We define a classifying stack for the bounded derived category associated to any scheme X. When X is projective, we show that this stack is locally geometric, i.e., we can treat it as a slight abstraction of a scheme. We will also provide some applications of this result.

One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered.

In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.

Shimura varieties are algebraic varieties that play an important role in number theory and the Langlands program. I will discuss constructions of models of Shimura varieties over the integers and recent results about the singularities of their reductions modulo primes that divide the level.

Let $\Delta$ be a triangulation of a connected region in the real plane. Let $C(r,d,\Delta)$ be the space of piecewise polynomial functions of degree $\leq d$ and smoothness $r$. A major question in Approximation Theory is to find the dimension of this space, which is not known even for the case when $d=3$ and $r=1$. Alfeld and Schumaker give a formula for this dimension, when $d\geq 3r+1$ and any $\Delta$. Using homological algebra, this problem can be translated into finding the Hilbert function of a graded module (the ``homogenization'' of $C(r,d,\Delta)$). I will discuss about this approach and about the Schenck-Stiller conjecture that says that Alfeld-Schumaker formula holds for any $d\geq 2r+1$. I will present a very recent project with Jan Minac where we prove this conjecture for a triangulation that is not trivial, in the sense that the formula does not hold if $d=2r$.