Equivariant aspects in cohomology, K-theory and index theory

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The aim of the project is to study symmetries and their mathematical incarnations, group actions. We are mostly interested in manifolds with rotational symmetries, which correspond to actions of tori. Specifically, we will look at the effect of the symmetry on algebraic invariants of these spaces, notably on equivariant cohomology and equivariant K-theory. A major theme is centred around syzygies. The notion of syzygy from commutative algebra interpolates between torsion-free and free modules. In a joint project with Chris Allday (Hawaii) and Volker Puppe (Konstanz) I have initiated the study of syzygies in the context of torus-equivariant cohomology. Our approach unifies and generalizes many results concerning the freeness or torsion-freeness of equivariant cohomology, and it provides a better understanding of the equivariant Poincaré pairing and the so-called "GKM method" for computing equivariant cohomology. One objective of the present project is to extend this theory to other groups (including p-tori and non-commutative groups) and also to equivariant K-theory. Part of this will be carried out with Volker Puppe and Reyer Sjamaar (Cornell). I will also investigate a new class of torus manifolds which are emerging from the previous work and generalize the "polygon spaces" studied by many authors. These spaces are intersections of real quadrics, which provides connections to singularity theory, complex analysis and non-commutative geometry. Many spaces are what is called "equivariantly formal"; they often come up in combinatorics and representation theory. Here I want to find a common proof of equivariant formality for the two main classes of examples. Another objective is to explore the connections between our work on syzygies and recent work of De Concini, Procesi and Vergne about vector partition functions, splines and the equivariant index. Both are based on the same work of Atiyah and Singer. While there are similar patterns, the precise connection between the two is not understood so far. A smaller sub-project focuses on a topological approach to the Galois theory of Pythagorean fields via actions of 2-tori.