Ultra-high-energy cosmic rays remain one of the main scientific objectives of astroparticle physics. On
the astrophysics side, the sources accelerating the most energetic Galactic and extragalactic particles are
still unknown. On the particle physics side, we do not understand why hadronic interaction models
predict too few muons in the atmospheric particle cascades initiated when cosmic-ray nuclei impinge the
atmosphere. To address these questions, new techniques and a higher measurement accuracy for the
properties of the primary cosmic rays is required. One way to increase the accuracy is to add radio
antennas to existing arrays of particle detectors, such as at the Pierre Auger Observatory in Argentina,
the largest air-shower array in the world. For IceTop, the surface array of the IceCube Neutrino
Observatory at the South Pole, an enhancement by scintillation panels and radio antennas is planned. In
January 2020, we completed a prototype station at the South Pole and now are analyzing its air-shower
data. This station also serves as prototype for the surface array of IceCube-Gen2, a planned expansion of
IceCube by an order of magnitude. In addition to new instrumentation, new analysis methods, such as
neural networks, will further increase the accuracy for the properties of the primary cosmic-ray particles.
This talk will summarize the state of the field, my work in the past years, and the plans for the next
decade.