Singularities of Schubert varieties

对称的研究几乎出现在数学的所有领域，尤其是几何领域。在数学语言中，对称性通常用一群作用于一个物体的方式来描述。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一组欧几里得运动。我们可以推广这幅图来研究李群或（在这个提议的情况下）代数群在向量空间（如平面）或更复杂的几何物体上的作用。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）。

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).