2,965 research outputs found
Tits Geometry and Positive Curvature
There is a well known link between (maximal) polar representations and
isotropy representations of symmetric spaces provided by Dadok. Moreover, the
theory by Tits and Burns-Spatzier provides a link between irreducible symmetric
spaces of non-compact type of rank at least three and irreducible topological
spherical buildings of rank at least three.
We discover and exploit a rich structure of a (connected) chamber system of
finite (Coxeter) type M associated with any polar action of cohomogeneity at
least two on any simply connected closed positively curved manifold. Although
this chamber system is typically not a Tits geometry of type M, we prove that
in all cases but two that its universal Tits cover indeed is a building. We
construct a topology on this universal cover making it into a compact spherical
building in the sense of Burns and Spatzier. Using this structure we classify
up to equivariant diffeomorphism all polar actions on (simply connected)
positively curved manifolds of cohomogeneity at least two.Comment: 43 pages, to appear in Acta Mathematic
Atomic Classification of 6D SCFTs
We use F-theory to classify possibly all six-dimensional superconformal field
theories (SCFTs). This involves a two step process: We first classify all
possible tensor branches allowed in F-theory (which correspond to allowed
collections of contractible spheres) and then classify all possible
configurations of seven-branes wrapped over them. We describe the first step in
terms of "atoms" joined into "radicals" and "molecules," using an analogy from
chemistry. The second step has an interpretation via quiver-type gauge theories
constrained by anomaly cancellation. A very surprising outcome of our analysis
is that all of these tensor branches have the structure of a linear chain of
intersecting spheres with a small amount of possible decoration at the two
ends. The resulting structure of these SCFTs takes the form of a generalized
quiver consisting of ADE-type nodes joined by conformal matter. A collection of
highly non-trivial examples involving E8 small instantons probing an ADE
singularity is shown to have an F-theory realization. This yields a
classification of homomorphisms from ADE subgroups of SU(2) into E8 in purely
geometric terms, largely matching results obtained in the mathematics
literature from an intricate group theory analysis.Comment: v3: 123 pages, 23 figures, typos corrected. Included with the
submission are the Mathematica notebooks "Bases.nb" and
"Fiber_Enhancements.nb
From isovists to visibility graphs: a methodology for the analysis of architectural space
An isovist, or viewshed, is the area in a spatial environment directly visible from a location within the space. Here we show how a set of isovists can be used to generate a graph of mutual visibility between locations. We demonstrate that this graph can also be constructed without reference to isovists and that we are in fact invoking the more general concept of a visibility graph. Using the visibility graph, we can extend both isovist and current graph-based analyses of architectural space to form a new methodology for the investigation of configurational relationships. The measurement of local and global characteristics of the graph, for each vertex or for the system as a whole, is of interest from an architectural perspective, allowing us to describe a configuration with reference to accessibility and visibility, to compare from location to location within a system, and to compare systems with different geometries. Finally we show that visibility graph properties may be closely related to manifestations of spatial perception, such as way-finding, movement, and space use
Urban identity through quantifiable spatial attributes: coherence and dispersion of local identity through the automated comparative analysis of building block plans
This analysis investigates whether and to what degree quantifiable spatial attrib-utes, as expressed in plan representations, can capture elements related to the ex-perience of spatial identity. By combining different methods of shape and spatial analysis it attempts to quantify spatial attributes, predominantly derived from plans, in order to illustrate patterns of interrelations between spaces through an ob-jective automated process. The study focuses on the scale of the urban block as the basic modular unit for the formation of urban configurations and the issue of spa-tial identity is perceived through consistency and differentiation within and amongst urban neighbourhoods
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