Mathematical Sciences: Global Differential Geometry and Singularities

主要研究人员将研究涉及以下方面的问题： 代数和微分几何中的奇点。特别是 他们将考虑高斯映射和拉格朗日的奇点 结构，切锥的奇性轨迹， 和奇异真实的解析簇的Stiefel-Whitney类。 六名调查人员的广泛背景将使他们能够 应用一系列通常不被关注的方法 这一系列的问题。特别是他们将应用层理论 方法，雅可比行列式的流动，以及积分几何， research. 几何学中的奇异性提出了一系列问题， 比那些研究表面光滑的更难。 因此，人们对这一案件的了解要少得多， 需要先进的技术才能取得进展。另 我们的物理世界充满了空间的例子， 奇异性很重要--从材料的断裂到 金属的液相和固相之间的边界。描述 理解这些奇点对我们来说 实践和理论观点。主要研究者 将利用他们不同的背景来研究这一重要的 问题的集合。

The principal investigators will study problems involving singularities in algebraic and differential geometry. In particular they will consider the singularities of Gauss maps and Lagrangian structures, tangent cones of the Thom-Boardman singularity loci, and Stiefel -Whitney classes of singular real analytic varieties. The broad background of the six investigators will allow them to apply a range of methods which might not ordinarily be focused on this set of problems. In particular they will apply sheaf theoretic methods, flows on Jacobians, and integral geometry in their research. Singularities in geometry present a set of problems much more difficult than those in which the surfaces studied are smooth. Consequently much less is known about this case and more sophisticated techniques are needed to make progress. On the other hand our physical world is full of examples of spaces in which singularities are important - from fractures in materials to boundaries between liquid and solid phases of metals. Describing and understanding these singularities is important from both a practical and theoretical standpoint. The principal investigators will use their varied backgrounds to study this important collection of problems.