We discuss determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We also explore the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.

The mass of neutrinos is a fundamental open question in physics, impacting both the Standard Model of elementary particles and cosmology. Precision measurements of weak decay kinematics offer a laboratory-based, model-independent method to probe the absolute neutrino mass scale. The KArlsruhe TRItium Neutrino experiment (KATRIN) seeks to detect the subtle imprint of the neutrino mass in the endpoint region of the tritium beta-decay spectrum. Combining a high-intensity gaseous molecular tritium source with a high-resolution electrostatic filter using magnetic adiabatic collimation, KATRIN has set the most stringent direct neutrino-mass limit at m(ν) < 0.8 eV/c2 (90% C.L.) based on an initial data set. Ongoing data collection and analysis aim for a sensitivity below 0.3 eV/c2. Additionally, KATRIN’s precision beta-spectrum data facilitates searches for new physics beyond the neutrino mass, including sterile neutrinos, Lorentz invariance violation, and non-standard weak interactions. This talk presents the latest results and future prospects of the experiment.

Over the last twenty years, physicists have learned to manipulate individual quantum objects: atoms, ions, molecules, quantum circuits, electronic spins... It is now possible to build "atom by atom" a synthetic quantum matter. By controlling the interactions between atoms, one can study the properties of these elementary many-body systems: quantum magnetism, transport of excitations, superconductivity... and thus understand more deeply the N-body problem. More recently, it was realized that these quantum machines may find applications in the industry, such as finding the solution of combinatorial optimization problems. This seminar will present an example of a synthetic quantum system, based on laser-cooled ensembles of individual atoms trapped in microscopic optical tweezer arrays. By exciting the atoms into Rydberg states, we make them interact, even at distances of more than ten micrometers. In this way, we study the magnetic properties of an ensemble of more than a hundred interacting ½ spins, in a regime in which simulations by usual numerical methods are already very challenging. Some aspects of this research led to the creation of a startup, Pasqal.