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

Examination of the effect of reduction of probiotic species Lactobacillus due to broad spectrum antibiotic treatment on oral tolerance

Antibiotic usage is on the rise in industrialized countries and as a result the prevalence of autoimmune and atopic diseases has risen. The use of antibiotics is connected to a depletion of the microflora located within the gastrointestinal tract. The microflora contains a variety of different bacterial species, including some that are probiotic species, Lactobacilli and Bifidobacteria, which have a beneficial effect on the host. Probiotic species of bacteria are important for immune function due to their ability to regulate oral tolerance, a state of unresponsiveness to antigens that have been introduced orally to the host. The goal of this study was to assess the effect of broad spectrum antibiotic treatment on the probiotic species Lactobacilli and the resulting effect on the induction of oral tolerization to the antigen ovalbumin.Department of BiologyThesis (M.S.

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

Complex determinantal processes and H1 noise

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes with intensity rho dnu, where nu is the corresponding invariant measure. We show that as rho converges to infinity, after centering, these processes converge to invariant H1 noise. More precisely, for all functions f in the interesection of H1(nu) and L1(nu) the distribution of sum f(z) - rho/pi integral f dnu converges to Gaussian with mean 0 and variance given by ||f||_H1^2 / (4 pi).Comment: 22 pages, 1 figur

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

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.Comment: Published in at http://dx.doi.org/10.1214/13-AAP958 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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