Spectral and Geometry 2025, Krakow

Schedule

29.09, 13:00
Bergfinnur Durhuus
Loop models on causal dynamical triangulations and height-coupled trees. A class of statistical models of loops on causal dynamical triangulations, related to the Ising model, will be discussed together with a bijective correspondence with planar trees with Boltzmann weights depending on the size as well as the height of trees. It is shown that such trees exhibit an interesting phase diagram with phases characterized by different values of the Hausdorff dimension of corresponding generic trees, and with implications for the critical behaviour of the mentioned loop models. Some open problems relating to the phase diagram of the loop models will be described.

29.09, 14:00
Eva-Maria Hekkelman
Connes' integration formula from 1915 The noncommutative integral in NCG is based on Connes' trace theorem from 1988. I will attempt to convince you that Szegő already proved a special version of Connes' theorem in 1915, at least on the circle, and only if you really use your imagination. More useful is that Szegő's version can be generalised way beyond the circle case, leading to a noncommutative version of Szegő's limit theorem. This is intimately connected to the Connes--Van Suijlekom study of spectrally truncated spectral triples. If time permits, I will also briefly highlight how the mathematics involved establishes a connection between NCG and the field of Quantum Ergodicity. Based on joint work with Ed McDonald.

(15:00 coffee break)
29.09, 15:30
Bruno Iochum
Towards Gibbs semigroup on Banach spaces Gibbs semigroups acting on Hilbert spaces are fundamental tools in quantum statistical mechanics and in the computation of actions across classical, quantum, and non-commutative frameworks. This talk presents a pathway for extending the theory of such semigroups to the Banach space setting.

29.09, 16:30
Raimar Wulkenhaar
Stochastic quantisation of scalar QFT on noncommutative spaces We present a joint work with C. Song and H. Weber in which we treat the stochastic quantisation equation of the $\lambda\phi^4$ QFT model on 2-dimensional non-commutative Moyal space. We adapt the Da Prato--Debussche formulation to a matrix setting and corresponding spaces of distributions. Special care is necessary to estimate a mixed term which does not have a commutative analogue. We prove existence and uniqueness of the solution up to some small stochastic time and give an a priori estimate for large time. The tools from stochastic analysis then allow to deduce existence of an invariant measure. In the second part we outline our project to combine (1) stochastic quantisation, (2) the exact solution of the planar sector of $\lambda\phi^4$ model on 4-dimensional Moyal space and (3) free probabilty to construct an interacting QFT in four dimensions.

30.09, 9:00
Giovanni Landi
Solutions to the Quantum Yang Baxter Equation and related deformations We report on a class of noncommutative products of finite-dimensional Euclidean spaces. They are described by natural families of coordinate algebras which are quadratic and are associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. Notably, we have eight-dimensional noncommutative euclidean spaces having deformation parameter in a sphere. Quotients include seven-spheres as well as noncommutative quaternionic tori. An invariance for $SU(2) \times SU(2)$ parallels the action of $U(1) \times U(1)$ on a `complex' noncommutative torus $T^2_\theta$ which allows one to construct quaternionic toric noncommutative manifolds and noncommutative principal fibrations.

30.09, 10:00
Shahn Majid
Geometric Realisation of Pre Spectral Triples Dividing Connes’s axioms into a `local tensorial part’ and an `operator algebraic part’, the first part can be addressed at an algebraic or smooth level without taking completions. This allows, for example, to take forward the Lorentzian case focusing on tensorial structures rather than the unsolved problems on the operator side for this case. These revolve around the `geometric realisation’ problem of factoring D through a Clifford action and a bimodule connection on a suitable spinor bundle over the differential algebra (A, Omega^1, d), along with an additional metric compatibility that connects it to a quantum Riemannian geometry (QRG) on the latter. We recap recent work where any QRG with a certain symmetry condition on the metric leads to a pre-spectral triple at this level, and, if time, the case of a classical manifold extended by a fuzzy sphere. We end with discussion of what some issues could then be for the operator level.

30.09, 11:00
Reamonn O'Buachalla
One-cross bundles for quantum homogeneous spaces We introduce a special type of quantum principal bundle, over a quantum homogeneous space B, known as a one-cross bundle. The calculus on the bases is shown to have a very rich structure, including metrics, Levi-Civita connections, complex structures, and holomorphic structures. We will also present a number of examples.

(12-13 lunch break)
30.09, 13:00
Thomas Weber
Quantization of infinitesimal braidings Quasitriangular bialgebras appear as symmetries of quantum spaces. Their universal R-matrix determines a braiding, which is an essential operation for many constructions in quantum algebra and noncommutative geometry. In this talk we consider first-order deformations of braided monoidal categories, meaning we introduce infinitesimal braidings, and discuss the emerging quantization problem. This is done parallel to the algebraic picture, where infinitesimal R-matrices and so-called pre-Cartier bialgebras appear. It turns out that the quantization problem admits a solution in terms of Drinfeld associators, resulting in quasitriangular quasi-bialgebras. As an explicit example, the deformation of the pointed E(n) Hopf algebras is discussed. This is based on a collaboration with Chiara Esposito, Jonas Schnitzer and Andrea Rivezzi.

30.09, 14:00
Tomasz Brzezinski
Remarks on differential structures on quantum spheres

(15:00 coffee break)
30.09, 15:30
Ludwik Dąbrowski
Spectral functionals for geometric invariants Given Laplace type operator and using Wodzicki residue I will recall certain spectral functionals of vector fields, the densities of which reproduce volume (integral), metric, scalar curvature and Einstein tensors. Alternatively, given Dirac type operator, I will describe analogous functionals of differential forms which yield the dual tensors. In the latter setup we recently introduced also spectral torsion functional which recovers the torsion tensor. for the canonical Dirac operator coupled to torsion. These functionals often generalize to noncommutative geometry. In particular, the conformally rescaled noncommutative torus, Einstein-Yang-Mills and quantum SU(2)-group spectral triples are torsion free, while the quantum 2-sheeted space and the almost commutataive geometry of the Standard Model do have torsion. I will comment on relation to the algebraic notion of torsion and Levi-Civita connection, and present impact on the other spectral functionals. Based on [Adv.Math. 427,2023; Commun.Math.Phys. 130,2024; JNCG 2024, Phys.Rev.Lett. 134,2025; arXiv:2412.19949] with A. Sitarz, P. Zalecki, A. Bochniak, Y. Liu, S. Mukhopadhyay and F. Pozar.

30.09, 16:30
Pierre Bieliavski
Deformation quantization for actions of non-Abelian Lie groups. In 1994, Marc Reiffel gave a deformation machinery for actions of the Lie group R^d, extending the formal notion of Drinfel'd twist to the one of universal deformation formula (UDF) internal to the category of Fréchet or even C*-algebras which the Lie group R^d acts on. Important examples of non-commutative manifolds can be interpreted as by-products of his construction such as the quantum tori, and more generally theta-manifolds. In the case of an isometric action of R^d on a Riemannian compact spin manifold, Rieffel's machinery yields non-commutative isospectral deformation of the commutative spectral triple coding the Riemannian manifold at hand. Rieffel addressed the natural question of defining such types of non-formal UDF's for action of non-Abelian Lie groups. In this talk, I will review in some details our joint work with V. Gayral, appeared as a Memoir of the AMS in 2015, where we defined such UDF's for actions of Khaelerian Lie groups (such as Iwasawa factors of Hermitean type real non-compact simple Lie groups).

1.10, 9:00
Arkadiusz Bochniak
Geometry of Quantum Graphs: Unknowns and Why They Matter The study of noisy quantum information channels has led to the notion of quantum graphs. Certain characteristics of these objects can reveal a wealth of interesting information. Investigating their geometric properties through spectral data may further deepen our understanding. In this talk, I will provide a brief overview of the topic, highlight open problems, and present some ongoing work.

1.10, 10:00
Fedele Lizzi
Mixed in Translations A Lie group is a manifold, associated to each of its points there is an element of the group, represented as a transformation of a vector space. Geometrically points are the pure states of the commutative algebra of the functions on the manifold. Mixed states are instead represented by positive normalised probability densities. We consider the transformations connected to these mixed states, and find that they form a semigroup. We will then investigate the simplest case of one-dimensional translations, and find interesting surprises, including a novel way to see the connections between temperature and time for thermal states. We then analyse the case of the quantum spacetime described by the Moyal product, and its Hopf algebra quantum symmetry. Joint work with Gaetano Fiore.

1.10, 11:00
Christoph Stephan
Noncommutative Geometry: Quo vadis? Noncommutative Geometry has developed remarkable mathematical depth, yet its dialogue with physics has grown a bit quiet. This talk will ask how we might challenge the field anew and search for future directions that reconnect ideas and observations.