Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

Hyperplane Sections and Derived Categories

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

Actions of vector groupoids

In this work we deal with actions of vector groupoid which is a new concept in the literature‎. ‎After we give the definition of the action of a vector groupoid on a vector space‎, ‎we obtain some results related to actions of vector groupoids‎. ‎We also apply some characterizations of the category and groupoid theory to vector groupoids‎. ‎As the second part of the work‎, ‎we define the notion...

Picard Groupoids and Spectra

The goal of this lecture is to explain in detail the statement that a Picard groupoid is “the same thing as” an Ω-spectrum E with πi(E) = 0 for i 6= 0, 1. Along the way we introduce the notion of a (very special) Γ-space, which provides one of the possible ways of formalising the concept of an “abelian group structure defined up to all the higher homotopies”, and we present an approach to K-the...

Picard Groups for Derived Module Categories

Random Walk and Hyperplane Arrangements

Let C be the set of chambers of a real hyperplane arrangement. We study a random walk on C introduced by Bidigare, Hanlon, and Rockmore. This includes various shuuing schemes used in computer science, biology, and card games. It also includes random walks on zonotopes and zonotopal tilings. We nd the stationary distributions of these Markov chains, give good bounds on the rate of convergence to...