In this project the PI will study questions in algebraic geometry and commutative algebra. Algebraic geometry is the study of algebraic varieties, which are solution sets for systems of polynomial equations. For example, in the xy-plane, the solutions for y=0 consist of all points along the x-axis, while the solutions for xy=0 consist of all points along both coordinate axes. Since the tangent line at the origin (0,0) is not defined for the algebraic variety defined by xy=0, we say that this variety has a singularity at the origin (0,0). A central focus in the proposed research is that studying singularities is indispensable even when the objects of interest are smooth manifolds, which are not singular. In another direction, certain analytic objects defined using possibly divergent power series are often unavoidable as well. This contrasts with the usual situation on smooth manifolds, where all smooth functions have Taylor series expansions that converge in some neighborhood of a point. The PI will study the analogues of these situations in the field of algebraic geometry -- in particular, birational geometry -- and its interactions with other fields of mathematics, such as commutative algebra, complex geometry, and arithmetic geometry. The project will also provide research training opportunities for students. This project unifies and expands the classical boundaries of algebraic geometry and commutative algebra in multiple interconnected directions. First, the PI will e