The Bernstein-Gelfand-Gelfand (BGG) correspondence is a an equivalence between the derived categories of graded modules over a polynomial ring and exterior algebra, respectively. Eisenbud, Fl\/oystad and Schreyer (2003) derive an explicit version of the BGG correspondence that, among other things, clarifies the relationship between free resolutions of graded modules over the exterior algebra and the cohomology of coherent sheaves on projective space.

On the other hand, work of Jan-Erik Roos and Maurice Auslander gives a filtration of a graded module over a polynomial ring with supports that decrease in dimension, via a suitable Grothendieck spectral sequence. I will describe some joint work with Hal Schenck in which we consider the interplay between these two constructions. We find applications to some topological spaces with cohomology rings generated in degree 1.

To motivate the stable homotopy category of CW-complexes, I will talk about generalized cohomology theories and Brown representability in more detail.

Recently it has been proved that any (not necesssarily affine) normal algebraic monoid M is a closed submonoid of the endomophism monoid of a conveniently chosen homogeneous vector bundle over A(M), the Albanese variety of M. This result suggests that the category of homogeneous vector bundles over the abelian variety A(M) is the natural setting for a representation theory of M. Such a theory has not yet been developed, even in the case when M is an algebraic group.

The main goal of this talk is to present the basic facts about the structure of the endomorphism monoid of a homogeneous vector bundle over an abelian variety. These endomorphism monoids play a role similar to the endomorphism monoid of a vector space (the monoid of n by n matrices) in the affine case.

This is a joint work with Leticia Brambila-Paz.

This talk combines two ideas: inverse functions and matrix functions.

Given a function f(z), it is a standard procedure to define an inverse function through the solution of the equation f(z)=w. If we denote the inverse function by invf, then we can write z=invf(w). The definition is often complicated by the fact that there are multiple solutions of the equation.

Given a scalar function f(z), there are standard procedures for extending the definition to a matrix argument. For example, integer powers of a matrix, or the exponential of a matrix.

There are two ways in which one can arrive at a definition of the inverse of a matrix function. First, one can define the scalar inverse and then extend the definition to a matrix argument; second, one can extend the definition of the equation f(z)=w to matrices and then consider its solutions.

These definitions are discussed in the context of the Lambert W function, which is the inverse of the function f(z) = z*exp(z).