Singularities of Schubert varieties

对称性的研究出现在几乎所有的数学领域，特别是在几何。在数学语言中，对称性通常通过作用于对象的群来描述。作为一个基本的例子，我们可以考虑平面上由旋转和平移产生的欧几里得运动群。人们可以将这幅图推广到研究李群或（如本建议的情况）代数群在向量空间（如平面）或更复杂的几何对象上的作用。对称群出现在许多自然科学中，无论是物理学（如量子力学或经典力学），化学（如晶体的对称性），还是生物学（如蛋白质结构的对称性）。然而，对称性在数学本身也扮演着非常重要的角色。这个建议的主题福尔斯属于代数几何领域。有人可能会说，代数几何中最基本的对象是射影空间（例如，平面中通过原点的一组直线）。它是齐次的，也就是说，它看起来到处都是一样的，因为对于任何两个点，都有一个射影空间的对称性，从一个点移动到另一个点（很像球面上的任何一点都可以旋转到同一球面上的任何其他点）。可以推广这个概念，得到一大类代数簇，即所谓的投射齐性空间。这些空间与用来定义它们的代数群紧密相连，而代数群在这些空间上充当对称。理解这些空间（的几何）提供了对这些群的结构的洞察，并且与它们的表示论等相关，这是一个具有广泛应用的重要课题。使用这些空间作为模型空间，希望这也会导致更好地理解其他代数几何对象。在这个拟议的项目中，我们试图理解这些空间的某些子集（称为舒伯特簇）的几何形状，这些子集自然产生，并且可以被证明是理解这些齐次空间的基础。

The study of symmetries arises in almost all areas of mathematics, and in particular in geometry. In the mathematical language, symmetries usually are described by means of a group acting on an object. As an elementary example one could consider the group of Euclidean motions on the plane, generated by rotations and translations. One can generalize this picture to study the action of Lie groups or (as in the case of this proposal) algebraic groups on vector spaces (like the plane) or more complicated geometric objects. Symmetry groups arise in numerous natural sciences, be it physics (e.g. quantum or classical mechanics), or chemistry (e.g. the symmetries of crystals), or biology (e.g. the symmetries of protein structures).Symmetries, however, also play a very important role in mathematics itself. The topic of this proposal falls into the realms of algebraic geometry. One could argue that the most fundamental object in algebraic geometry is projective space (e.g. the set of lines through the origin in a plane). It is homogeneous, i.e. it looks everywhere the same, because for any two points there is a symmetry of projective space moving one point to the other (much like any point on a sphere may be rotated to any other point on the same sphere). It is possible to generalize this notion and obtain a large class of algebraic varieties, the so called projective homogeneous spaces. These spaces are tightly connected to the algebraic group used to define them, and the group acts as symmetries on these spaces.Understanding (the geometry of) these spaces provides insight into the structure of these groups, and is connected to e.g. their representation theory among other things, an important subject with a wide array of applications. Using these spaces as model spaces, hopefully this also leads to a better understanding of other algebraic geometric objects. In this proposed project we try to understand the geometry of certain subsets (called Schubert varieties) of these spaces which arise naturally and can be shown to be fundamental for understanding these homogeneous spaces.