Titles and abstracts

Réamonn Ó Buachalla (Charles University) - The Noncommutativity Geometry of the Quantum Flag Manifolds: An Overview

The Podleś sphere is a prototypical examples of a noncommutativebmanifold. It is a fundamental object of study in Beggs and Majid's noncommutative Riemannian geometry, Connes' spectral triples, and Rieffel's compact quantum metric spaces. The Podleś sphere is the simplest example of a fascinating class of quantum homogeneous spaces known as the quantum flag manifolds. For the last two decades this class of quantum homogeneous spaces has been the focus of intense study, as the noncommutative geometry community has tried to extend our understanding of the Podleś sphere to this general class of examples. In this talk I will give an overview of our current understanding of the noncommutative geometry of the quantum flag manifolds, emphasising how this rich class of examples serves as a point of contact between the various approaches to noncommutative geometry. In addition to the programs of Beggs--Majid, Connes, and Rieffel, we will discuss connections to noncommutative projective algebraic geometry, Nichols algebras, and Lusztig's quantum root vectors.

Shahn Majid (Queen Mary, University of London) - Quantum geodesics on graphs 

Quantum geodesics are a coordinate invariant tool to study the properties of a quantum Riemannian geometry. The latter is a framework for noncommutative geometry starting from a quantum metric in the tensor square of a bimodule of 1-forms over the algebra. Associated to such data and a quantum Levi-Civita (or other) connection, quantum geodesics are solutions of a certain flow equation which classically would be an evolution equation for a wave function whose modulus square behaves like a fluid density for particles moving on geodesics. I will describe the quantum geometry and solutions of the quantum geodesic flow equations on a small graph taken from recent work with Beggs.  If time, I will mention some applications of quantum Riemannian geometry to physics and potentially to quantum computing.

Joakim Arnlind (Linköping University) - Noncommutative Riemannian geometry of Kronecker algebras

A fundamental result in Riemannian geometry states that there exists a unique torsion free connection that is compatible with the metric. Over the last decade, the corresponding question in noncommutative geometry has been studied from different points of view. In this talk I will approach this question in a derivation based noncommutative calculus for a finite dimensional algebra arising as the path algebra of a generalized Kronecker quiver.

Sugato Mukhopadhyay (SISSA) - TBA

TBA

Marco Matassa (Oslo Metropolitan University) - Equivariant quantizations of the positive nilradical and covariant differential calculi

We consider the problem of quantizing the positive nilradical of a complex semisimple Lie algebra of finite rank, together with a certain fixed direct sum decomposition. The decompositions we consider are in one-to-one correspondence with total orders on the simple roots, and exhibit the nilradical as a direct sum of graded modules for appropriate Levi factors. We show that this situation can be quantized equivariantly as a finite-dimensional subspace within the positive part of the corresponding quantized enveloping algebra. Furthermore, we show that such subspaces give rise to left coideals, with the possible exception of components corresponding to some exceptional Lie algebras, and this property singles them out uniquely. Finally, we discuss how to use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.

Suvrajit Bhattacharjee (University of Oslo) - Braided quantum groups and their actions

Hopf algebra objects in braided monoidal categories or braided Hopf algebras, as they are called today following Majid, were introduced and studied in various physical contexts in the early nineties. Although quantum groups in the analytic setting emerged more or less at the same time, their braided analogue has only been recently introduced. Allowing the deformation parameter q in SUq(2) to be any nonzero complex number, one obtains the first example of a braided compact quantum group constructed by Kasprzak-Meyer-Roy-Woronowicz, which is

also shown to appear naturally as a Cuntz-Pimsner algebra by Habbestad-Neshveyev. We, in a series of previous works, constructed other examples of braided compact quantum groups and presented them as symmetry objects of various (C*-)algebraic structures. In this talk, I will report on some further developments along this direction, based on ongoing work with Sergey Neshveyev.

Bram Mesland (Leiden University) - TBA
TBA

Mike Whittaker (University of Glasgow) - Limit spaces of Katsura groupoids

Katsura proved that every Kirchberg algebra arises, up to strong Morita equivalence, from a C*-algebra associated to two integer matrices. Exel and Pardo realised that Katsura's construction gives rise to a self-similar group(oid) action on the path space of a graph, and that Katsura's C*-algebra is the associated Cuntz-Pimsner algebra of the self-similar action. In this talk, we introduce Katsura groupoids and construct their limit spaces, in the sense of Nekrashevych. Under mild conditions, these are circle bundles fibred over a totally disconnected space and we prove these embed in the plane. We also show how this is related to a recent construction of Putnam, and use the limit space viewpoint to answer a question he asked. This is joint work with Jeremy Hume.

Jeremy Hume (University of Glasgow) - Contracting C*-correspondences

Given a C*-correspondence from a C*-algebra to itself, when does the associated Cuntz-Pimsner algebra have a unique KMS state? We consider a condition, inspired by Nekrashevych's self-similar group theory and Ruelle's thermodynamic formalism, that guarantees this. We show a variety of correspondences satisfy this property and discuss some work in progress generalizing the thermodynamic formalism to C*-correspondences.

Tatiana Shulman (Chalmers University of Technology) - On residually finite-dimensional C*-algebras in dynamical context

A C*-algebra is residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. The property of a C*-algebra of being RFD is central in C*-algebra theory and has connections with other important notions and problems. The topic of this talk will be the RFD property in dynamical context, namely we will discuss the RFD property of crossed products by amenable action and, if time permits,  of C*-algebras of amenable etale groupoids. We will present consequences of our results to residual properties of groups and to approximations of representations in spirit of Exel and Loring, and we will discuss examples. Joint work with Adam Skalski.   

Sven Raum (University of Potsdam) - Nuclear dimension of crossed products by free actions of csc nilpotent Lie groups

Driven by the success of Elliott's classification programme for amenable C*-algebras, there has been interest in understanding which topological dynamical systems give rise to classifiable C*-algebras via the crossed product construction. Regularity, e.g. in the incarnation of finite nuclear dimension of the crossed product C*-algebra, is typically the hardest aspect to understand. Addressing this problem for connected groups for the first time, work of Hirshberg-Szabo-Winter-Wu from 2017 showed that crossed products by free actions of the real numbers have nuclear dimension controlled by the covering dimension of the space acted on. Later, Hirschberg-Wu could generalise this result, dropping the freeness assumption. I will report on joint work with Ulrik Enstad and Gabriel Favre extending the nuclear dimension bounds of Hirschberg-Szabo-Winter-Wu to free actions of connected simply connected nilpotent Lie groups. Prior to this work, no larger classes of Lie groups than the real numbers could be dealt with.

Dimitris Gerontogiannis (Leiden University) - Heat operators and isometry groups on Cuntz–Krieger algebras

In this talk, we explore the heat semigroups of Cuntz–Krieger algebras using spectral noncommutative geometry. The key tool is the logarithmic Dirichlet Laplacian for Ahlfors regular metric measure spaces, which produces spectral triples on Cuntz–Krieger algebras from singular integral operators. These spectral triples exhaust the K-homology and for Cuntz algebras their heat operators turn out to be Riesz potential operators. Moreover, the isometry groups of the spectral triples admit a concrete description in terms of symmetries of the associated directed graph of the Cuntz–Krieger algebra. Finally, Voiculescu's noncommutative topological entropy vanishes on those isometry groups. This is joint work with Magnus Goffeng (Lund) and Bram Mesland (Leiden).

Magnus Fries (Lund University) - Localizing spectral triples to ideals and manifolds with boundary

The notion of a spectral triple can be extended to higher-order operators to accommodate several types of hypoelliptic differential operators. Using that differential operators act locally, we can localize these spectral triples to a domain and obtain a relative K-homology class. With relative K-homology it is feasible to calculate the boundary map in K-homology and we obtain a class on the boundary of the domain. For elliptic operators, rewriting the class on the boundary provides generalizations of the Baum-Douglas-Taylor index theorem and Boutet de Monvel's index theorem, as well as a Poincaré-dual description of Aityah-Bott's obstruction of the existence of elliptic boundary conditions.

Sanaz Pooya (University of Potsdam) - Higher Kazhdan projections, K-theory and applications to delocalised L^2 Betti numbers

For a discrete group G, higher Kazhdan projections are projections constructed from the reduced cohomology of G with coefficients in unitary representations. If a unitary representation has spectral gap, then such projections lie in the group C*-algebra associated with the unitary representation and give rise to K-theory elements. In this talk I introduce the construction of higher Kazhdan projections and compute the associated K-theory classes for certain groups, such as PSL_2(Z). This leads to several vanishing and non vanishing results for delocalised L²-Betti numbers. This talk is based on joint work with Piotr Nowak and Kang Li as well as joint work with Hang Wang.

Valerio Proietti (University of Oslo) - The rational HK conjecture

Building on previous work by Davis and Lück, and recent constructions of KK-theory as a stable ∞-category, I will sketch the construction of a Chern character running from the left-hand side of the Baum–Connes conjecture for ample groupoids with torsion-free isotropy to the periodicized homology groups of the given groupoid. This map is a rational isomorphism, thereby establishing a modified form of Matui’s HK conjecture (after integral counterexamples have been found). This construction also computes the rational homotopy type of the algebraic K-theory spectrum of ample groupoids as defined in a recent work by X. Li. This is joint work-in-progress with M. Yamashita.

Anna Duwenig (KU Leuven) - Smooth Cartan triples and Lie groupoid twists

Kumjian–Renault theory allows us to reconstruct an étale groupoid twist from a Cartan pair of C*-algebras. I will explain what additional functional analytic data is needed to reconstruct a smooth structure on said twist. Our result gives rise to a correspondence between so-called "Lie groupoid twists" and "smooth Cartan triples", and it ties in Kumjian and Renault's theorems with Connes' reconstruction of a smooth manifold structure from certain spectral triples over commutative C*-algebras. Based on joint work with Aidan Sims (University of Wollongong).

Mario Klisse (KU Leuven) - The Choquet-Deny property for groupoids

A countable discrete group is called Choquet-Deny if, for any irreducible probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Despite many partial results, a characterization of this property in terms of the underlying structure of the group remained an open question for a long time. Only recently, building on the previous work of Jaworski, a complete characterization of Choquet-Deny groups was achieved by Firsch, Hartman, Tamuz, and Ferdowski. In this talk I will give a brief introduction into Choquet-Deny groups and then sketch how to define a suitable analogue of the Choquet-Deny property within the framework of discrete measured groupoids. Finally, I will explain our primary result, which offers a complete characterization of this property in terms of the isotropy groups and the equivalence relation associated with the given groupoid. This talk is based on joint work with Tey Berendschot, Soham Chakraborty, Milan Donvil, and Sam Kim.

Walter van Suijlekom (Radboud University) - Noncommutative geometry and operator systems

We give an overview of the recent interactions between noncommutative geometry and operator systems. We will see that the structure of an operator system is the minimal structure required to be able to speak of positive elements, states, pure states, etc. After presenting the general theory, we will illustrate this by many examples, ranging from spectral truncations of geometric spaces, to metric spaces up to a finite resolution. We will also present a general approach to analyzing (Gromov-Hausdorff) convergence results, which we will illustrate once again in these examples.

Marc Rieffel (UC Berkeley) - Dirac operators for quantum Hamming metrics

Given the set of words of a given length for a given alphabet, the Hamming metric between two such words is the number of positions where the two words differ. A quantum version of Hamming metrics was introduced in 2021 by De Palma, Marvian, Trevisan and Lloyd. For the quantum version the alphabet is replaced by a full matrix algebra, and the set of words is replaced by the tensor product of a corresponding number of copies of that full matrix algebra. While De Palma et al. work primarily at the level of states, they do obtain the corresponding seminorm (the Hamming metric) on the algebra of observables that plays the role of assigning Lipschitz constants to functions. A suitable such seminorm on a unital C*-algebra is the current common method for defining a quantum metric on a C*-algebra. In many important cases such seminorms can be obtained from spectral triples. 

I will indicate how quantum Hamming metrics can be obtained from spectral triples, that is, from a representation of the C*-algebra on a Hilbert space together with a self-adjoint operator  D  on the Hilbert space such that the value of the seminorm on an element  a  of the algebra is given by the operator norm of the commutator  [D, a]. One consequence of this is that the seminorm will be strongly Leibniz, i.e. satisfy a Leibniz inequality and more.

This is part of a project in progress.

Frederic Latrémolière (University of Denver) - The spectral propinquity

The spectral propinquity is a distance over the class of metric spectral triples, up to unitary equivalence. It is constructed as a special form of the propinquity between compact quantum metric spaces, itself rooted in the work on the Gromov-Hausdorff distance and its noncommutative analogues, in the more general framework of Rieffel's work in noncommutative metric geometry. In this talk, we wish to present the spectral propinquity and discuss various properties and examples of convergence in the sense of this metric: for instance, we will discuss how the spectrum of Dirac operators is continuous with respect to the spectral propinquity, and how various spectral triples on quantum tori, quantum solenoids, and Bunce-Deddens algebras are limits of other spectral triples.

Konrad Aguilar (Pomona College) - The study of AF algebras as quantum metric spaces and some of its byproducts

In 2006, Christensen and Ivan initiated the study of AF algebras as quantum metric spaces (in the sense of Rieffel), where they placed quantum metrics induced by spectral triples on AF algebras. In this talk, we summarize the continuation of the study of AF algebras as quantum metric spaces with an eye towards recent results, which includes a direct continuation of the results of Christensen and Ivan where we establish continuity of AF algebras in the Gromov-Hausdorff propinquity (a noncommutative analogue to the Gromov-Hausdorff distance introduced by Latr\'emoli\`ere) using their quantum metrics. Moreover, we discuss some of the byproducts of this study, which include but are not limited to: the development of a Gromov-Hausdorff propinquity that acknowledges a quotient rule-type property (the strongly Leibniz Gromov-Hausdorff propinquity) and a characterization of the dimensionality of a quantum metric space using the domain of an L-seminorm. This work is partially supported by NSF grant DMS-2316892. (This talk includes joint work from various different articles with C. Adams, E. Ayala, K. von Bornemann Hjelmborg, S. R. Garcia, E. Kim, E. Knight, F. Latrémolière, and C. Marple).

Yvann Gaudillot-Estrada (École normale supérieure) - Convergence of state spaces for truncations or filtrations of a quantum metric space

The notion of quantum metric space (due to Rieffel) is the right framework to deal with finite dimensional approximations of (noncommutative) metric spaces. These approximations are often constructed from a spectral triple by truncating the algebra with a spectral projection or by taking a subspace of it. We give sufficient conditions ensuring that the state spaces associated to these approximations converge, in Gromov-Hausdorff sense, to the full state space. We present some examples, such as Laplacian filtrations and Peter-Weyl truncations.

Christian Voigt (University of Glasgow) - Self-similarity and quantum symmetry

Self-similar groups form a fascinating class of groups with links to dynamics and geometry. In this talk I’ll explain how to combine the combinatorics underpinning self-similarity with the idea of quantum symmetry. The latter has recently attracted attention in connection with nonlocal games and quantum information theory.

(Based on joint work with N. Brownlowe, D. Robertson and M. Whittaker) 

Stefan Wagner (Blekinge Institute of Technology) - Spin structures in quantum geometry: A bundle theoretic approach

Riemannian spin geometry is a special and important topic within differential geometry which is mainly based on principal bundle theory and has objects such as spin structures and Dirac operators. It also has wide applications to mathematical physics, in particular to quantum field theory, where spin structures are an essential ingredient in the definition of any theory with uncharged fermions. In the noncommutative setting the notion of a spectral triple provides a natural framework for noncommutative Riemannian spin geometry. However, unlike in the classical setting, the axiomatic description of a noncommutative Riemannian spin geometry does not incorporate noncommutative principal bundles. In this talk we give a novel perspective on spin structures in quantum geometry by means of noncommutative principal bundles.

Teun van Nuland (TU Delft) - Multiple operator integrals and the abstract pseudodifferential calculus of Connes and Moscovici

Multiple operator integrals (MOIs) appear in various areas of noncommutative geometry, like the theories of cyclic cohomology (cf. the JLO-cocycle), spectral flow, spectral shift, the spectral action, and heat trace asymptotics. Recognizing MOIs where they appear 'undercover' enables a unified view, and often leads to new connections and generalizations. However, sometimes the integral expressions that you want to uncover as MOIs are unbounded, which does not match the traditional MOI formalism as pioneered by Peller in 2006. Our proposed solution is to merge multiple operator integration with the abstract pseudodifferential calculus of Connes and Moscovici. Based on joint work with Eva-Maria Hekkelman and Edward McDonald.

Francesca Arici (Leiden University) - TBA

TBA

Sophie Zegers (Delft University of Technology) - Explicit KK-equivalences for quantum flag manifolds

In the study of noncommutative geometry, various classical spaces have been given a quantum analogue. Examples include Drinfeld-Jimbo quantum flag manifolds for which the C*-completions have recently been described as graph C*-algebras by Brzeziński, Krähmer, Ó Buachalla and Strung. The quantum complex projective space $C(\mathbb{C}P_q^n)$ falls into this class of examples and was already known to be a graph C*-algebra due to Hong and Szymański.

In this talk, I will recall the explicit KK-equivalence between $C(\mathbb{C}P_q^n)$ and $\mathbb{C}^{n+1}$ constructed in collaboration with Francesca Arici by finding an explicit splitting for a short exact sequence. In the construction, the graph C*-algebraic description is crucial. Secondly, we will see how this approach can be used to construct KK-equivalences in the more general framework of quantum flag manifolds.

Wojciech Szymanski (University of Southern Denmark) - TBA

TBA