Rational equivariant cohomology theories

上同调理论将几何问题转化为代数问题，通常允许求解。也许更重要的是，它们通常体现了情况的几何形状，代数结构通常暴露了重要的组织原则。例如，加性群和乘性群引起普通上同调和k理论，椭圆曲线引起椭圆上同调。这些理论都集中在流形几何的各个方面，具体体现在特征，a帽格和椭圆格。即使在合理化之后，这种几何形状的大部分仍然存在。非等价地，有理上同调理论本身非常简单：表示对象的范畴等价于分级有理向量空间的范畴，所有的上同调理论都是普通的。PI猜想，对于每一个紧李群G，都存在一个阿贝尔范畴A(G)，使得有理G谱的同伦范畴等价于A(G)的派生范畴：这个猜想描述了A(G)的各种性质，特别是断言它的内射维数等于G的秩。实际上，这使得人们可以进行完全的计算，并且可以对所有这样的上同调理论进行分类。更重要的是，我们可以通过在a (G)中写下一个对象来构造上同调理论：这就是圆-等变椭圆上同调的构造方法，以及等变格的构造方法。这个建议是扩展已知猜想的群的类别，并以各种方式利用结果：(1)通过对上同调理论进行分类(2)通过研究它们所体现的通用de Rham模型(3)通过研究一般G的G等变椭圆上同调(4)通过展示高属曲线如何产生上同调理论，并利用与函数相关的属。(5)通过计算几何项上同调理论对一组环面变异。

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.