Title: Classification of Projective Varieties and the Minimal Model Program

Abstract: A projective variety X over a field k is the zero locus in projective space Pnk

of a finite set of homogeneous polynomials in n+1 variables. Two projective varieties X and Y are

said to be birationally equivalent if there exist open subsets U ⊆ X and V ⊆ Y such

that U \simeq V . The classification of projective varieties up to birational equivalence is one

of the main problems of algebraic geometry. The Minimal Model Program (MMP) is an

algorithm that, given a projective variety, aims to construct a minimal model, a “nice”

representative of each birational equivalance class, given a projective variety. We first

motivate the rich mathematics involved in the MMP by investigating the classification of

curves and surfaces. Next, we examine the general MMP and some variations, such as the

log MMP. Finally, we survey some recent results regarding the MMP and its analogues

in positive and mixed characteristic.