A unified approach on Springer fibers in the hook, two-row and two-column cases

We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections.Comment: 42 page

ERrrr…Where are the Progenitors? Hormone Receptors and Mammary Cell Heterogeneity

The mammary epithelium is a highly heterogenous and dynamic tissue that includes a range of cell types with varying levels of proliferative capacity and differentiation potential, from stem to committed progenitor and mature cells. Generation of mature cells through expansion and specification of immature precursors is driven by hormonal and local stimuli. Intriguingly, although circulating hormones can be directly sensed only by a subset of mammary cells, they also regulate the behaviour of cells lacking their cognate receptors through paracrine mechanisms. Thus, mapping the hormonal signalling network on to the emerging mammary cell hierarchy appears to be a difficult task. Nevertheless, a first step towards a better understanding is the characterization of the hormone receptor expression pattern across individual cell types in the mammary epithelium. Here we review the most relevant findings on the cellular distribution of hormone receptors in the mammary gland, taking into account differences between mice and humans, the methods employed to assess receptor expression as well as the variety of approaches used to resolve the mammary cell heterogeneity