While polynomial equations with multiple variables are among the most elementary of equations, requiring only addition and multiplication of variables, the set of points which satisfies a given system of equations carries rich algebraic and geometric structure. The set of points with complex coordinates which satisfies a fixed system of equations gives a geometric object is called an algebraic variety. The classification of algebraic varieties is the main motivation of the PI's field of birational geometry. On the other hand, arithmetic geometry is the study of polynomial equations whose coefficients are integers. This allows more tools from number theory to be used to study the corresponding varieties. The field of arithmetic geometry has recently seen an explosion of new techniques leading to spectacular progress. The PI's research on this project will apply these new techniques to problems in birational geometry which were previously out of reach. The educational component of the project involves several initiatives designed to increase access to a career in mathematics. This includes creating a summer school introducing high school students to mathematical proofs using the Lean proof assistant. The project centers on interactions between birational and arithmetic geometry. New techniques from arithmetic geometry involving perfectoids and prisms have made it possible to overcome significant difficulties in mixed characteristic birational geometry, mainly related to t