Álvaro Pelayo                                                    











A theory is more impressive the greater the simplicity of its premises, the more kinds of things it relates, and the more extended its area of applicability.

A. Einstein

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Summary of G3 : Group Actions


In 1982 Atiyah and independently Guillemin & Sternberg proved one of the most influential results in the equivariant symplectic theory of group actions: the momentum map (introduced by Lie, Kostant, and Souriau) image of the action of a compact connected abelian Lie group (i.e. a torus) equals the convex hull of the images of the action fixed points, in particular it is convex. A school was created around this topic which flourished, and the area of Hamiltonian group actions is now a mainstream topic in equivariant geometry.

Several classification results have been achieved since, which involve some form of the momentum map image as an invariant. If one drops the condition on the action to be Hamiltonian (that is we no longer have a momentum map) the convexity result does not hold. However, it is still possibly to classify such actions in some cases which are of interest both in complex algebraic geometry (for instance the Kodaira variety falls into this class), and in symplectic topology. I worked on several of these cases.

Shortly after the work of Atiyah and Guillemin & Sternberg, Duistermaat and Heckman (1982) proved the DH Theorem which has extensive applications in geometry and analysis. This influential result expresses the Fourier transform of the push-forward of the Liouville measure by the momentum map (DH measure) of a Hamiltonian torus action in terms of the behavior around the fixed points (assumed isolated).

In 1984 Atiyah and Bott generalized the DH Theorem (to include non isolated points) and presented it using the far reaching setting of equivariant cohomology. Several of the problems I have worked on G3 concern the so called logarithmic concavity conjecture, which asks when the logarithm of the DH measure is a concave function. Although this is not in general true (proved by Karshon), it holds in important cases.

I have worked on: