Francesca Arici (University of Leiden)
Title: Quadratic subproduct systems and their algebras
Abstract: Quadratic algebras provide a natural framework for studying quantum spaces and deformations arising from the theory of quantum groups, as pioneered in Manin’s programme for noncommutative geometry. These algebras can be studied using the tools of noncommutative geometry and operator algebras, by realizing them in terms of a suitable subproduct systems of Hilbert spaces. I shall discuss recent results concerning the extension of various operations on quadratic algebras to their C*- counterparts, focusing on free products and Veronese powers.
Joakim Arnlind (Linköping University)
Title: Noncommutative Levi-Civita connections in derivation based calculi
Abstract: We study the existence of Levi-Civita connections, i.e torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over *-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a generalized symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry.
Are Austad (University of Oslo)
Title: Generating hermitian algebras from noncommutative geometry
Abstract: Using tools from noncommutative geometry we establish new classes of hermitian convolution algebras associated with groups and groupoids. A group $G$ is said to be hermitian if $L^1(G)$ is a hermitian Banach $*$-algebra, i.e. if every self-adjoint element of $L^1 (G)$ has real spectrum. Understanding which groups are hermitian is an important problem in harmonic analysis with roots in Wiener's lemma. The study of hermitian groups stretched over several decades, culminating in Losert's landmark result in 2001 which established the property for compactly generated groups of polynomial growth. Reframing hermitianness in terms of the study of a derivation on $B(L^2(G))$, we recover Losert's result. We are simultaneously able to show that étale groupoids with polynomial growth are hermitian, thus providing the first examples of hermitian étale groupoids which are not just discrete groups.
Sara Azzali (University of Bari)
Title: Traces in KK-theory and index pairings
Abstract: Traces on C*-algebras play an important role in index theory, for instance they allow to extract numerical invariants from classes in K-theory. By introducing real coefficients in Kasparov bivariant K-theory, traces give classes in $KK_\mathbb{R}$. The process of applying a trace then corresponds to taking the Kasparov product. In this talk, we explain these constructions and some applications. In particular, we give a natural $KK_\mathbb{R}$-class that represents the Godbillon-Vey invariant of a foliation of codimension one. This work is in collaboration with Paolo Antonini (Università del Salento) and Georges Skandalis (Université Paris Cité).
Christian Bönicke (University of Newcastle)
Title: The triangulated category approach to groupoid equivariant KK-theory
Abstract: The usefulness of viewing KK-theory as a category rather than a functor was recognised quite early on in the subject. Later, Meyer and Nest described a natural triangulated structure on KK-theory and used it to study the Baum-Connes conjecture for groups. I will explain a generalisation of this approach to groupoid equivariant KK-theory and outline its many applications. This talk is based on joint work with Valerio Proietti.
Réamonn Ó'Buachalla (Charles University Prague)
Title: Quantum exterior algebras, torsion-free bimodule connections, and the Cactus Group
Abstract: A spectral triple, or a noncommutative manifold in the sense of Connes, naturally gives rise to a first-order differential calculus that generalises the space of one-forms on a Riemannian manifold. Extending this structure to a full differential graded algebra — generalising the classical extension to the de Rham complex — admits an abstract description, but explicit computations remain subtle and technically challenging. In this talk, we show how the bimodule map associated to a torsion-free bimodule connection offers a compact, tractable construction of these extensions. We apply this framework to the Heckenberger–Kolb differential calculi over the irreducible quantum flag manifolds. Remarkably, this leads to an unexpected link with Drinfeld’s normalized braiding, which yields representations of the cactus group — rather than the usual braid group — on the category of finite-dimensional $U_q(\frak{g})$--modules. Time permitting, we will discuss extensions of this result beyond the Heckenberger--Kolb setting. (Joint work with Alessandro Carotenuto and Junaid Razzaq).
Koen van den Dungen (University of Bonn)
Title: Index theory and spectral flow of Toeplitz operators
Abstract: Classical Dirac-Schrödinger operators are given by Dirac-type operators on a smooth manifold, together with a potential. I will describe a general notion of Dirac-Schrödinger operators with arbitrary signatures (with or without gradings), which allows us to study index pairings and spectral flow simultaneously. I will first describe a general Callias Theorem, which computes the index (or the spectral flow) of Dirac-Schrödinger operators in terms of index pairings on a compact hypersurface. Associated to each Dirac-Schrödinger operator is also a Toeplitz operator, which is obtained by compressing the potential to the kernel of the Dirac operator. I will then explain how the index or spectral flow of these Toeplitz operators is related to the index or spectral flow of Toeplitz operators on the compact hypersurface. These results generalise various known results from the literature, while presenting them in a common unified framework.
Ulrik Enstad (University of Oslo):
Title: Z-stability of twisted group C*-algebras of nilpotent groups
Abstract: Z-stability is a regularity property central to the classification program for nuclear C*-algebras. In this talk, I will present a characterization of when a twisted group C*-algebra of a finitely generated nilpotent group is Z-stable. I will also discuss appliations of this result to frame theory. The talk is based on joint work with Eduard Vilalta.
Eske Ellen Ewert (University of Hannover)
Title: Partial group actions and boundary value problems
Abstract: Let G be a discrete group which acts on a manifold W. Suppose M contained in W is a compact submanifold with boundary which is not G-invariant. Then the interior of M carries a partial G-action and one can study the partial shifts U_g for g in G on L^2(M) given by U_g f(x)=f(g^{-1}.x) if g^{-1}.x is in M and zero otherwise.
In this talk, I will describe an algebra of operators $A\subseteq\mathbb B(L^2M\oplus L^2\partial M)$ generated by boundary value problems on M and the partial shifts U_g for g in G (under suitable assumptions on the action). As in the classical Boutet de Monvel calculus, there are two principal symbol maps defined on A: one associated with the interior and one with the boundary. Here, they take values in crossed product algebras of corresponding partial actions. Then an operator in A is Fredholm if and only these principal symbols are invertible. I will discuss how one can classify the stable homotopy classe of elliptic operators over A in terms of K-theory. This talk is based on joint work with Anton Savin and Elmar Schrohe.
Marzieh Forough (Czech Technical University in Prague)
Title: Groupoid equivariant KK-theory and C*-extensions
Abstract: Kasparov's KK-theory admits a description in terms of extensions of C*-algebras. Thomsen generalized this description to the case of equivariant KK-theory when the underlying groups are locally compact and second countable. Le Gall extended the equivariant KK-theory to the actions of Hausdorff groupoids then Tu used it to study Baum-Connes conjecture. The relation between groupoid equivariant KK-theory and C*-extensions has been remained untouched. The first step to investigate this problem is to study the relation between RKK-theory and extension of C*-algebras. I begin my talk with establishing this relation. I will also discuss one of our primary motivations to study this problem which came from our joint work on quasi-invariant lifts of completely positive maps for groupoid actions. Finally, I will explain the relation between groupoid equivariant KK-theory and C*- extensions. This talk is based on ongoing joint work with Suvrajit Bhattacharji.
Dimitris Gerontogiannis (IMPAN)
Title: Ideal quantum metrics from fractional Laplacians
Abstract: This talk presents a new framework for constructing computable Monge–Kantorovich metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. These “ideal” metrics admit explicit spectral formulas and naturally respect underlying dynamics. Our methods introduce new tools in noncommutative geometry, including a fractional Weyl law and Schatten-class commutator estimates. As an application, we extend the construction to expansive Z^m-actions and their associated C*-algebras, illustrating the reach of fractional analysis across dynamics, fractal geometry, and quantum metric spaces. This is joint work with Bram Mesland.
Victor Hildebrandsson (Linköping University)
Title: Levi-Civita connections for noncommutative tori in derivation based calculi
Abstract: We consider noncommutative Riemannian geometry via the derivation based calculi approach, introduced by Michel Dubois-Violette. After presenting the motivating background and necessary definitions, we present a necessary and sufficient condition for the existence of Levi-Civita connections on free modules. We then construct Levi-Civita connections on the 2- and 3-tori. This is joint work with Joakim Arnlind.
Jens Kaad (University of Southern Denmark)
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 geometric data yields a twisted derivation on the coordinate algebra whereas the vertical geometric data produces a derivation in the usual sense. Nonetheless we are able to 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.
Evgenios Kakariadis (University of Newcastle)
Title: Morita equivalence for operator systems
Abstract: In ring theory, Morita equivalence preserves many properties of the objects, and generalizes the isomorphism equivalence between commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphism, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.
Aaron Kettner (Czech Academy of Sciences/Charles University)
Title: Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles
Abstract: We discuss how to associate a C*-algebra to a partial action of the integers acting on the base space of a vector bundle, using the framework of Cuntz--Pimsner algebras. We investigate the structure of the fixed point algebra under the canonical gauge action. We also analyse the ideal structure, and give conditions under which the Cuntz--Pimsner algebra is simple. Finally we establish a bijective corrrespondence between tracial states and invariant measures on the base space. This generalizes results about the C*-algebras associated to homeomorphisms twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung and Viola. If time permits, we will discuss nuclear dimension estimates.
Jacek Krajczok (Vrije Universiteit Brussel)
Title: Invariants of fusion algebras of compact quantum groups
Abstract: Representation theory of compact quantum groups closely resembles its classical counterpart. Classes of finite dimensional representations, together with the operations of taking tensor product, direct sum, contragradient representation and a choice of dimension function, form an algebraic data called the fusion algebra. Typically it is much easier to study then the quantum group itself, but nonetheless carry interesting information. I will report on a joint work with Adam Skalski, in which we study certain asymptotic invariants of fusion algebras: Folner constant (describing the level of non-amenability) and uniform growth rate. This is joint work with Adam Skalski.
David Kyed (University of Southern Denmark)
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. Based on joint works with Konrad Aguilar, Are Austad and Jens Kaad.
Xin Li (University of Glasgow)
Title: On Hausdorff covers for non-Hausdorff groupoids
Abstract: Many important examples of C*-algebras are constructed from topological groupoids, which arise naturally in a variety of areas such as dynamics, topology, geometry and group theory. While a satisfactory theory of groupoid C*-algebras has been developed for groupoids which are Hausdorff, mysterious phenomena and new challenges arise in the setting of non-Hausdorff groupoids. This talk is about a new approach to non-Hausdorff groupoids and their algebras based on Timmermann’s construction of Hausdorff covers (joint work with Brix, Gonzales and Hume).
Xiaoqi Lu (University of Glasgow)
Title: Hilbert transforms on Coxeter groups and groups acting on buildings
Abstract: Cotlar identities were used as a tool to show the Lp (1 < p < ∞) boundedness of Hilbert transforms on the real line. In 2017, Mei and Ricard first generalized Cotlar identities to the noncommutative setting and showed the Lp-boundedness (1 < p < ∞) of Hilbert transforms on free groups. Then in 2022, González-Pérez, Parcet and Xia generalized the result on groups acting on R-trees by using a geometric model. In this talk, we will further discuss this topic on Coxeter groups (abstract reflection groups) and groups which admit actions on buildings. This is a joint work with Runlian Xia.
Bram Mesland (University of Leiden)
Title: E-mergence of the spectral localiser
Abstract: The spectral localiser of Loring and Schulz-Baldes gives a method to compute the index pairing between an odd K-theory class and an odd spectral triple as the signature of a certain finite dimensional matrix. A spectral triple determines a natural asymptotic morphism, and thus an element in the E-theory of Connes-Higson. In this talk I will show how the spectral localiser emerges from computing the pairing of the K-theory class with this asymptotic morphism. This new proof sheds some light on the parameters appearing in the spectral localiser formula. Based on joint work in progress with my Ph.D. student Yuezhao Li.
Teun van Nuland (University of Delft)
Title: The Feynman rules of spectral QED
Abstract: I will report some recent progress in the perturbative quantization program of the spectral action. This is based on ongoing joint work with Eva-Maria Hekkelman and Jesse Reimann.
Francesco Pagliuca (University of Glasgow)
Title: Equivariant periodic cyclic homology
Abstract: Periodic cyclic homology is a variation of cyclic homology and it is a fundamental tool in noncommutative geometry because it plays the same role as de Rham cohomology in commutative geometry. This theory has been widely studied and also extended in an equivariant context. In this talk, we aim to go beyond the group case and introduce the category of modules over the convolution algebra of an ample groupoid. Then, we will present all the objects needed to define equivariant periodic cyclic homology for algebras in this category. Finally, we will discuss the main properties of this homology, as homotopy invariance and stability. This is a joint work with C. Voigt.
Jesse Reimann (Delft University of Technology)
Title: On the best constants of Schur multipliers of second order divided difference functions
Abstract: We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko’s conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the Hörmander-Mikhlin-Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. Joint work with Martijn Caspers.
Karen Strung (Institute of Mathematics of the Czech Academy of Sciences)
Title: Fell bundles over unital based rings
Abstract: A common strategy for understanding the structure of a C*-algebra is to decompose it into more manageable subcomponents—such as its lattice of ideals or subspaces arising from a grading—with the aim of reconstructing global information from local data and the way these pieces interact. Conversely, one can begin with structured pieces and ask how to assemble them into a C*-algebra. A well-studied example of this approach is the theory of Fell bundles over discrete groups.
In this talk, we present a generalization of Fell bundles from discrete groups to unital based rings, a broader algebraic framework that accommodates richer types of gradings. We describe how to construct a reduced C*-algebra of sections associated to such a bundle and discuss examples arising from group gradings and coactions of compact quantum groups. This is joint work in progress with Suvrajit Bhattacharjee, Bhishan Jacelon, and Réamonn Ó Buachalla.
Haluk Sengun (University of Sheffield)
Title: New applications of C*-algebras in representation theory
Abstract: In this expository talk, I will discuss the recent applications of the notions of Rieffel induction and strong Morita equivalence in the theory of theta correspondence, a prominent theme in representation theory of reductive groups and in the theory of automorphic forms. The results will be based on my joint works with Bram Mesland and Magnus Goffeng.
Walter van Suijlekom (University of Nijmegen)
Title: A generalization of K-theory to operator systems
Abstract: We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by C*-algebras and inspired by the realization of the K-theory of a C*-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the K0-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For C*-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.
Bob Yunken (University of Lorainne)
Title: A K-theoretic approach to the minimal $K$-types of representations of real reductive groups
Abstract: Let G be a real reductive group and K a maximal compact subgroup, say G=SL(n,R) and K=SO(n). A famous theorem of Vogan in unitary representation theory states that there is a one-to-one correspondence between irreducible representations of K and irreducible tempered representations of G with real infinitesimal character, given by the minimal K-type. We will show how a K-theoretic version of this theorem can be proven by studying the deformation of G to its Cartan motion group thanks to the Connes-Kasparov conjecture.