Mathematical Sciences: Global Differential Geometry and Singularities

主要研究者将研究代数和微分几何中涉及奇点的问题。特别地，他们将考虑高斯映射和拉格朗日结构的奇点，汤姆-博德曼奇点轨迹的切锥，以及奇异实解析变的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会集中在这组问题上的方法。特别地，他们将在他们的研究中应用束理论方法、雅可比矩阵流和积分几何。几何中的奇点所提出的一系列问题比研究光滑表面的问题要困难得多。因此，对这种情况的了解要少得多，需要更复杂的技术才能取得进展。另一方面，我们的物理世界充满了奇点很重要的空间例子——从材料的裂缝到金属的液体和固体相之间的边界。从实践和理论的角度来看，描述和理解这些奇点都很重要。主要研究人员将利用他们不同的背景来研究这一系列重要的问题。

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.