This project is concerned with a number of issues in algebraic geometry, the geometric theory of noncommutative rings, the differential topology of complex algebraic varieties, and arithmetic algebraic geometry. These issues include the following: (1) the enumeration of the multiple-points of a finite map that is birational onto its image; (2) Whitney equisingularity and the Buchsbaum-Rim multiplicity; (3) the inseparablility of the Gauss map of a smooth complete interaction; (4) the enumeration of the rational normal curves on the generic quintic hypersurface in 4-space, and the singularity of the Hilbert scheme component of the rational normal n-ics; (5) the autoduality of the compactified Jacobian; (6) the theory of a transition map in the theory of residues and duality; (7) quantum projective planes; (8) the foundations of noncommutative projective geometry; (9) noncommutative projective schemes of dimension 1; (10) maximal orders with p-fold ramification in characteristic p; (11) special polyhedral filtrations; (12) higher Chow groups as derived functors; (13) bivariant sheaves and motives; (14) applications of geometry to the study of class groups of number fields. This is research in the field of algebraic geometry, which is one of the oldest parts of modern mathematics. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but also from analysis and topology. It is finding application in those areas of mathematics as well as in theoretical computer science and in robotics.
