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The East Midlands in 2010: sub-regional profiles
The East Midlands in 2010 is the updated evidence base that supports the regional economic strategy, A Flourishing Region. This document presents detailed analysis of the East Midlands' sub-regions.
The Topology of Parabolic Character Varieties of Free Groups
Let G be a complex affine algebraic reductive group, and let K be a maximal
compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or
equal to 0, let X (respectively, Y) be the space of equivalence classes of
representations of the free group of m+n generators in G (respectively, K) such
that for each i between 1 and m, the image of the i-th free generator is
conjugate to h_i. These spaces are parabolic analogues of character varieties
of free groups. We prove that Y is a strong deformation retraction of X. In
particular, X and Y are homotopy equivalent. We also describe explicit examples
relating X to relative character varieties.Comment: 16 pages, version 2 includes minor revisions and some modified
proofs, accepted for publication in Geometriae Dedicat
Entrepreneurship, innovation and the triple helix model: evidence from Oxfordshire and Cambridgeshire
This paper focuses on how regions become entrepreneurial and the extent to which the actors in the triple helix model are dominant at particular stages in development. It uses the case studies of Oxfordshire and Cambridgeshire in the UK to explore this theme. Both can now be described as âregional triple helix spacesâ (Etzkowitz 2008), and form two points of the Golden Triangle of Oxford, Cambridge and London universities. As entrepreneurial regions, however, they differ in a number of respects. This is not surprising given their differing geo-historical contexts. However, by comparing the two similar counties but which have their own distinctive features we are able to explore different dynamics which lead to the inception, implementation, consolidation and renewal (Etzkowitz and Klofsten 2005) of regions characterised by very high levels of technology-based entrepreneurship
Recommended from our members
The East Midlands in 2010
The East Midlands in 2010 is the updated evidence base that supports the regional economic strategy, A Flourishing Region. It presents a statistical portrait of the region covering demography, housing, economy & productivity, the labour market, deprivation & economic inclusion, transport, infrastructure & development, the environment and the spatial structure of the economy along with a series of sub-regional profiles
The topology of moduli spaces of free group representations
For any complex affine reductive group G and a fixed choice of maximal
compact subgroup K, we show that the G-character variety of a free group
strongly deformation retracts to the corresponding K-character space, which is
a real semi-algebraic set. Combining this with constructive invariant theory
and classical topological methods, we show that the SL(3,C)-character variety
of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character
variety of a rank 3 free group is homotopic to a 6 sphere.Comment: 37 pages, 2 figures, version 2 corrects typos, generalizes context,
and adds a corollary, it is the accepted version to appear in Mathematische
Annale
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