncommutative metric geometry of quantum spheres. 

Abstract: In this talk we investigate the noncommutative metric geometry of the higher Vaksman-Soibelman quantum spheres. More precisely, we shall see how to endow a given quantum sphere with the structure of a compact quantum metric space by means of a seminorm arising from noncommutative differential geometric data. We view our quantum sphere as a noncommutative circle bundle over the corresponding quantum projective space. Using techniques from unbounded KK-theory this point of view allows us to construct vertical and horizontal differential geometric data on the quantum sphere in question. The vertical data comes from the generator of the circle action and the horizontal data comes from the unital spectral triple on quantum projective space introduced by D’Andrea and Dabrowski. An interesting feature of our setting is that the horizontal geo-metric data yields a twisted derivation on the coordinate algebra whereas the vertical geometric data produces a derivation in the usual sense. Nonetheless we can assemble these two (twisted) derivations into a single seminorm on our quantum sphere and show that the corresponding metric on the state space metrizes the weak*-topology. 

Title: Kazhdan-type rigidity for groups and algebras acting on Banach spaces

Abstract: Property (TE) of Bader, Furman, Gelander, and Monod is a Kazhdan-type rigidity property concerning how a group may act on Banach spaces belonging to a class, E. In this talk, we explore different generalizations of this property to C*-algebras and Banach algebras and discuss how Property (TE) for a discrete or locally compact group is related to Property (TE) for its associated pseudofunction algebras. We are specifically interested in the case where E is the class of Lp-spaces. This talk is based on joint work with Sanaz Pooya.

Title: Constructing a C*-diagonal in the Jiang-Su algebra

Abstract: Cartan subalgebras and C*-diagonals in C*-algebras (motivated by Cartans in von Neumann algebras) build a bridge between C*-algebras and topological dynamics, which led to striking results regarding the existence of Cartans/C*-diagonals in classifiable C*-algebras. In particular, this connection was used to construct examples of C*-diagonals in the Jiang-Su algebra Z. In this talk, I will construct a new C*-diagonal in Z using solely C*-algebraic methods. This is based on joint work with Wilhelm Winter.

Title: Uniform property Gamma for diagonal pairs

Abstract: Uniform property Gamma is an adaptation of the well-known notion of property Gamma, from tra-cial von Neumann algebras, to the framework of tracial C*-algebras. In this talk, after giving an overview on uniform property Gamma and on its role in the Toms-Winter conjecture, I will introduce an adaptation of this concept to the context of diagonal pairs, and discuss a tight connection that this property shares with the no-tion of small boundary property and mean dimension zero for topological dynamical systems.

Title: A unified approach for classifying simple nuclear C*-algebras

Abstract: The classification program of C*-algebras aims to classify simple, separable, nuclear C*-algebras by their K-theory and traces, inspired by analogous results obtained for von Neumann algebras. A landmark result in this project was obtained in 2015, building upon the work of numerous researchers over the past 20 years. More recently, Carrión, Gabe, Schafhauser, Tikuisis, and White developed a new, more abstract approach to classification, which connects more explicitly to the von Neumann algebraic classification results. In their paper, they carry out this approach in the stably finite setting, while for the purely infinite case, they refer to the original result obtained by Kirchberg and Phillips. In this talk, I provide an overview of how the new approach can be adapted to classify purely infinite C*-algebras, recovering the Kirchberg-Phillips classification by K-theory and obtaining Kirchberg's absorption theorems as corollaries of classification rather than (pivotal) ingredients. This is joint work with Jamie Gabe.

Title: The noncommutative Choquet boundary and approximation theory

Abstract: In recent years, the study of operator systems and non self-adjoint operator algebras have flourished due to the successful realization of Arveson's noncommutative counterpart to the Choquet boundary. Originally introduced in the 1960s, this invariant serves as a building block for the C*-envelope, noncommutative convexity theory, as well as being a tool for encoding the C*-algebraic information that is intrinsic to an operator space. One of the most well-studied questions in this program concerned the noncommutative Choquet boundary and its connections to approximation theory. In this talk, I will discuss the recent advancements in this direction. This is based on joint work with Raphaël Clouâtre.

Title: Symbolic dynamics through the lens of C*-algebras

Abstract: In symbolic dynamics, Subshifts of Finite Type (SFTs) are often used as discretized models for various dynamical systems. Two-sided SFTs are bi-infinite paths of a directed graph together with a natural bilateral left shift on them, making them particularly amenable to study via combinatorial and matrix-theoretical techniques.

Despite their apparent simplicity, we still do not know whether the conjugacy problem for SFTs is decidable, and conjugacy for very simple examples still remains mysterious. In work of Williams from 1973, conjugacy of SFTs was shown to have an equivalent matrix-theoretic formulation in terms of adjacency matrices, and was conjectured to coincide with eventual conjugacy. This led to the discovery of various invariants that distinguish SFTs up to conjugacy, and eventually to a counterexample to Williams conjecture in 1999 by Kim and Roush.

Together with early attacks on Williams conjecture, Cuntz and Krieger found a construction of C*-algebras associated to SFTs that recover several standard invariants of SFTs, and has led to the discovery of new invariants. This makes several classification problems for Cuntz-Krieger C*-algebras particularly relevant to the discovery of new (and computable) obstructions to conjugacy of SFTs. 

In this talk, I will showcase some old and new invariants of SFTs, the relationship between them, how to associate algebras to SFTs, and how to recover some of these invariants from the algebras. I will explain how we answered a question of Eilers, showing that eventual conjugacy of SFTs coincides with stable graded homotopy equivalence of the associated C*-algebras. Our proof relies on bimodule theory for C*-algebras, as well as on a new (and surprisingly necessary) bicategorical approach for bimodules initiated by Ralf Meyer and his students.

*Based on joint work with Boris Bilich and Efren Ruiz.

Title: Generic Central Sequence Algebras in II_1 Factors

Abstract: When does a tracial von Neumann algebra admit factorial relative commutant in its own ultrapower? Is there a special class of algebras with a distinguished object M so that eve-ry algebra of the class admits an embedding into an ultrapower of M with factorial relative commutant? We consider these questions and give some partial answers via a central se-quence property we call "uniformly super McDuff". This is based on a joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.

Title: Strict comparison of the reduced C*-algebra of the free group

Title: K-theory invariance of $L^p$-operator algebras associated with étale groupoids of strong subexponential growth

Abstract: Given a group or a groupoid, we can associate to it a reduced $L^p$-operator algebra. It is a natural problem to investigate to which extent the structure of these algebras differ for various exponents p. We show that for étale groupoids which have strong subexponential growth with respect to a locally bounded length function, the K-theory of the associated reduced $L^p$-operator algebras is independent of $p \in [1, \infty)$. 

This is joint work with Eduard Ortega and Mathias Palmstrøm.

Title: Quantum metrics from length functions on quantum groups

Abstract: I will report on an ongoing (and far from completed) joint project with Are Austad, investigating the metric information contained in spectral triples arising naturally within a quantum group context.  

Title: Dynamical Covers

Abstract: A method to understand a dynamical system is to study simpler and better behaved systems that factor onto it, which we will call "covers". For instance, this method can be successfully employed to prove the existence of unique measures of maximal entropy and provide combinatorial classifications for symbolic dynamical systems known as sofic sub-shifts. 

The talk is based on the pre-print https://arxiv.org/abs/2408.11917 (joint work with Kevin Brix and Xin Li), where we construct covers that generalize the ones in symbolic dynamics responsible for the two mentioned successes. We will discuss the functorial and universal properties of the constructions which make the covers computable in practice. We will also describe some relationships between properties of the original systems and properties of their covers, and provide applications to C*-algebras of semi-etale groupoids.

Title: Homology and K-theory for self-similar group actions

Abstract: Self-similar groups are groups of automorphisms of infinite rooted trees obeying a simple but powerful rule. Under this rule, groups with exotic properties can be generated from very basic starting data, most famously the Grigorchuk group which was the first example of a group with intermediate growth.

Nekrashevych introduced a groupoid and a C*-algebra for a self-similar group action on a tree as models for some underlying noncommutative space for the system. Our goal is to compute the K-theory of the C*-algebra and the homology of the groupoid. Our main theorem provides long exact sequences which reduce the problems to group theory. I will demonstrate how to apply this theorem to fully compute homology and K-theory through the example of the Grigorchuk group.

This is joint work with Benjamin Steinberg.

Title: Building Completely Positive Inductive Sequences

Abstract: Generalized inductive sequence constructions promise a new perspective for viewing and building (nuclear) C*-algebras. In this construction, one replaces the *-homomorphisms from classic inductive sequences with completely positive contractive (cpc) maps, which somehow still yield a C*-algebra in the limit. To guarantee a C*-structure on the limit, one must place conditions on the cpc maps in these systems: asymptotic multiplicativity, asymptotic orthogonality preserving, or C*-encoding. The first two conditions, though structurally interesting, are often too stringent to allow for many natural and/or constructive examples. On the other hand, the C*-encoding criteria is easier (and in fact necessary) to satisfy, opening up the possibility of relatively hands-on inductive limit constructions of broad classes of nuclear C*-algebras. 

To demonstrate this, I will describe how one can use Følner(-like) sequences to build inductive limit constructions of C*-algebras arising from amenable group (actions). 

Title: Classifiability of crossed products

Abstract: To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C* algebra in terms of the dynamical system. In this talk, we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. 

Parts of this talk are joint works with Geffen, Kranz, and Naryshkin, and with Geffen, Gesing, Kopsacheilis, and Naryshkin.

Title: Is the index of sub-Laplacians interesting?

Abstract: A remarkable index theorem of Baum-van Erp from last decade computes the index of sub-Laplacians on contact manifolds. Little is known beyond contact manifolds, even to what extent the index can be nontrivial. We discuss how to rephrase the problem in terms of representations of nilpotent Lie groups and show that the struggle is real to find a nonzero index. Based on joint work with Bernard Helffer. 

Title: Insplitting for textile systems and 2-graphs

Abstract: Introduced by Nasu, textile systems use two directed graphs and homomorphisms between them to encode a 2-dimensional shift of finite type. Johnson and Madden observed in 1996 that any 2-dimensional SFT is conjugate to one arising from a textile system.  Moreover, they proved that every conjugacy of textile systems arises from an insplitting, outsplitting, or inversion.

Many 2-dimensional shifts of finite type can also be described via a higher-rank graph of rank 2, and in the context of higher-rank graphs, insplitting was introduced by Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales, and Pask.  Unfortunately, the Johnson-Madden definition of insplitting is not compatible with the EFGGGP definition: the result of Johnson-Madden insplitting never yields a 2-graph-compatible textile system.

However, our joint work with S. Brooker, P. Ganesan, Y.-F. Lin, D. Pask, and J. Plavnik has uncovered a description of 2-graph insplitting (at the level of the infinite path space, or the SFT) in terms of Johnson-Madden insplits and amalgamations.  As a corollary, we conclude that 2-graph insplitting yields a conjugacy of the associated SFT.  (This is consistent with the known result [EFGGGP] that 2-graph insplitting yields an isomorphism of C*-algebras.)

Title: Nonlocal games with quantum inputs and outputs

Abstract: In this talk, I will present an operator algebraic generalization of nonlocal games and quantum correlation classes to a setting where the inputs/outputs are allowed to be quantum states. This framework will then be used to discuss different types of homomorphisms for a quantum graph.

Title. Asymmetric graphs with quantum symmetry

Abstract. Quantum isomorphisms and automorphisms of graphs form a bridge between noncommutative geometry (NCG) and quantum information theory (QIT), as they can be motivated from the point of view of both quantum groups and nonlocal games. This allows us to use techniques from QIT in NCG and vice versa. In this talk, I will explain this connection and use it to prove a surprising result in NCG. Using a construction similar to the Mermin–Peres magic square from QIT, we construct graphs with trivial automorphism group and non-trivial quantum automorphism group, which shows that quantum symmetry can even occur in completely asymmetric classical spaces. To our knowledge, these are the first known examples of any kind of commutative spaces in NCG having this property.

This talk is based on joint work with David E. Roberson (DTU) and Simon Schmidt (Ruhr University Bochum; formerly KU).

Title: The Haagerup property for Thompson-like groups

Abstract: A decade ago, Vaughan Jones developed a powerful technology for constructing representations of discrete groups. Using this technology, Brothier and Jones gave a simple and analytical proof that Richard Thompson’s groups F and T have the Haagerup property (first proved geometrically by Farley).

We show that the proof can be adapted to the larger Thompson’s group V, and discuss how our techniques lead to analytical proofs that other Thompson-like groups have the Haagerup property.

This is joint work with Dilshan Wijesena.

Title: Ideal structure of group C*-algebras

Abstract: Fundamental work of Kalantar and Kennedy (and later with Breuillard and Ozawa) disclosed precisely when a discrete group is C*-simple (i.e. the reduced group C*-algebra is simple), and one important observation is the connection with the Furstenberg boundary. I will discuss this result as well as the more general problem of clarifying the ideal structure of group C*-algebras to which I have made some recent contributions with Chris Bruce, Kang Li, and Eduardo Scarparo.

Title: Domains of quantum metrics on AF-algebras

Abstract: Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite-dimensional. Intuitively, this should allow for different quantum metrics with distinct domains when A is infinite-dimensional, and we show how to explicitly build such quantum metrics. Lastly, we also provide a strategy for the calculation of distance between certain states in these quantum metrics in the special cases of the quantized interval and the Cantor space. This is joint work with Konrad Aguilar and Frederic Latremoliere.

Title: The Levi-Civita connection on noncommutative differential forms

Abstract: I will discuss the main results and consequences of recent work with Bram Mesland. Using mostly algebraic techniques, with just a little Hilbert module analysis, we present necessary and sufficient conditions for the existence of Hermitian torsion-free connections on noncommutative  one-forms, such as those arising from spectral triples. Various consequences and examples will be discussed also.

Title: Classification of simple C*-algebras from singular graphs

Abstract: There is a rich classification theory for unital simple C*-algebras associated to finite graphs with no sinks, allowing to decide by invariants when two such C*-algebras are isomorphic, not just in their own right, but equipped with their natural diagonals (Cartan subalgebras) or their natural circle actions. All of these results were proved by involving results from symbolic dynamics, even though some now are special cases of much more general results.

When one allows singular vertices — sinks or infinite emitters — the connection to the rich theory of shifts of finite type is lost, but the classical classification results have natural generalizations which one may aspire to show by other means. I will discuss and compare different notions of sameness of such graph C*-algebras; all fully understood in the regular case, both only half resolved in general. All work presented is joint with Efren Ruiz, and some also with Aidan Sims.

Title: Pure C*-algebras and *-homomorphisms

Abstract: The notion of (m,n)-pure C*-algebras was introduced by Winter in his seminal work on separable, simple, unital C*-algebras of finite nuclear dimension. Although a lot of effort has been put on understanding (0,0)-pureness (often simply called pureness), much less is known about the apparently weaker notion of (m,n)-pureness for m,n>0. This is especially the case in the non-simple setting.

I will begin the talk by recalling the Toms-Winter conjecture and the importance of pureness in its study. I will then discuss results from two different ongoing projects: Pure *-homomorphisms and their properties (joint with J. Bosa), and (m,n)-pure C*-algebras (joint with R. Antoine, F. Perera, and H. Thiel).

Title: Dynamic asymptotic dimension and applications

Abstract: Dynamic asymptotic dimension is a dimension theory for topological dynamical systems introduced by Guentner, Willett, and Yu in 2017. Since its inception it has found numerous applications to the structure theory of C*-algebras as well as homology and K-theory. In this talk I will discuss some aspects of this dimension theory and its applications to C*-Algebras.

Title: Stein’s groups as topological full groups

Abstract: Topological full groups are a framework which allow us to understand simple groups with finiteness properties through ample groupoids. In particular, purely infinite groupoids give us interesting generalisations of Thompson’s group V. In this talk, I will describe how I have used this framework to better understand a class of groups introduced by Melanie Stein in 1992. 

Title: The dynamical Kirchberg-Phillips theorem

Abstract:  I will talk about the problem of classifying C*-dynamical systems for which the underlying C*-algebra is purely infinite and simple, such as the Cuntz algebras O_n. I will try to make the talk accessible with focus on some elementary examples. This is joint work with Gabor Szabo.

Title: A spectral gap for twisted Dolbeault–Dirac operators over irreducible quantum flag manifolds

Abstract: In this talk we present a geometric approach to understanding the analytic properties of twisted Dolbeault–Dirac operators D_{\overline{\partial}} over the irreducible quantum flag manifolds \mathcal{O}_q(G/L_S). We deduce from a noncommutative Akizuki–Nakano identity that twisting a Dolbeault–Dirac operator by a positive line bundle results in a spectral gap for these operators. We then conclude that each positively twisted D_{\overline{\partial}} is an (unbounded) Fredholm operator, and calculate its index using the recently established noncommutative Borel–Weil theorem. (Joint work with Biswarup Das and Petr Somberg)

Title: Admissible KMS bundles on classifiable C*-algebras

Abstract: Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C*-algebra admitting a continuous action of the real numbers. The collection of KMS states for a given flow on a C*-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C*-algebras.

Title: Classification of crossed product C*-algebras arising from essentially free actions

Abstract: Essentially free actions play an important role in ergodic theory and group measure space II1-factors. In this talk, I will explain why (uniform) essential freeness is a better prerequisite than topological freeness in the classification of crossed product C*-algebras, and provide a first general result on uniform property Γ and Z-stability, assuming only topological freeness, in this direction. 

Title: The primitive-ideal spaces of some groupoid C*-algebras

Abstract: The primitive-ideal space of the universal C*-algebra of a dynamical system is an important bit of structural information, but very hard to compute. For reversible, abelian dynamics it can be computed using powerful results going back to Mackey, Green and Rieffel. For rank-1 irreversible dynamics, it can be computed using recent results Katsura, building on earlier work of an Huef-Raeburn and then Hong-Szymanski, that rely on reducing key components of the dynamics to reversible subsystems. For rank two or more, such reductions are not available in general. I’ll discuss an approach using groupoids that applies to C*-algebras of large classes of actions of free abelian monoids, including all 2-graph C*-algebras. This is joint work with Kevin Brix and Toke Carlsen.

Title: C*-envelopes of semigroup operator algebras

Abstract: I will present on joint work with Kevin Aguyar Brix and Adam Dor-On in which we prove that every normal coaction of a discrete group on an operator algebra extends to a (unique) normal coaction on the C*-envelope of the operator algebra. As an application, we identify the C*-envelopes operator algebras of many (e.g., groupoid-embeddable) small categories with the boundary quotient C*-algebra of the category, answering a question posed by Xin Li. I will focus on the case where the category is a monoid, where we obtain the first results outside of the group-embeddable case.

Title: Pure largeness and the Elliott-Kucerovsky theorem

Abstract: The concept of absorbing extensions is a ubiquitous tool in Brown-Douglas-Filmore theory of extensions, with applications in (modern) C*-classification results and structure of C*-algebras. In their 2001 paper, Elliott and Kucerovsky characterize this abstract and a priori extrinsic property by an intrinsic condition which they call being "purely large". During this talk, we will introduce both concepts, explore them in greater detail, and highlight some of the main ideas of the proof. If time permits, we will also briefly discuss its connection with modern C*-classification theory, in the framework of separably inheritable properties.

Title: Quantum metric Choquet simplices

Abstract: I will describe some work in progress on a special class of compact quantum metric spaces. Abstractly, the additional requirement on the underlying order unit spaces is the Riesz interpolation property. In practice, I am interested in Lipschitz structures associated to metrics on the *trace* spaces of separable, unital C*-algebras. This added structure is designed for witnessing metric features of morphisms into the category of classifiable C*-algebras. I will give some examples and explain how to build (continuous fields of) new ones by forming 'quantum crossed products' associated to dynamical systems on compact, connected metric spaces.

Title: NC Convexity and Operator System Duality

Abstract: Operator systems are linear spaces of bounded operators on a fixed Hilbert space that is closed under the involution operation. If the operators in this operator system commute, this is what is known in the literature as a function system. A foundational result of Kadison demonstrates that function systems are categorically dual to the category of compact convex sets. Here the function system can be thought of as the scalar valued affine functions over this convex set. We may therefore think of operator systems in general as a kind of non-commutative function system.

NC convex sets were introduced first in the matricial case by Webster--Winkler in 2008, and in a more general unital setting by Davidson--Kennedy in 2019, and finally in general by myself, Matt Kennedy, and Nicholas Manor. These nc convex sets are geometric objects closed under a certain matricial analogue of convex combinations. Surprisingly, Kadison's orginal duality theorem goes through completely in this far more general setting, allowing us use intuition that we have for convex sets in this new category. In this talk we will explore this duality in more detail, and explain how one may use commutative intuition to derive facts about operator systems using this nc convex algebraic geometry. 

Title: Ideals in groupoid C*-algebras and their isotropy fibers 

Abstract: In this talk I will report on a joint project with Sergey Neshveyev where we investigate how any ideal in a locally compact étale groupoid C*-algebra defines a family of ideals in the group C*-algebras of isotropy groups and to which extent these families determine the ideal. I will illustrate how this question is connected to the study of certain (non) exotic norms on the isotropy groups of the groupoid. As an application I will among other things describe the maximal ideals of any étale groupoid C*-algebra and classify the primitive ideals of a class of graded groupoids. 

Title: Embezzlement and the classification of type III factors

Abstract: In quantum information science, embezzlement refers to the counterintuitive possibility of extracting arbitrary entangled states from a reference state (the "embezzler") via local unitary operations while hardly perturbing the latter. Basic conservation laws imply that an embezzler has infinite entanglement.

To study this phenomenon, we consider a pair (M, M’) of a von Neumann algebra and its commutant in standard form as the bipartite system and study how well a given pure state, i.e., a unit vector Ω in the positive cone, performs at the task of embezzling arbitrary entangled states.

We find a connection to the flow of weights on M: Ω is good at embezzling if and only if the dual state ω̂ is an approximate fixed point of the flow of weights. In particular, embezzling states correspond to fixed points of the flow of weights, and the type III_1 factor can be uniquely characterized by the property that all states are embezzling. For type III_λ factors, 0<λ<=1, the value of λ can be recovered from the worst-case embezzlement performance of all bipartite pure states. The λ=0 is open.

This is joint work with A. Stottmeister, H. Wilming, and R.F. Werner.

Title: Generalized inductive systems, convergence of dynamics, and renormalization

Abstract: I will discuss a generalized notion of inductive systems of Banach spaces, coined soft inductive systems, that can serve as a flexible tool to describe the limits of physical theories, e.g., in the context of the thermodynamic limit and phase transitions or renormalization of quantum field theories.

General criteria for the convergence of dynamics can be obtained in this setting, which I will illustrate using the example of renormalization of free fermion theories on a one-dimensional spatial lattice. Specifically, this example can be used to prove the conformal invariance of various correlation functions of the classical Ising model using operator algebraic methods.

This talk is based on work with L. van Luijk, T.J. Osborne, and R.F. Werner.

Title: A type III_1 factor with the smallest outer automorphism group

Abstract: Outer automorphism groups have played a crucial role as invariants in understanding II_1 factors for a long time. After Connes showed that property (T) factors have a countable outer automorphism group in 1980, it remained open for many years to determine if there is a II_1 factor with a trivial outer automorphism group. Finally, in 2004, such a II_1 factor was shown to exist by Ioana, Peterson and Popa and then explicitly constructed by Popa and Vaes in 2008. In the case of type III factors, the modular automorphisms unavoidably intersect the outer automorphism group and the natural analogous question is: can one construct a factor with the `smallest' outer automorphism group? In this talk, we shall give an explicit ergodic theoretic construction of a full type III factor M with Out(M) equal to the real line.

Title: Local Hilbert-Schmidt stability

Abstract: We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS-) stable, answering a question by Lubotzky. This is based on joint work with Francesco Fournier-Facio and Maria Gerasimova.

Title: The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this new operator is intrinsically defined, has properties analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden). 

Title: Crossed products as compact quantum metric spaces

Abstract: In 2013 Hawkins, Skalski, White and Zacharias constructed and investigated certain spectral triples on crossed product C*-algebras by actions of discrete groups which are in a natural sense equicontinuous. Following Connes, one of the ingredients of their construction are certain multiplication operators associated with length functions on the group. In their article they further formulated the question for whether their triples turn the corresponding crossed product C*-algebras into compact quantum metric spaces. The aim of this talk is to give a short introduction into Rieffel's theory of compact quantum metric spaces. By combining his ideas on horofunction boundaries of groups with results from metric geometry, I will further answer the question of Hawkins, Skalski, White and Zacharias in the affirmative in the case of virtually abelian groups, equipped with suitable length functions.

Title: Groupoid models of semigroup C*-algebras

Abstract: For a left-cancellative semigroup, $S$, one always has an associated reduced C*-algebra $C_r^*(S)$,

but defining a universal counterpart is usually challenging. One approach is to find an étale groupoid, $G(S)$,

such that $C_r^*(S)$ is isomorphic to $C_r^*(G(S))$- the reduced C*-algebra of $G(S)$. Then $G(S)$ is called a

groupoid model for $S$ and once we have it, it makes sense to define the universal C*-algebra of $S$ as the

universal C*-algebra of $G(S)$. In this talk we go through the construction of a so called reduced Paterson

groupoid associated to $S$, $G(S)$, and give rather weak conditions on $S$ which guarantees that

$C_r^*(S)\simeq C_r^*(G(S))$. We also present a groupoid model for the boundary quotient of $S$, which is a

quotient of $C_r^*(S)$. This is based on joint work with Sergey Neshveyev. 

Title: Nuclearity and Inductive Limits

Abstract: Inductive limits are a central construction in operator algebras because they allow one to use well-understood building blocks to naturally construct more complicated objects whose properties remain tractable. In the classical setting, few nuclear C*-algebras arise as inductive limits of finite-dimensional C*-algebras. However, by generalizing our notion of an inductive system, Blackadar and Kirchberg were able to characterize quasidiagonal nuclear C*-algebras as those arising as (generalized) inductive limits of finite dimensional C*-algebras. In this talk, I will describe new notions of inductive systems which allow us to realize any nuclear C*-algebra as the limit of a system of finite dimensional C*-algebras. Though seemingly abstract, these systems correspond naturally to completely positive approximations of nuclear C*-algebras. This is based in part on joint work with Wilhelm Winter.

Title: From directed graphs of groups to Kirchberg algebras 

Abstract: Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their K_1 group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts – directed graphs of groups – and describe how their algebras inherit the best properties of its parents’, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory. 

Title: Generalized dynamics yielding nuclear crossed products

Abstract: Given a (generalized) dynamical system defined on a commutative C*-algebra, one can construct a suitable notion of reduced crossed product. This crossed product, however, may fail to admit a conditional expectation onto the original C*-algebra, but it does admit a weak variant. In this talk we will define these systems, and construct their associated C*-algebras. We will then give a sufficient condition for the nuclearity of these algebras that generalizes amenability for group actions on C*-algebras and twisted étale groupoids. This talk will be based on joint work with Alcides Buss.

Title: Finite Approximations of Dynamical Systems

Abstract:  I will give an introduction to residually finite and quasidiagonal group actions first defined by Kerr and Nowak.  These are group actions which admit finite (dimensional) approximations and, as a result, have reduced crossed products which can be approximated by finite dimensional algebras whenever the reduced algebra of the acting group can be.  We will see that isometric actions on compact metric spaces have especially nice approximation properties.  Toward the end, there will be some discussion of connections to approximate representations and semiprojective C^*-algebras. 

Title: Passing Z-stability in finite index inclusions

Abstract: The question of under what conditions a crossed product of a Z-stable C*-algebra by a group action remains Z-stable has been the focus of much recent research, with many techniques developed to address this. In this talk, I will discuss actions of tensor categories on C*-algebras and how key concepts and techniques from the group action setting can be adapted to this broader context. Through these adaptations, we aim to address the question of what conditions ensure that Z-stability is preserved in inclusions of finite index. This talk is based on joint work with Samuel Evington, Corey Jones and Stuart White.

Title: Towards noncommutative dynamics

Abstract:  Starting with work of Muhly, Pask, and Tomforde fundamental notions (such as Strong Shift Equivalence) from symbolic dynamics were imported into C*-correspondences and their Cuntz-Pimsner algebras. The analogy goes deeper: C*-correspondences may be viewed as noncommutative symbolic systems and I will describe how to perform state splittings that give (all?) examples of Strong Shift Equivalences. Joint with Alex Mundey and Adam Rennie.

Title: Morphisms between étale groupoids that lift to C*-algebras

Abstract: The obvious choice of arrow between two groupoids is a homomorphism (or functor). This works well enough until one wants to induce morphisms of groupoid C*-algebras. The problem that arises is that homomorphisms of groupoids would be covariant on the source and range fibres of the groupoid, but contravariant on the unit space, so a *-homomorphism of the C*-algebras would have to face both directions.

Buneci and Stachura introduce a different morphism of groupoids by considering actions of groupoids on other groupoids. They show that such morphisms covariantly induce *-homomorphisms of the groupoid C*-algebras. Meyer and Zhu call these morphisms 'actors' and build a categorical framework around the category of groupoids with actors as morphisms.

In this talk, I will define actors for étale (twists over) étale groupoids and show that any *-homomorphism between Cartan pairs which entwines all of the Cartan structure must come from an actor between the underlying groupoids. In this way an equivalence of categories is realised.

Title: Tracially amenable actions and purely infinite crossed products

Abstract: Amenable group actions on C*-algebras are actions that behave like actions of amenable groups. I will review the theory of amenable actions and report on applications to the classification of C*-algebras. I will also discuss how the search for new examples resulted in the concept of tracially amenable actions and how these give rise to new examples of purely infinite crossed products. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

Title: Equivariant Jiang-Su stability and equivariant property Gamma

Abstract: Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain what this property is and how it arises naturally in the context of the classification of C*-algebras and their actions. By work of Matui-Sato, it turns out that equivariant Jiang-Su stability can be captured in terms of uniform tracial data. In the rest of the talk I will explain how one can use this to establish equivariant Jiang-Su stability for certain actions, and how this leads to the recent result of Gábor Szabó and myself showing that equivariant Jiang-Su stability is equivalent to equivariant property Gamma under certain conditions.

Title: Simple AF embeddability for unimodular group C*-algebras

Abstract: For any locally compact group G, the left regular (unitary) representation generates a C*-algebra of bounded operators on the Hilbert space L^2(G). J. Rosenberg proved in the 80's that a discrete group G is amenable provided its induced C*-algebra forms a quasidiagonal set of operators on L^2(G), and he conjectured that the converse also holds. The conjecture was confirmed in 2015 by Tikuisis, White, and Winter, and using methods of Ozawa, Rørdam, and Sato for elementary amenable groups, they showed the stronger result that such group C*-algebras embed into a simple approximately finite-dimensional (AF) C*-algebra. I will report on some developements on how to extend this result to unimodular groups.

Title: A groupoid approach to sampling and interpolation in unimodular groups

Abstract: In this talk I will show how groupoids associated to point sets in unimodular locally compact groups and their von Neumann algebras can be used to approach sampling and interpolation problems in harmonic analysis. The talk is based on joint work with Sven Raum.

Title:  K-theory of noncommutative Bernoulli shifts

Abstract: For any discrete group G, for any G-set Z and for any unital C*-algebra A, the Bernoulli shifts G-action on the (minimal) tensor product of A over Z can be defined naturally. In a joint work with Sayan Chakraborty, Siegfried Echterhoff and Julian Kranz, we developed a new technique in operator K-theory to study the K-theory of the reduced crossed product of the Bernoulli shifts G-actions, assuming the Baum-Connes conjecture with coefficients for G. In this talk, I aim to describe this technique and some of our main results. I also plan to briefly describe the analogous results for algebraic K-theory. The latter is a joint work in progress with Julian Kranz.

Title: Finitely generated simple groups of homeomorphisms of R

Abstract: I will give an introduction to Thompson's groups F and T and then build and discuss three families of examples of finitely generated simple groups of homeomorphisms of the real line. This represents joint work with Yash Lodha and Cristobal Rivas.

Title:  The bundle of KMS states for flows on simple C*-algebras.   

Abstract: I will describe what is presently known about the structure of the collection of KMS states for flows on a simple separable unital C*-algebra, and I will try to sketch both the history behind and the proofs of the latest results. 

Title: Equivariant isomorphisms of quantum lens spaces

Abstract: In the study of noncommutative geometry many classical spaces have been given a quantum analogue. An example is quantum lens spaces, which are defined as fixed point algebras of the quantum sphere by Vaksman and Soibelman under the actions of finite cyclic groups. Quantum lens spaces have been given a graph $C^*$-algebraic description which has made it possible to work on classifying them.

Every quantum lens space comes with a natural circle action, leading to an equivariant isomorphisms problem. In this talk, I will present some recent work on the existence and construction of equivariant isomorphisms of low dimensional quantum lens spaces. Contrary to the isomorphism problem, we can no longer use the graph $C^*$-algebraic description to solve the equivariant isomorphism problem. This is due to the fact that full corners of graph $C^*$-algebras are not always equivariantly isomorphic to graph $C^*$-algebras themselves.

This is based on joint work with Søren Eilers.

Title: Homology of ample groupoids

Abstract: I will give a brief overview of the history of ample groupoid homology as used in operator algebras. I’ll introduce ample groupoid homology with the module approach, and sketch a cute proof that the homology is preserved by Morita equivalences of groupoids.

Title: A comparison of two quantum distances

Abstract:  Rieffel’s theory of compact quantum metric spaces comes equipped with the so-called quantum Gromov-Hausdorff distance, which allows one to measure the distance between two compact quantum metric spaces. In fact, there exist several competing versions of the quantum Gromov-Hausdorff distance, whose internal relationships are reasonably well understood, but in a recent work by van Suijlekom, a new candidate entered the scene.  Although this version is very closely related to Rieffel’s original definition,  it turns out that that the two are indeed different and, in my talk, I will explain a proof of this fact.  The talk is based on a joint paper with Jens Kaad. 

Title: An index pairing in scattering theory

Abstract: In this talk I will begin by introducing some concepts from mathematical scattering theory, including the wave and scattering operators and Levinson's theorem. I will review some recent work of Kellendonk and Richard, relating scattering theory to index theory and discuss how we can interpret Levinson's Theorem as an index pairing between the class of the scattering operator and an appropriate spectral triple.