Dualities and dual pairs in Heyting algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction

Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts

We study elliptically fibered K3 surfaces, with sections, in toric Fano threefolds which satisfy certain combinatorial properties relevant to F-theory/Heterotic duality. We show that some of these conditions are equivalent to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the elliptic K3 surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.Comment: References adde

Learning architectures and negotiation of meaning in European trade unions

As networked learning becomes familiar at all levels and in all sectors of education, cross-fertilisation of innovative methods can usefully inform the lifelong learning agenda. Development of the pedagogical architectures and social processes, which afford learning, is a major challenge for educators as they strive to address the varied needs of a wide range of learners. One area in which this challenge is taken very seriously is that of trade unions, where recent large-scale projects have aimed to address many of these issues at a European level. This paper describes one such project, which targeted not only online courses, but also the wider political potential of virtual communities of practice. By analysing findings in relation to Wengers learning architecture, the paper investigates further the relationships between communities of practice and communities of learners in the trade union context. The findings suggest that a focus on these relationships rather than on the technologies that support them should inform future developments

Dualities in Convex Algebraic Geometry

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre's representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.Comment: 48 pages, 11 figure