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.[-]

Recent developments in quantum information led to a generalised notion of reference frames transformations, relevant when reference frames are associated to quantum systems. In this talk, I discuss whether such quantum reference frame transformations could realise a notion of deformed symmetries formalised as quantum group transformations. In particular, I show the correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at the first order in the quantum deformation parameter. This correspondence is found once the group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame.[-]

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.[-]

Cosmological N-body simulations are one of the most versatile tools for studying the evolution of large-scale structure in the Universe. While the Newtonian limit of general relativity can be used for most purposes within the basic LCDM model, the true nature of the dark components (dark matter and dark energy) is unknown and may ultimately require a relativistic description. Also the neutrinos from the standard model are relativistic for most of the cosmic history if they have a mass within the range allowed by cosmological and laboratory constraints. In this course I will introduce a framework for relativistic N-body simulations that can treat any relativistic degrees of freedom self-consistently. Furthermore, I will discuss the important aspect of how the simulation data are mapped to observables by constructing the past light cone of an observation event. As an instructive example the effect of massive neutrinos will be the subject of two Hands-on sessions, where I will demonstrate how to use the relativistic N-body code gevolution.
• Lecture 1 — The spacetime of N-body simulations
• Lecture 2 — Observables on the light cone
• Hands-on 1 — The matter power spectrum with massive neutrinos
• Hands-on 2 — Weak lensing with massive neutrinos[-]

Current estimations of cosmological parameters depend often on assumptions about the cosmological model itself. The estimates of $H_{0}$ or $\Omega_{m}$ from CMB surveys, for instance, are valid only assuming a particular model, typically the standard $\Lambda$CDM. Similar model-dependent results are obtained also from analyses of large-scale structure. In some case it is however possible to combine observations in such a way to get estimates of physical quantities that are valid regardless (to some extent) of the underlying model. Here I will discuss how one can determine the cosmological expansion rate $H(z)$ and the deviation from Einstein gravity $\eta$ by combining in a model-independent way several observational probes, from redshift distortions, to lensing, to matter clustering.[-]

In these lectures we give a general introduction to the theory of gravitational waves and the analytic approximation methods in general relativity. More precisely we focus on the theory which is necessary to accurately and reliably predict the gravitational waves generated by compact binary systems, made of black holes or neutron stars. The predictions are used in the form of gravitational-wave “templates” in the data analysis of the detectors LIGO, VIRGO, ... LISA. In particular we present the state-of-the-art on the post-Newtonian approximation in general relativity, which is the main tool for describing the famous gravitational wave “chirp” of compact binary systems. The outline of the lectures is :
1. Gravitational wave events
2. Methods to compute gravitational waves
3. Einstein quadrupole formalism
4. Post-Newtonian parameters
5. Finite size effects in compact binaries
6. Synergy with the effective field theory
7. Radiation reaction and balance equations. [-]

In this course I will give an overview of different line of sight and environmental effects that are expected to modify/distort a gravitational wave (GW) signal emitted by an astrophysical source at cosmological distance. After an overview of the state of the art of GW observations, I will review the derivation of the waveform emitted by a binary system of compact objects in a flat background, in the newtonian approximation. I will then consider a cosmological context and introduce the concept of standard siren. Finally I will derive which is the expected distortion of an emitted waveform induced by the presence of peculiar velocities and (strong) gravitational lensing.

Lecture 1 — Gravitational waves–introduction
Lecture 2 — Gravitational waves–propagation effects in waveform modeling[-]

Recent years have seen the development of a range of modified gravity theories to tackle the dark energy and cosmological constant problem. An interesting class of such models are those in which the graviton effectively becomes massive, modifying gravity at large (cosmological) distances without changing physics at solar system and smaller scales. Such effective field theories have the Vainshtein mechanism built in, and are closely associated with Galileon theories.

In these lectures I will give a brief general overview of both soft and hard massive gravity theories, their origin from extra dimensional models, and the special class of ghost-free massive gravity models and their extensions with multiple massive (and massless) spin-2 states. I will review the Vainshtein mechanism, and the decoupling limits of these theories and how they are related to Galileons. I will then discuss a variety of implications of such theories.

• lecture 1: I will introduce the description of massless and massive spin-2 states, and explain how theseemerge naturally in extra dimensional models, and go on to give the description of so-called ghost-freeor $\Lambda_{3}$ theories of interacting massive spin-2 fields, aka massive gravity and its multi-gravity extensions.

• lecture 2: I will review some aspects of the phenomenology of the general class of massive gravity theories: screening - cosmology - black hole solutions, and if time go on to discuss more recent extensions which attempt to raise the cutoff, connections with UV completions such as TTbar deformations, and the significance of ‘positivity bounds' applied to these effective theories.[-]