Rational equivariant cohomology theories

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Cohomology theories convert geometric problems to algebraic problems, often allowing solutions. Perhaps more significantly, they often embody the geometry of the situation and algebraic structures often expose important organizational principles. For instance the additive and multiplicative groups give rise to ordinary cohomology and K-theory, and elliptic curves give rise to elliptic cohomology. These theories each focus on aspects of the geometry of manifolds, embodied by the signature, A-hat genus and elliptic genus. Much of this geometry remains, even after rationalization.Non-equivariantly, rational cohomology theories themselves are very simple: the category of representing objects are equivalent to the category of graded rational vector spaces, and all cohomology theories are ordinary. The PI has conjectured that for each compact Lie group G, there is an abelian category A(G) so that the homotopy category of rational G-spectra is equivalent to the derived category of A (G): the conjecture describes various properties of A(G), and in particular asserts that its injective dimension is equal to the rank of G. In practical terms, this allowsone to make complete calculations, and one can classify all such cohomology theories. More important though, one can construct a cohomology theoryby writing down an object in A (G): this is how circle-equivariant elliptic cohomology was constructed, and the equivariant sigma genus can be constructed. This proposal is to extend the class of groups for which the conjecture is known and to exploit the result in various ways: (1) by classifying cohomology theories(2) by studying the universal de Rham model they embody(3) by studying G-equivariant elliptic cohomology for general G(4) by showing how curves of higher genus give rise to cohomology theories, and exploiting the genera associated to theta functions.(5) by calculating the cohomology theories in geometric terms for a range of toric varieties.