Basic gerbe over non simply connected compact groups

We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of E. Meinrenken \cite{Meinr} restricted to the case of simply connected groups.Comment: 27 pages, latex, 8 incorporated figure

Strictification of etale stacky Lie groups

We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.Comment: 25 page

On the representation theory of Galois and Atomic Topoi

We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic]. These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups. We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of "points". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi). We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.Comment: This is a revised version of arXiv.org/math.CT/02008222 to appear in JPA

Managed ecosystems of networked objects

Small embedded devices such as sensors and actuators will become the cornerstone of the Future Internet. To this end, generic, open and secure communication and service platforms are needed in order to be able to exploit the new business opportunities these devices bring. In this paper, we evaluate the current efforts to integrate sensors and actuators into the Internet and identify the limitations at the level of cooperation of these Internet-connected objects and the possible intelligence at the end points. As a solution, we propose the concept of Managed Ecosystem of Networked Objects, which aims to create a smart network architecture for groups of Internet-connected objects by combining network virtualization and clean-slate end-to-end protocol design. The concept maps to many real-life scenarios and should empower application developers to use sensor data in an easy and natural way. At the same time, the concept introduces many new challenging research problems, but their realization could offer a meaningful contribution to the realization of the Internet of Things

Wonderful varieties of type D

Let G be a complex connected semisimple group, whose simple components have type A or D. We prove that wonderful G-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type A; thanks to another result of Luna, this implies also the classification of all spherical G-varieties for the groups G we are considering. For these G we also prove the smoothness of the embedding of Demazure.Comment: 60 pages, AMSLaTeX, 11 eps file

Minimal Envy and Popular Matchings

We study ex-post fairness in the object allocation problem where objects are valuable and commonly owned. A matching is fair from individual perspective if it has only inevitable envy towards agents who received most preferred objects -- minimal envy matching. A matching is fair from social perspective if it is supported by majority against any other matching -- popular matching. Surprisingly, the two perspectives give the same outcome: when a popular matching exists it is equivalent to a minimal envy matching. We show the equivalence between global and local popularity: a matching is popular if and only if there does not exist a group of size up to 3 agents that decides to exchange their objects by majority, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected to a popular matching where along the path groups of up to 3 agents exchange their objects by majority. A market where random groups exchange objects by majority converges to a popular matching given such matching exists. When popular matching might not exist we define most popular matching as a matching that is popular among the largest subset of agents. We show that each minimal envy matching is a most popular matching and propose a polynomial-time algorithm to find them