Singularities in Mixed Characteristic

Singularities are local properties of algebraic varieties. For example, the solution set of a polynomial in several variables such as $y^2=x^3$ has a singularity at the origin $(0,0)$. In algebra, one often studies singularities through the quotient ring, such as $C[x,y]/y^2-x^3$ for this example. In this talk, we will give an introduction to recent progress in the study of singularities in mixed characteristic, roughly speaking, these correspond to quotients of $Z[x_1,...,x_n]$ by polynomials with integer coefficients. A central theme in this study is the use of so-called big Cohen-Macaulay algebras, these are often "large" (non-Noetherian) rings that nevertheless enjoys remarkably nice homological properties  (and whose existence is related to recent developments in p-adic Hodge theory). We also discuss some applications of the mixed characteristic singularity theory to questions in birational geometry.