Linquan Ma of the Institute for Advanced Study gives a Members' Colloquium talk on singularities in mixed characteristic algebraic geometry. He opens with the classical picture, singularities as local properties of varieties such as the cusp at the origin of y squared equals x cubed, studied through quotient rings like C[x,y] over that polynomial. He then moves to mixed characteristic, where the analogous objects are quotients of Z[x1,...,xn] by integer polynomials, and surveys recent progress in the field. The central tool is big Cohen-Macaulay algebras, large non-Noetherian rings with strong homological properties whose existence connects to developments in p-adic Hodge theory. Ma closes by discussing how this singularity theory feeds into questions in birational geometry. The talk runs 57 minutes and assumes graduate-level familiarity with commutative algebra and algebraic geometry.