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

 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:

• Lagrangian Actions:

studying symplectic nonnecessarily Hamiltonian actions with Lagrangian orbits (which includes all Hamiltonian torus actions with maximal dimensional orbits); more generally study (and classify) symplectic actions whose orbits are coisotropic manifolds; such manifolds include the Kodaira variety (also known as Kodaira-Thurston manifold); the study of such actions involves classical geometry going back to the foundational results of Cartan, Haefliger, Koszul, and Weinstein as well as recent work of Benoist and others;

• Logarithmic Concavity Conjecture:

studying in which settings the logarithmic concavity conjecture holds; the main tool is Hodge theory;

• Kosnowski Conjecture:

this conjecture (1979) relates the existence of fixed points to the Hamiltonian character of a circle action on a compact manifold, when the fixed points are isolated. Full solutions are due to Frankel for compact Kahler manifolds and to McDuff for compact symplectic 4-manifolds. I have worked on the remaining cases of this conjecture from several angles with different collaborators: using Hodge theory, Atiyah-Bott localization in equivariant cohomology, and most recently methods originating in equivariant K-theory;

• Toric Poisson Geometry:

going back to the pioneer work of Weinstein in Poisson geometry, I have been working with my collaborators on extending to the Poisson setting the theory of symplectic toric manifolds;

• Symplectic Fiber Bundles:

proving that symplectic non-Hamiltonian actions whose orbits are symplectic can be viewed as symplectic orbifold bundles over an orbifold. The base of the orbifold and the monodromy of the flat connection over it (given by the symplectic orthogonal complements to the tangent spaces to the orbits), essentially classify them;

• Existence of invariant Kahler and complex structures:

Delzant
classified in 1988 Hamiltonian actions of n-tori on compact 2n-dimensional manifolds (for any natural number n); I extended this result to symplectic, non-necessarily Hamiltonian actions of tori when 2n=4. As a consequence, one can determine which compact symplectic 4-manifolds endowed with torus actions admit complex and/or Kahler structures which are also invariant under the action; this is done by using a dictionary betwen the symplectic and complex analytic theory, using Kodaira's classification of complex analytic surfaces.