Title:Intersections of Shubert Varieties
Abstract: Shubert varieties are subvarieties of the Grassmannian. Many interesting
geometric problems can be interpreted in terms of intersections of these
varieties. The intersection can be studied by associating to a Shubert
variety an element of a ring where the multiplication can be viewed as
intersection. The multiplication in the ring may easily be performed by a
simple combinatorial formula due to Pieri.
In this talk I will discuss the zero-dimensional intersection of Shubert
varieties. The classical Shubert problem of four lines, which asks how
many lines in space meet four fixed lines, will be examined. I'll show how
to model this problem as an intersection of Shubert varities, then use
Pieri's formula to calculate the answer. This theory requires working over
an algebraically closed field. If the Shubert problem is posed in real
terms, the results above may include complex points of intersection.
Lastly, I'll describe some special cases where the intersection points are
all real, which includes some current work.