9800807 Shokurov This award support a project of Professors Shokurov and Kachi. The project is focused on problems in the Log Minimal Model Program (LMMP) in higher dimensions, mainly, on the existence of the log flips in dimension 4 and higher. The aim is to obtain such flips inductively from the LMMP itself and some terminations in lower dimensions. In particular, the aim is to construct the log flips and even to complete the LMMP in dimension 4. In addition, a geometric description of flips in families will be given. Shokurov and Kachi will also start to investigate needed terminations and to demonstrate further applications of the LMMP. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials, or be given in the 3-space by the simplest geometric constructions, e. g., conic sections. Traditionally, the field interacts with most of branches of mathematics. Nowadays the field makes use of methods not only from algebra, but from analysis, topology and mathematical physics, and conversely is finding application in those fields as well as in number theory, physics, theoretical computer science, and robotics.
