Higher holonomies: comparing two constructions

We compare two different constructions of higher dimensional parallel transport. On the one hand, there is the two dimensional parallel transport associated to 2-connections on 2-bundles studied by Baez-Schreiber, Faria Martins-Picken and Schreiber-Waldorf. On the other hand, there are the higher holonomies associated to flat superconnections as studied by Igusa, Block-Smith and Arias Abad-Schaetz. We first explain how by truncating the latter construction one obtains examples of the former. Then we prove that the 2-dimensional holonomies provided by the two approaches coincide.Comment: comments are welcome

The A∞ de Rham Theorem and Integration of Representations up to Homotopy

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an version of de Rham's theorem due to Gugenheim [15]. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa [17] and Block-Smith [6], to the case of certain differential graded manifold

Holonomies for connections with values in L-infinity algebras

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's A-infinity version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space, and to generalizations of the holonomy representations of braid groups.Comment: Two missing diagrams added. Comments still welcome

User Interface Design Recommendations for Small Unmanned Aircraft Systems (sUAS)

The number of small unmanned aircraft systems (sUAS) has dramatically increased in recent years. As a consequence, the number of incidents involving manned and unmanned aircraft has soared. For this reason, the Federal Aviation Administration has released a notice of proposed rulemaking to delineate the operational limitations for sUAS. Many efforts have been introduced to regulate the operations of these systems and educate operators. Despite these efforts, there are no clear standards related to the type of information that should be available to operators, or how this information should be conveyed during flight operations. For this reason we present a series of design recommendations in order to encourage a discussion within the aviation community about the need to develop visual displays that could aid sUAS users unfamiliar with the intricacies of a complex air system to operate in a safe and efficient manner

Deformations of Lie brackets and representations up to homotopy

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.Comment: 28 page