Quantizations of conical symplectic resolutions I: local and global structure

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.Comment: 74 pages; v4: minor changes based on referee comments; v5: minor adjustment in numbering to match published versio

Hypertoric category O

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a "hypertoric enveloping algebra." We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, compute its center, and study its cell structure. We also consider a collection of derived auto-equivalences analogous to the shuffling and twisting functors for BGG category O.Comment: 65 pages, TikZ figures (PDF is recommended; DVI will not display correctly on all computers); v3: switched terminology for twisting and shuffling; final version; v4: small correction in definition of standard module

Inducement Prizes and Innovation.

We examine the effect of prizes on innovation using data on awards for technological development offered by the Royal Agricultural Society of England at annual competitions between 1839 and 1939. We find large effects of the prizes on competitive entry and we also detect an impact of the prizes on the quality of contemporaneous patents, especially when prize categories were set by a strict rotation scheme, thereby mitigating the potentially confounding effect that they targeted only “hot” technology sectors. Prizes encouraged competition and medals were more important than monetary awards. The boost to innovation we observe cannot be explained by the re-direction of existing inventive activity.Awards; Patents; Contests.

Dynamical Interactions with Electronic Instruments

This paper examines electronic instruments that incorporate dynamical systems, where the behaviour of the instrument depends not only upon the immediate input to the instrument, but also on the past input. Five instruments are presented as case studies: Michel Waisvisz’ Crackle-box, Dylan Menzies’ Spiro, no-input mixing desk, the author’s Feedback Joypad, and microphone-loudspeaker feedback. Links are suggested between the sonic affordances of each instrument and the dynamical mechanisms embedded in them. These affordances are contrasted with those of non-dynamical instruments such as the Theremin and sample-based instruments. This is discussed in the context of contemporary, material-oriented approaches to composition and particularly to free improvisation where elements such as unpredictability and instability are often of interest, and the process of exploration and discovery is an important part of the practice

Gale duality and Koszul duality

Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.Comment: 55 pages; v2 contains significant revisions to proofs and to some of the results. Section 7 has been deleted; that material will be incorporated into a later paper by the same author