In 1995, de Concini and Procesi investigated certain iterative blowups of affine space along intersections of linear subspaces, their wonderful models, a fundamental example being the Fulton-Macpherson configuration space compactification. In doing so, they developed suitable combinatorics to describe, among other things, the cohomology of the wonderful models.

In 2006, Feichtner and Yuzvinsky constructed smooth toric varieties from de Concini and Procesi's combinatorial data, and found that, for any arrangement of hyperplanes, the cohomology ring of the de Concini-Procesi wonderful model is isomorphic to the Chow ring of their toric variety. Their argument is indirect, via the combinatorics defining the rings in question.

I will outline a toric construction of de Concini and Procesi's wonderful models for hyperplane arrangements. This is an example of Tevelev's notion of a tropical compactification. One advantage is that it provides a geometric explanation of Feichtner and Yuzvinsky's isomorphism.

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.

Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome!

When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations

eventually showed that the extrinsically defined curvature of a

surface can be expressed entirely in terms of its intrinsic metric (= the

first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem).

Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for

all of differential geometry, as it was later shown by Riemann in 1859

that the curvature of higher dimensional manifolds can be understood

purely in terms of curvatures of its two dimensional submanifolds.

Theorema Egregium can also be regarded as the infinitesimal form of,

and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.

An embedding $\phi: S \to \mathbb R^4$ from a real surface is called Lagrangian if $\phi^* \omega =0$, where $\omega$ is the standard symplectic form on $\mathbb R^4$. Gromov's theorem (1985) on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of $\mathbb C^n$ provides topological obstructions for Lagrangian embeddings (or immersions) of compact surfaces. However, Givental (1986) showed that such maps always exist if we allow singularities that are either self-intersections or open Whitney umbrellas.

Existence of holomorphic discs attached to a submanifold $X$ of $\mathbb C^n$ is related to the question of polynomial convexity of $X$. I will discuss the joint work with Alexandre Sukhov, where we show that if a Lagrangian surface $X \subset \mathbb C^2$ has an isolated singularity which is a Whitney umbrella, then near the singularity the surface $X$ is locally polynomially convex.

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.

In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.

This is joint work with Chris Allday and Volker Puppe.