Abstract: Even when trying to classify smooth varieties, it is natural to stumble upon singular varieties. Recently, there has been a lot of progress in the classification of varieties over fields of positive characteristics and over mixed characteristic DVR’s like \(\mathbb{Z}_p \). This has been possible partly thanks to the introduction of new notions of singularities related to Frobenius splittings and perfectoid methods, respectively. Given a hypersurface in a projective space over the complex numbers, we can measure how singular it is with an invariant called the « log canonical threshold ». Similarly, in positive characteristic, we can define the « F-pure threshold » and in mixed characteristic the « plus-pure threshold ». In this talk, we will explore some examples of how to compute the plus-pure threshold and how this relates to the invariants in positive characteristic and in characteristic 0. This is based on joint work with V. Jagathese, V. Pandey, P. Ramírez-Moreno, K. Schwede, P. Sridhar.