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，它们通常体现了形势的几何形状，而代数结构往往揭示了重要的组织原则。例如，加法和乘法群产生普通上同调和K-理论，而椭圆曲线产生椭圆上同调。这些理论都集中在流形几何的各个方面，体现在签名、A-HAT亏格和椭圆亏格上。不同的是，有理上同调理论本身是非常简单的：表示对象的范畴等价于分次有理向量空间的范畴，并且所有上同调理论都是普通的。PI猜想，对于每个紧李群G，都有一个交换范畴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度等于G的秩.实际上，这使得一个人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，人们可以通过在A(G)中写下一个对象来构造上同调理论：这就是构造圆等变椭圆上同调的方法，并且可以构造等变sigma亏格。这一建议是为了扩展已知猜想的群类，并以不同的方式利用其结果：(1)通过分类上同调理论(2)通过研究它们所体现的普适De Rham模型(3)通过研究一般G(4)的G-等变椭圆上同调来证明更高亏格的曲线如何产生上同调理论，并利用与theta函数相关的亏格。(5)通过计算一系列Toric簇的几何形式的上同调理论。

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.