This project involves a variety of topics. Those concerning regular finitely presented graded noncommutative algebras include the investigation of these topics: (1) the correspondence between algebras of dimension 3 and automorphisms of elliptic curves, (2) the dualizing module, (3) the discovery, with the aid of a computer, of some new algebras of dimension 4 having two generators, (4) the constructability of the condition that an algebra be finite and of rank n over its center, and (5) the classification of those algebras via almost split sequences and via tilting equivalences. In geometry, the projects include the investigation of these topics: (1) the multiple-point theory of maps via the Hilbert scheme, (2) the structure and autoduality of the relative compactified jacobian, (3) the classical enumerative geometry of smooth plane cubics, (4) the Halphen intersection ring of a symmetric variety, (5) the degeneration of the tangent space and Whitney stratification, (6) the positivity of the coefficients of Kazdan-Lusztig polynomials via intersection homology and the geometry of Schubert varieties (7) the stratum of the miniversal deformation space of a 2-dimensional hypersurface singularity as the hull of Wall's EF-functor, (8) the study of surfaces as projections, as subvarieties of 3-folds of small degree, and as the carriers of important double structures, and (9) the study of L-functions of certain elliptic curves over function fields via crytalline cohomology. This project concerns research on a variety of topics in algebraic geometry. After a long period of dormancy, this area of mathematics has flourished with problems that are centuries old being resolved. The results of this research will be of interest to many different areas of mathematics.
