Georgia Institute of Technology College of Sciences

The complete flag variety is a fundamental object at the confluence of algebraic geometry, representation theory, and algebra. It is defined to be the space parametrizing certain chains of vector subspaces, and is intimately linked to Grassmannians, incidence varieties, and other important geometric objects of a representation-theoretic flavor. The problem of computing the cohomology of any line bundle on a flag variety in characteristic 0 was solved in the 1950's, culminating in the celebrated Borel--Weil--Bott theorem. The situation in positive characteristic is wildly different, and remains a wide-open problem despite many decades of study. After surveying this topic, I will speak about recent progress on a characteristic-free analogue of the Borel--Weil--Bott theorem through the lens of representation stability and the theory of polynomial functors. This "stabilization" of cohomology, combined with certain universal categorifications of the Jacobi-Trudi identity, has opened the door to concrete computational techniques whose applications include effective vanishing results for Koszul modules, yielding an algebraic counterpart for the failure of Green's conjecture for generic curves in arbitrary characteristic.