Affine Buildings and Tropical Convexity

The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure

\Lambda-buildings and base change functors

We prove an analog of the base change functor of \Lambda-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine building is closed under ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.Comment: revised version, 29 pages, to appear in Geom. Dedicat

Economic impact of reduced mortality due to increased cycling.

Increasing regular physical activity is a key public health goal. One strategy is to change the physical environment to encourage walking and cycling, requiring partnerships with the transport and urban planning sectors. Economic evaluation is an important factor in the decision to fund any new transport scheme, but techniques for assessing the economic value of the health benefits of cycling and walking have tended to be less sophisticated than the approaches used for assessing other benefits. This study aimed to produce a practical tool for estimating the economic impact of reduced mortality due to increased cycling. The tool was intended to be transparent, easy to use, reliable, and based on conservative assumptions and default values, which can be used in the absence of local data. It addressed the question: For a given volume of cycling within a defined population, what is the economic value of the health benefits? The authors used published estimates of relative risk of all-cause mortality among regular cyclists and applied these to levels of cycling defined by the user to produce an estimate of the number of deaths potentially averted because of regular cycling. The tool then calculates the economic value of the deaths averted using the "value of a statistical life." The outputs of the tool support decision making on cycle infrastructure or policies, or can be used as part of an integrated economic appraisal. The tool's unique contribution is that it takes a public health approach to a transport problem, addresses it in epidemiologic terms, and places the results back into the transport context. Examples of its use include its adoption by the English and Swedish departments of transport as the recommended methodologic approach for estimating the health impact of walking and cycling

Isometric embeddings of Johnson graphs in Grassmann graphs

Let $V$ be an $n$-dimensional vector space ($4\le n <\infty$) and let ${\mathcal G}_{k}(V)$ be the Grassmannian formed by all $k$-dimensional subspaces of $V$. The corresponding Grassmann graph will be denoted by $\Gamma_{k}(V)$. We describe all isometric embeddings of Johnson graphs $J(l,m)$, $1<m<l-1$ in $\Gamma_{k}(V)$, $1<k<n-1$ (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of $J(n,k)$ in $\Gamma_{k}(V)$ is an apartment of ${\mathcal G}_{k}(V)$ if and only if $n=2k$. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in $\Gamma_{k}(V)$, $1<k<n-1$.Comment: New version -- 14 pages accepted to Journal of Algebraic Combinatoric

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all Q-rank one lattices in semisimple groups of R-rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.Comment: 7 figure