**Sudhir Ghorpade**

Indian Institute of Technology, Mumbai

April 22, 2010

**Maximal linear sections of Grassmannians**:
Consider a finite dimensional vector space over a field and the collection of all its subspaces of a fixed dimension. It is well-known that this constitutes a nice geometric object, namely, the Grassmannian with its canonical Plucker embedding. Linear sections of Grassmannians, of which Schubert varieties in Grassmannians are special cases, are interesting objects from algebraic, topological and
combinatorial viewpoints.
We consider the following question: Among the sections of Grassmannians by linear subspaces of a fixed dimension of the Plucker
projective space, which are "maximal"?
The term "maximal" can be interpreted in several ways and we will be
particularly interested in maximality with respect to the number of
points, when working over a finite field. In general, this is an open
problem. We will describe some of the known results and a connection
with some basic questions of multilinear algebra. If time permits we
will discuss certain extensions and generalizations as well.