Singularities in Mixed Characteristic

Linquan Ma introduces recent progress on singularities in mixed characteristic. Starting from singular points of algebraic varieties, such as the cusp y² = x³, the talk explains their algebraic study through quotient rings and the mixed-characteristic setting of integer-coefficient polynomial rings. A central tool is the big Cohen–Macaulay algebra: a typically non-Noetherian ring with strong homological properties whose existence connects to advances in p-adic Hodge theory. The presentation also discusses applications of this singularity theory to birational geometry.