Abstract: The Borel-Bott-Weil theorem describes the cohomology of line bundles on homogeneous varieties in terms of the representation theory linear algebraic groups. This talk will be a gentle introduction to this theorem in the case of the general linear group.

The first part of the talk will be an extensive motivation for why one is interested in line bundles on projective varieties and their cohomologies. In an nutshell, line bundles turn the extrinsic problem of studying maps to projective space into an intrinsic one.

The second part of the talk will discuss the representation theory of the general linear group and will end with a statement of the theorem in this special case.