9 research outputs found
Perverse poisson sheaves on the nilpotent cone
For a reductive complex algebraic group, the associated nilpotent cone is the variety of nilpotent elements in the corresponding Lie algebra. Understanding the nilpotent cone is of central importance in representation theory. For example, the nilpotent cone plays a prominent role in classifying the representations of finite groups of Lie type. More recently, the nilpotent cone has been shown to have a close connection with the affine flag variety and this has been exploited in the Geometric Langlands Program. We make use of the following important fact. The nilpotent cone is invariant under the coadjoint action of G on the dual Lie algebra and admits a canonical Poisson structure which is compatible in a strong way with the action of G. We exploit this connection to develop a theory of perverse sheaves on the nilpotent cone that is suitable for the G-equivariant Poisson setting. Building on the work of Beilinson--Bernstein--Deligne and Deligne--Bezrukavnikov, we define a new category, the equivariant Poisson derived category and endow it with a new semiorthogonal filtration, the perverse Poisson t-structure. In order to construct the perverse Poisson t-structure, we also prove an axiomatized gluing theorem for semiorthogonal filtrations in the general setting of triangulated categories which generalizes the construction of the perverse coherent sheaves of Deligne--Bezrukavnikov
A categorical foundation for Bayesian probability
Given two measurable spaces and with countably generated
-algebras, a perfect prior probability measure on and a
sampling distribution , there is a corresponding inference
map which is unique up to a set of measure zero. Thus,
given a data measurement , a posterior probability
can be computed. This procedure is iterative: with
each updated probability , we obtain a new joint distribution which in
turn yields a new inference map and the process repeats with each
additional measurement. The main result uses an existence theorem for regular
conditional probabilities by Faden, which holds in more generality than the
setting of Polish spaces. This less stringent setting then allows for
non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as
non finite) spaces, and also provides for a common framework for decision
theory and Bayesian probability.Comment: 15 pages; revised setting to more clearly explain how to incorporate
perfect measures and the Giry monad; to appear in Applied Categorical
Structure
AN INTRODUCTION TO PEAKLESS FUNCTIONS
Abstract. Convex functions have been studied both from a geometrical and analytical standpoint. The goal of this paper is to introduce the idea of peaklessness as a generalization of convexity
FALK’S DESCRIPTION OF THE FIRST RESONANCE VARIETY
In the late nineties, M. Falk described in [1] a method for computing the first resonance variety of an arrangement based solely on the combinatorics of the arrangement. Since this resonance variety is an invariant of the arrangement, this can be very helpful in distinguishing arrangements that otherwise look the same unde