Crepant resolutions of a Slodowy slice in a nilpotent orbit closure in slN(C)\mathfrak{sl}_N(\mathbb{C})

One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$ can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.Comment: 22 page

Spontaneous symmetry breaking and the formation of columnar structures in the primary visual cortex II --- Local organization of orientation modules

Self-organization of orientation-wheels observed in the visual cortex is discussed from the view point of topology. We argue in a generalized model of Kohonen's feature mappings that the existence of the orientation-wheels is a consequence of Riemann-Hurwitz formula from topology. In the same line, we estimate partition function of the model, and show that regardless of the total number N of the orientation-modules per hypercolumn the modules are self-organized, without fine-tuning of parameters, into definite number of orientation-wheels per hypercolumn if N is large.Comment: 36 pages Latex2.09 and eps figures. Needs epsf.sty, amssym.def, and Type1 TeX-fonts of BlueSky Res. for correct typo in graphics file