Noncommutative geometry: metric and spectral aspects

Contribution List

1. Atiyah sequences of braided Lie algebras and their splittings

Giovanni Landi

9/28/22, 9:00 AM

28-morning

2. Dual of the dual formula for the distance in noncommutative geometry

Pierre Martinetti

9/28/22, 10:00 AM

28-morning

Connes's spectral distance is an extended metric on the state space of a C*-algebra, generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dua formula - as an infimum - generalizing Beckmann's ``dual of the dual'' formulation of the Wasserstein distance.

7. Spectral gaps for higher Laplacians and group cohomology

Piotr Mizerka

9/28/22, 11:20 AM

28-morning

I will focus on the cohomology of finitely presented groups. I plan to investigate two conditions concerning it: vanishing and reducibility (for all unitary representations). These conditions are related to Kazhdan's property (T): vanishing and reducibility coincide in degree one and are equivalent to this property. It is already known, due to the work of Dymara and Januszkiewicz, that this...

24. Spectral triple with real structure on fuzzy sphere

Anwesha Chakraborty

9/28/22, 11:45 AM

28-morning

Here we have illustrated the construction of a real structure on a fuzzy sphere S^*_2 in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S^∗_2 given by U. C. Watamura et.al. [Commun. Math. Phys. 183, 365–382 (1997)], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition,...

25. Hopf algebroids on quantum projective spaces

Jacopo Zanchettin

9/28/22, 12:10 PM

28-morning

We construct the Ehresmann-Schauenburg bialgebroid for a family of U(1)-quantum principal bundles over quantum projective spaces, showing that another antipode (related to K-theory on the base algebra) exists besides the "classical" flip. Moreover, we show how the theory of twists (generalized characters) applies in this situation.

20. Localization and quantum observers in some Lie-algebra type noncommutative spacetimes.

Fedele Lizzi

9/28/22, 2:30 PM

28-afternoon

I will discuss how the observers in a noncommutative spacetime need to be considered quantum objects as well, and discuss issues of localization of states for kappa-Minkowski spaces and some (angular type) variants of it.

5. Curvature for modules over spectral triples

Bram Mesland

9/28/22, 3:30 PM

28-afternoon

In this talk we introduce the curvature of densely defined universal connections on Hilbert C*-modules relative to a spectral triple, obtaining a well-defined curvature operator. Algebraically, this curvature can be interpeted as the defect of the unbounded Kasparov product to commute with the operation of taking squares. The definition recovers the represented curvature of finitely generated...

6. The Gromov-Hausdorff distance in noncommutative geometry: convergence of quantum compact metric spaces and their modules

Frederic Latremoliere

9/28/22, 4:30 PM

28-afternoon

Connes' Spectral triples have emerged as the preferred tool to encode geometric information over possibly noncommutative C*-algebras. We present, in two lectures, a distance on the space of metric spectral triples, which then enables us to formally discuss ideas such as approximations or perturbations of spectral triples, and opens the possibility to study the geometry of spaces of spectral...

12. Noncommutative spaces at finite resolution

Walter van Suijlekom

9/29/22, 9:00 AM

29-morning

We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to a finite resolution. In our approach the traditional role played by C*-algebras is taken over by so-called operator systems. We consider...

11. Tolerance relations and quantization

Francesco D'Andrea

9/29/22, 10:00 AM

29-morning

It is well known that “bad” quotient spaces can be studied by associating to them the groupoid C*-algebra of an equivalence relation. A similar procedure for relations that are reflexive and symmetric but fail to be transitive leads to a non associative algebra. I will discuss some of its properties based on a recent joint work with G. Landi and F. Lizzi.

14. Spectral metric and Einstein tensors.

Ludwik Dabrowski

9/29/22, 11:20 AM

29-morning

We define bilinear functionals of vector fields and differential forms, the densities of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. We generalise these concepts in non-commutative geometry and, in particular, we prove that for the conformally rescaled geometry of the noncommutative two-torus the Einstein functional vanishes.

17. Causality in noncommutative deformation spacetimes

Nicolas Franco

9/29/22, 12:20 PM

29-morning

A notion of causality specific to noncommutative geometry was introduced in 2013 by Franco and Eckstein, leading at the same time to a notion of Lorentzian metric. This notion has been widely explored with regards to almost-commutative space-times. In this talk, we present a more difficult exploration concerning "truly" noncommutative spacetimes, i.e. deformation spacetimes as Moyal spacetime...

15. Lorentzian spectral zeta functions

Michał Wrochna

9/29/22, 2:30 PM

29-afternoon

The spectral theory of the Laplace–Beltrami operator on Riemannian manifolds is known to be intimately related to geometric invariants such as the Einstein-Hilbert action. These relationships have inspired many developments in physics including the Chamseddine–Connes action principle in the non-commutative geometry programme. However, a priori they do only apply to the case of Euclidean...

16. Quantum metric structures on q-deformed spaces

David Kyed

9/29/22, 3:30 PM

29-afternoon

I will survey recent results concerning the quantum metric structures on the Podleś sphere and quantum SU(2). Along the way, I will provide a brief introduction to Rieffel’s theory of compact quantum metric spaces and the notion of quantum Gromov-Hausdorff convergence. The talk is based on joint works with Konrad Aguilar, Thomas Gotfredsen and Jens Kaad.

13. The Gromov-Hausdorff distance in noncommutative geometry: convergence of spectral triples

Frederic Latremoliere

9/29/22, 4:30 PM

29-afternoon

We present a distance on the class of metric spectral triples. It thus becomes possible to formally discuss the idea of approximating a spectral triple with others: for instance, to approximate a spectral triple on the 2-torus by means of natural spectral triples on the finite dimensional so-called fuzzy tori.

A spectral triple induces an extended pseudo-metric on the state space of its...

19. Geometrically realised spectral triples and dynamical mass

Shahn Majid

9/30/22, 9:00 AM

30-morning

We apply the geometric realisation programme for spectral triples within quantum Riemannian geometry (QRG) to the algebra of 2x2 matrices with a Lorentzian metric to find an S^1 moduli of almost spectral triples where the Dirac operator is not hermitian but has natural hermitian and antihermitian parts (based on joint work with E. Lira-Torres). I will also explain how Kaluza Klein ideas look...

4. Around heat kernel

Bruno Iochum

9/30/22, 10:00 AM

30-morning

First, I present a few properties of heat kernel and its trace and then, revisit the way of computing the coefficients of the heat trace asymptotics for a differential operator acting on a fiber bundle over a Riemannian manifold in a way which avoids entering within the pseudodifferential theory.

21. Sequences of Noncommutative Gauge Field Theories on AF-algebras

Thierry Masson

9/30/22, 11:00 AM

30-morning

With Gaston Nieuviarts we have built and started to study a new framework to define sequences of Noncommutative Gauge Field Theories (NCGFT) on top of the defining sequence of an AF algebra. The main objective of this construction is to manage the way these NCGFT are related to each other along the sequence. A notion of “compatibility” is then necessary to handle this problem. In my talk, I...

23. Differential nested pairs of quantum homogeneous spaces.

Alessandro Carotenuto

9/30/22, 12:20 PM

30-morning

TBA

8. Quantum Kaluza-Klein theory with M_2(C)

Chengcheng Liu

9/30/22, 12:45 PM

30-morning

We follow a quantum Kaluza-Klein formulation where we solve for the quantum Riemannian geometry on A = C∞(M) ⊗ M2(C) in terms of classical Riemannian geometry on M, the finite quantum geometry on M2(C) and gauge-field like cross term. We look at how scalar fields on the total space decompose into multiplets of fields in M differing in mass.

22. Spectral triple based Noncommutative Gauge Field Theories on AF-algebras

Gaston Nieuviarts

9/30/22, 1:10 PM

30-morning

In the continuity of the presentation made by Thierry Masson, in which the general framework was introduced to define Noncommutative Gauge Field Theories (NCGFT) on top of the sequence of an AF algebra, I will present the part of our work that focuses on the study of these NCGFTs using spectral triples. In particular, I will insist on the
"compatibility" relations on the defining structure of...

3. Spectral and metric aspects of the Dolbeault-Dirac spectral triple on quantum SO(5)/(SO(2)xSO(3)

Elmar Wagner

9/30/22, 1:35 PM

30-morning

The Berstein-Gelfand-Gelfand resolution for irreducible quantum flag manifolds gives an algebraic description of the Dolbeault complex of (anti-)holomorphic k-forms by actions of quantum tangent space. Requiring equivariance and compatibility with the real form of the quantum enveloping algebra, there is an essentially unique hermitian metric on the (0,k)-forms given by the Haar state. Using...

18. TBA

TBA