Singularities of Schubert varieties

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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.