This is a survey of the relationship between C *-algebraic deformation quan-tization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C *-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl–Moyal quantization may be described using the ta...

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid G is equivalent to the continuity of the implied convolution product f * g for all pairs of functions / and g. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual C*-algebra associated with a Haar measure on G cannot, in general, be constr...

These notes are an attempt to structure the author’s thoughts and conjectures related to higher relations and their quotients. We define the notion of a Yoneda Groupoid formally, the name of which is inspired by the relation to the Yoneda lemma, and show how a weak ω groupoid structure can be extracted. We also prove that, in the presence of bracket types (in the sense of Awodey & Bauer [3]), e...

In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox’s complete spreads. A ...

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

Five hundred million travelers enter the US yearly at a Port of Entry (POE) after an Immigration & Naturalization Service (INS) interview. We describe a general method for sampling from a flow, and summarize results from random reinspections of travelers at 20 POEs. Analyses reveal that 47 in 5614 travelers (0.8% ± 0.24%) were erroneously granted entry. Results suggest INS intercepts 9.3% to 16...

Problem Oriented Engineering (POE) is a Gentzen-style ‘natural’ framework for engineering design. As such, POE supports rather than guides its user as to the particular sequence of design steps that will be used; the sequencing is user determined as that most appropriate to the context of application. In this paper, however, we suggest a sequencing of steps and interactions with stake-holders t...

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.