A Localization Theorem for Finite W-algebras

Following the work of Beilinson-Bernstein and Kashiwara-Rouquier, we give a geometric interpretation of certain categories of modules over the finite W-algebra. As an application we reprove the Skryabin equivalence.Comment: Preliminary Version. Comments Welcom

\u27Dear Colleague\u27 letter

\u27Dear Colleague\u27 letter for the September 2006 briefing on Children\u27s Well-Being and the Role of Workplace Flexibility for Parents event. Prepared on behalf of Workplace Flexibility 2010 by Senator Mike DeWine and Senator Christopher J. Dodd

Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.Comment: Version 2. A number of minor edits and corrections. Comments welcom

Minimal Betti Numbers

We give conditions for determining the extremal behavior for the (graded) Betti numbers of squarefree monomial ideals. For the case of non-unique minima, we give several conditions which we use to produce infinite families, exponentially growing with dimension, of Hilbert functions which have no smallest (graded) Betti numbers among squarefree monomial ideals and all ideals. For the case of unique minima, we give two families of Hilbert functions, one with exponential and one with linear growth as dimension grows, that have unique minimal Betti numbers among squarefree monomial ideals

Nuremberg: Past, Present and Future

Through historical and personal reflection, these remarks detail the unprecedented conditions of the Nuremberg trials. In addition, the paper explores the historical and modern impact of the principles of the Nuremberg trials