Mathematical Sciences: Algebraic Geometry

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Professor Pinkham will study the m-regularity of smooth varieties in projective space via generic projection, and also the equations of canonical curves. Professor Friedman will continue to investigate Donaldson's new invariants for smooth 4- manifolds, especially elliptic surfaces and their blow-ups. The associate Principle Investigator Professor Morrison will study the slopes of effective divisors on the moduli space of stable curves, the stability of projective varieties via its state polytopes and the volume of simplices in spaces of constant curvature in terms of its dihedral angles along codimension two faces. The post doctoral assistant Dr. Cutkosky will study the birational geometry of threefolds concentrating on the analysis of the divisorial contraction of extremal rays. Finally the post doctoral assistant Dr. O'Grady will study the Kodaira dimension of the moduli space of polarized K3 surfaces of large degree. All of these topics are intermingled facets of the subject of algebraic geometry, the study of the geometric objects obtained as solution sets of systems of polynomial equations. The concentration in this proposal is on the geometric and topological aspects of these objects. This is an especially important area of modern mathematics with widespread applications. There is a rich mixture of young and mature investigators on this grant which will give the research an extra strength through their communication. There is a lot of promise in this research, much of which will surely be realized.