Mathematical Sciences: Algebraic Geometry

Pinkham教授将通过一般投影研究射影空间中光滑变量的m-正则性，以及正则曲线的方程。Friedman教授将继续研究Donaldson的光滑4-流形的新不变量，特别是椭圆曲面及其膨胀。副首席研究员Morrison教授将研究稳定曲线模空间上有效因子的斜率，通过其状态多面体的射影变的稳定性，以及恒曲率空间中简单体沿余维两面的二面角的体积。博士后助理库特考斯基博士将研究三倍的二分几何，重点分析极值射线的分形收缩。最后，博士后助理O'Grady博士将研究大阶极化K3曲面模空间的Kodaira维数。所有这些主题都是代数几何主题的混合方面，研究作为多项式方程系统解集的几何对象。该方案的重点是这些物体的几何和拓扑方面。这是现代数学中一个特别重要的领域，有着广泛的应用。年轻和成熟的研究人员都参与了这项资助，这将通过他们的交流为研究提供额外的力量。这项研究有很多希望，其中大部分肯定会实现。

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.