One of the most fundamental question in algebraic geometry is whether a certain algebraic variety is birational to a projective space or not. Even in the case of hypersurfaces in projective spaces, this question is far from being easy. Our toy in the talk is a certain quartic hypersurface in 7-dimensional projective space, called the 'Coble quartic'. It is the moduli space of (S-equivalent classes of) semi-stable vector bundles of rank 2 on a non-hyperelliptic curve of genus 3 with canonical determinant. We introduce the rationality problem and explain some geometry of this space.

We survey the theory of non-abelian Galois representations and its applications, as developed

out of the anabelian programme of Grothendieck from the 1980's.

Hessenberg varieties are a class of subvarieties of the flag variety which appear in many areas, e.g. in geometric representation theory. In order to generalize Schubert calculus to Hessenberg varieties, a first step is to construct computationally convenient module bases for the (equivariant) cohomology rings of Hessenberg varieties analogous to the famous Schubert classes which are a basis for the cohomology of flag varieties. Goresky-Kottwitz-MacPherson ("GKM") theory gives a concrete combinatorial description of the equivariant cohomology of spaces with torus action which satisfy certain conditions (usually called the GKM conditions). We propose a framework for approaching the problem of constructing module bases for Hessenberg varieties which uses GKM theory. The main conceptual challenge in this context is that conventional GKM theory requires a `sufficiently large-dimensional torus' action (to be made precise in the talk), while Hessenberg varieties generally have only a circle action. To resolve this, we define the notion of GKM-compatible subspaces of GKM spaces and give applications in some special cases of Hessenberg varieties. The talk will be intended for a wide audience, and in particular I will begin with a conceptual sketch of the main ideas in Schubert calculus and of classical GKM theory.

This is mainly joint work with Tymoczko; time permitting, I will mention joint work with Bayegan, and also with Dewitt.

Submodular functions are discrete analogues of convex functions.

Examples include cut capacity functions, matroid rank functions,

and entropy functions. Submodular functions can be minimized in

polynomial time, which provides a fairly general framework of

efficiently solvable combinatorial optimization problems.

In contrast, the maximization problems are NP-hard and several

approximation algorithms have been developed so far.

In this talk, I will review the above results in submodular

optimization and present recent approximation algorithms for

combinatorial optimization problems described in terms of

submodular functions.