Deviations from the Circular Law

Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe fluctuations about this {\em Circular Law}. First we obtain finite $N$ formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics

Formality for the nilpotent cone and a derived Springer correspondence

Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaves and the derived category of differential graded modules over a dg-ring related to the Weyl group.Comment: 29 pages, a few corrections and changes in notatio

Seeing Double: Assessing Kendall Walton’s Views on Painting and Photography

In this paper I consider Kendall Walton’s provocative views on the visual arts, including his approaches to understanding both figurative and nonfigurative painting. I introduce his central notion of fictionality, illustrating its advantages in explaining the phenomenon of ‘perceptual twofoldness’. I argue that Walton’s position treats abstract artwork reductively, and I outline two essential components of our aesthetic encounters with the nonfigurative that Walton excludes. I then offer some criticisms of his commitment to photographic realism, emphasising its theoretical inconsistencies with his account of representation. My own proposal is that in our apprehension of non-figurative artworks, our attention is drawn to the underlying structures of both emotive and perceptual experience. In this way, paintings, particularly abstract ones, disclose human cognition in a manner that makes fictionality an inappropriate tool for their analysis

Debating the key issues

A note on the W G Hart Legal Workshop 1999 entitled “Legal regulation of the employment relation”. Published in the First Page feature of Amicus Curiae - Journal of the Institute of Advanced Legal Studies and its Society for Advanced Legal Studies. The Journal is produced by the Society for Advanced Legal Studies at the Institute of Advanced Legal Studies, University of London

Small deviations for beta ensembles

We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately

Perverse Sheaves on the Nilpotent Cone and Lusztig's Generalized Springer Correspondence

In this note, we consider perverse sheaves on the nilpotent cone. We prove orthogonality relations for the equivariant category of sheaves on the nilpotent cone in a method similar to Lusztig's for character sheaves. We also consider cleanness for cuspidal perverse sheaves and the generalized Lusztig--Shoji algorithm.Comment: 18 pages, minor typos correcte