I will first introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This is a cohomology theory that refines the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred (by the algebraic geometers) in the cohomological study of real algebraic varieties. Since the construction of this bigraded cohomology and its associated characteristic classes relies on the sheaf exponential morphism, I will explain how to produce an arithmetic (or algebraic) variant of these cohomology groups, whose main advantage is toeliminate topological or transcendental conditions. I will conclude by comparing these two versions of bigraded cohomology.

I will define a notion of resolution of a proper action. Such resolutions always exist but are not canonical. However, for so-called polar actions I will describe a canonical construction of a resolution, which can be used to show that the leaf space has the structure of an orbifold. I will illustrate this construction with two examples: (i) the adjoint action, where it allows one to identify the classical Weyl group with the orbifold fundamental group; and (ii) toric manifolds, where the resolution can be described in terms of the real part of the toric manifold.

Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow generically performs at singular times. Shrinking Ricci solitons model these topological operations, and only the stable ones should arise generically.

I will present recent joint works with Olivier Biquard and Keaton Naff that determine the stability of all of the currently known shrinking Ricci solitons in dimension 4. The arguments use structural features unique to dimension four, in particular self-duality.

Choice of centers of blowup; descent in dimension; lexicographic decrease of invariant; transversality; obstructions in positive characteristic; resolution of planar vector fields.

First examples of resolutions; curves and surfaces; effect of blowups on polynomials; singularity invariants; lexicographic orderings.