Computing (co)homology of a space gets a lot easier if the space can be decomposed into a (locally finite) disjoint union of cells, each isomorphic to a complex affine space. Luckily, many important spaces admit such decompositions: Grassmanians, Flag varieties, smooth compact toric varieties. In fact, this is not just a lucky coincidence. All these cell decompositions can be described as Bialiniki-Birula decompositions into stable subvarieties of fixed points under actions of complex tori.
In fact, in all these cases varieties carry actions of multidimentional complex tori. Therefore, instead of one cell decomposition, we get a family of decompositions depending on a choice of a one dimensional subgroup in the corresponding torus. However, due to topological invariance of homology, numbers of cells of a given dimension should not depend on the choice of the subgroup.
In this talk I will explore the above ideas on the example of the Hilbert scheme of n points on the complex plane. This is a smooth 2n-dimensional variety with a natural action of a two dimensional torus. The fixed points are naturally enumerated by partitions of n, and the dimensions of stable subvarieties are given by combinatorial statistics on partitions, depending on the choice of a one dimensional subgroup. It follows that these statistics are equidistributed, which is very non-trivial from the combinatorial point of view.
This seminar gives an introduction to this weeks "QGM research seminar" - the seminar is meant for Master and PhD students - all are welcome.