I will first explain the joint work with Walter van Suijlekom on a new result about th zeros of the Fourier transform of extremal eigenvectors for quadratic forms associated to distributions on a bounded interval and its relation with the spectral action. Then I will explain how these results allow to advance in the joint work which I am doing with Consani and Moscovici on the zeta spectral triple. Finally, if time permits, I will discuss several ideas in connection with physics and non-commutative geometry.[-]

The quantisation of the spectral action for spectral triples remains a largely open problem. Even within a perturbative framework, serious challenges arise when in the presence of non-abelian gauge symmetries. This is precisely where the Batalin–Vilkovisky (BV) formalism comes into play: a powerful tool specifically designed to handle the perturbative quantisation of gauge theories. The central question I will address is whether it is possible to develop a BV formalism entirely within the framework of noncommutative geometry (NCG). After a brief introduction to the key ideas behind BV quantisation, I will report on recent progress toward this goal, showing that the BV formalism can be fully formulated within the language of NCG in the case of finite spectral triples. [-]

This talk introduces a class of Hopf algebras, called T -Poincaré, which represent, arguably, the simplest small scale/high energy quantum group deformations of the Poincaré group. Starting from some reasonable assumptions on the structure of the commutators, I am able to show that these models arise from a class of classical r-matrices on the Poincaré group. These have been known since the work of Zakrzewski and Tolstoy, and allow me to identify 16 multiparametric models. Each T -Poincaré model admits a canonical 4-dimensional quantum homogeneous spacetime, T -Minkowski, which is left invariant by the coaction of the group. A key result is the systematic unification provided by this framework, which incorporates well-established non-commutative spacetimes like Moyal, lightlike κ-Minkowski, and ρ-Minkowski as specific instances. I will then outline all the mathematical structures that are necessary in order to study field theory on these spaces: differential and integral calculus, noncommutative Fourier theory, and braided tensor products. I will then discuss how to describe (classical) Standard Model fields within this framework, and the challenges associated with quantum field theory. Particular focus is placed on the Poincar´e covariance of these models, with the goal of finding a mathematically consistent model of physics at the Planck scale that preserves the principle of Special Relativity while possessing a noncommutativity length scale.[-]