2024 Noncommutative Geometry Research Seminar

In the second semester of 2024, the noncommutative geometry group ran a local NCG research seminar.

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List of talks

14 Jun 2024: Dimitris M. Gerontogiannis (Leiden)

Title: Heat operators and isometry groups on Cuntz–Krieger algebras

Abstract: 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 and encode the 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).

Place and time: Snellius 174 , 15:15-16:15.

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1 May 2024: Torstein Ulsnaes (Leiden & SISSA)

Title: Boundary extensions of symmetric spaces and their KK-cycles

Place and time: DM 1.19. 15:15-16:15.

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27 Mar 2024: Ada Masters (Wollongong)

Title: Conformal geometry, dynamics, and equivariance in unbounded KK-theory

Abstract: I will discuss the problem of representing dynamical systems in unbounded KK-theory, one of the main technical devices of noncommutative geometry. In the development of unbounded KK-theory, an aspect which has been left unresolved is the definition of equivariance. One reason for this is that the natural definition fails to capture all the degrees of freedom available in the usual, bounded equivariant KK-theory. This is apparent already in the classical, commutative, case, as we shall see. In light of the forgoing, conformal geometry will be briefly introduced and I will discuss the definition of conformally equivariant unbounded KK-theory. I will also touch on ongoing work which uses these tools to analyse more general dynamical systems.

Place and time: DM 1.15. 15:15-16:15.

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20 Mar 2024: Bram Mesland (Leiden)

Title: The Friedrichs angle and alternating projections in Hilbert C*-modules

Abstract: In his foundational work on rings of operators, John von Neumann proved that given two projections P,Q on a Hilbert space, the sequence (PQ)^n converges in the *-strong topology to the projection onto the intersection of the ranges of P and Q. The finer convergence properties of the sequence (PQ)^n are detected by a numerical invariant called the Friedrichs angle between P and Q. In this talk I will discuss the generalization of this problem to the setting of Hilbert C*-modules, where various technicalities present itself. The main issue is the fact that the intersection of complemented submodules need not be complemented. Due to this issue, von Neumann’s convergence result does not hold for an arbitrary pair of projections, and the Friedrichs angle cannot be straightforwardly defined. A definition of the Friedrichs angle can be obtained using the local-global principle for Hilbert modules, and yields an invariant that effectively detects the properties of the sequence (PQ)^n. I will discuss (counter)examples and interpretations of this result in noncommutiative geometry.

Place and time: Snellius 3.12. 15:15-16:15.

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13 Mar 2024: Aquerman Kuczmenda Yanes (Radboud Universiteit Nijmegen)

Title: Equivariant Spectral Flow for Families of Dirac-type Operators

Abstract: In this talk we show how to construct an equivariant version of spectral flow for paths of Dirac-type operators in the setting of a proper, cocompact action by a locally compact, unimodular group $G$ on a Riemannian manifold. This takes values in the $K$-theory of the group $C^*$-algebra of $G$. Our goal is to show that our equivariant spectral flow refines classical spectral in the case where $G$ is the fundamentalg roup of a compact manifold. If time permits, we look into its relationship with higher index theory and secondary invariants. Joint work with Peter Hochs.

Place and time: DM 1.19. 15:15-16:15.

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6 Mar 2024: Teun van Nuland (TU Delft)

Title: Multiple operator integrals and the abstract pseudodifferential calculus of Connes and Moscovici

Abstract: 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 developed by Peller in 2006. Our proposed solution is to merge multiple operator integration with the abstract pseudodifferential calculus of Connes and Moscovici.Based on ongoing joint work with Eva-Maria Hekkelman and Edward McDonald.

Place and time: DM 1.15. 15:15-16:15.

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28 Feb 2024: Dimitris Gerontogiannis (Leiden)

Title: KK-duality for Temperley-Lieb subproduct systems

Abstract: The notion of KK-duality generalises Spanier-Whitehead duality in the noncommutative setting of C*-algebras. As C*-algebras are noncommutative topological spaces in view of the Gelfand duality, C*-algebras with KK-duals can be thought of as noncommutative manifolds. Therefore, a main utility of KK-duality is that it makes it possible to study C*-algebras by more classical analytic methods. Interestingly, several C*-algebras associated to dynamical systems appear to have KK-duals. This talk is about the KK-duality of Cuntz-Pimsner algebras associated to Temperley-Lieb subproduct systems, a class of C*-algebras with rich quantum group symmetries and relations to topological Markov chains. This is joint work with Francesca Arici (Leiden) and Sergey Neshveyev (Oslo).

Place and time: DM 1.15. 15:15-16:15.

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21 Feb 2024: Sophie Zegers (TU Delft)

Title: Split extensions and KK-equivalences for quantum flag manifolds

Abstract: In this talk, I will first present the explicit KK-equivalence between $C(\mathbb{C}P_q^n)$ and the commutative algebra $\mathbb{C}^{n+1}$ constructed in collaboration with Francesca Arici. The KK-equivalence is constructed by finding an explicit splitting for the short exact sequence $\mathcal{K}\to C(\mathbb{C}P_q^n)\to C(\mathbb{C}P_q^{n-1})$. In the construction of a splitting it is crucial that $C(\mathbb{C}P_q^n)$ can be described as a graph algebra. Secondly, I will present how this approach can be used to construct KK-equivalences in the more general framework of quantum flag manifolds which is based on ongoing work with Réamonn Ó Buachalla and Karen Strung.

Place and time: DM 1.15. 15:15-16:15.

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14 Feb 2024: Yufan Ge (Leiden)

Title: SU(2)-symmetries of C*-algebras: from bricks to buildings

Abstract: In this talk, we will consider subproduct systems coming from SU(2)-representations and discuss the associated C*-algebras. We will first review results concerning irreducible representations from Arici–Kaad, then provide some further results about more general cases. More specifically, we will discuss the structure of the SU(2)- subproduct systems associated to isotypic representations and multiplicity-free representations. Finally, we will provide results about the K-theory groups of their Toeplitz algebras. This is joint work in progress with Francesca Arici.

Place and time: DM 1.15. 15:15-16:15.

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7 Feb 2024: Adam Rennie (Wollongong)

Title: Using the Cayley transform to relate van Daele K-theory and KK

Abstract: I will start with a warm-up in the (friendly) complex case showing that $K_1(A)\cong KK^1(\mathbb{C},A)$ via the Cayley transform. Then I will show that the “same” thing works in real, Real, graded cases when we start with van Daele K-theory. This will involve some discussion of van Daele K-theory, and how one defines it…which is slightly intricate.

Why did we want to do all that? Make Kasparov products easy of course!!

Joint work with Chris Bourne and Johannes Kellendonk.

Place and time: DM 1.15. 15:15-16:15.

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24 Jan 2024: Francesca Arici (Leiden)

Title: Some results about the K-theory of C*-algebras of subproduct systems

Abstract: In this talk, we will consider subproduct systems of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras, and discuss their relation to the theory of polynomials in noncommuting variables. We will provide results about their topological invariants through K(K)-theory and discuss some open problems.

Place and time: DM 1.19. 15:30-16:30.