The International Muon Collider Collaboration (IMCC) [1] was established in 2020 following the recommendations of the European Strategy for Particle Physics (ESPP) and the implementation of the European Strategy for Particle Physics-Accelerator R D Roadmap by the Laboratory Directors Group [2], hereinafter referred to as the the European LDG roadmap. The Muon Collider Study (MuC) covers the accelerator complex, detectors and physics for a future muon collider. In 2023, European Commission support was obtained for a design study of a muon collider (MuCol) [3]. This project started on 1st March 2023, with work-packages aligned with the overall muon collider studies. In preparation of and during the 2021-22 U.S. Snowmass process, the muon collider project parameters, technical studies and physics performance studies were performed and presented in great detail. Recently, the P5 panel [4] in the U.S. recommended a muon collider R D, proposed to join the IMCC and envisages that the U.S. should prepare to host a muon collider, calling this their "muon shot". In the past, the U.S. Muon Accelerator Programme (MAP) [5] has been instrumental in studies of concepts and technologies for a muon collider.

We study the kinetic inductance in thin superconducting/normal metal (S/N) bilayer films where the penetration depth is greater than the S thickness. Using the two-coil mutual inductance method, we show that it is possible to estimate the dependence of kinetic inductance on sheet current density. It is found that, as expected from Ginzburg-Landau theory, the dependence is very weak in bare superconducting films. However in S/N bilayers of NbN/Al and Nb/Al the kinetic inductance can be smoothly varied over at least a 2:1 range by application of a dc transport current. It is proposed that this novel non-linearity could be exploited in some microwave applications.

The superconducting density of states for the cuprates, particularly for the Y-Ba-Cu-O compound, has been evaluated, add its dependence on temperature and the oxygen content has been analyzed. The analysis is based on the two-gap model. Moreover, the magnetic scattering and corresponding pair-breaking effect, correlated with the oxygen depletion, is taken into account as a key factor. The temperature dependencies of the energy gaps are calculated. Intensive magnetic scattering leads to gaplessness. The calculation allows us to describe various experimental data.

High T-c cuprates contain various magnetic scatterers. Their presence affects many features of the materials, including tunneling density of states. Correlation between magnetic moments leads to a novel ''recovery'' effect, and, consequently, to an unusual temperature dependence of the Josephson current.

When a pair of coils is positioned on opposite sides of a superconducting thin film, measurement of their mutual inductance may in principle be used to infer the penetration depth λ in the superconductor. We have studied how to optimize this measurement with respect to coil design, and have found that the approach that has been generally used is far from the optimum. Useful simplifications to the expression relating mutual inductance to penetration depth are derived. An analysis of the sources of uncertainty in determining λ is presented. For an optimized coil set, the major source of uncertainty often is uncertainty of the thickness of the film. The sensitivity to changes in λ is also studied; it is shown that this can approach 1 pm for a typical high temperature superconductor sample. Finally, it is shown that the analysis may be extended to normal metal films, with the skin depth playing a role similar to that of the penetration depth in superconductors. Measurement of a high conductivity normal metal foil can be used to check calculated calibration factors for a coil set or to determine the skin depth.

We have included in our two-gap model of superconductivity in the cuprates the effect of pair breaking caused by magnetic impurity scattering (pair breaking). This model explains many of the observed features of the cuprates including the temperature and frequency dependence of the surface resistance and penetration depth, the anomalous temperature dependence of the upper critical magnetic field, and the zero bias anomalies observed between cuprates and normal metal contacts. This model can also explain many of the phase coherence experiments in YBaCuO on the basis of magnetic scattering at an interface or grain boundary due to strain-induced oxygen loss. We have recently extended this model to calculate the electromagnetic properties and the density of states in the presence of magnetic scattering. The latter results will be presented here.