Nick Wright

Hierarchical spectral clustering of power grids

A power transmission system can be represented by a network with nodes and links representing buses and electrical transmission lines, respectively. Each line can be given a weight, representing some electrical property of the line, such as line admittance or average power flow at a given time. We use a hierarchical spectral clustering methodology to reveal the internal connectivity structure of such a network. Spectral clustering uses the eigenvalues and eigenvectors of a matrix associated to the network, it is computationally very efficient, and it works for any choice of weights. When using line admittances, it reveals the static internal connectivity structure of the underlying network, while using power flows highlights islands with minimal power flow disruption, and thus it naturally relates to controlled islanding. Our methodology goes beyond the standard k-means algorithm by instead representing the complete network substructure as a dendrogram. We provide a thorough theoretical justification of the use of spectral clustering in power systems, and we include the results of our methodology for several test systems of small, medium and large size, including a model of the Great Britain transmission network.

Finite asymptotic dimension for CAT(0) cube complexes

We prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups.

AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-\v{C}ech compactification of the group we obtain a cohomological characterisation of exactness for the group, answering a question of Higson.

A cohomological characterisation of Yu's property A for metric spaces

We introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invari- ant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A.

Spaces of Graphs, Boundary Groupoids and the Coarse Baum-Connes Conjecture

We introduce a new variant of the coarse Baum–Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum–Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum–Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum–Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discrete counterexamples to the coarse Baum–Connes conjecture in an elementary way.

Pairings, duality, amenability and bounded cohomology

We give a new perspective on the homological characterisations of amenability given by Johnson and Ringrose in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.

Expanders and property A

We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu’s property A.

Coarse medians and Property A

We prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have property A.

Simultaneous metrizability of coarse spaces

A metric space can be naturally endowed with both a topology and a coarse structure. We examine the converse to this. Given a topology and a coarse structure we give necessary and sufficient conditions for the existence of a metric giving rise to both of these. We conclude with an application to the construction of the coarse assembly map.

Partial translation algebras for trees

In [1] we introduced the notion of a partial translation C*-algebra for a discrete metric space. Here we demonstrate that several important classical C*-algebras and extensions arise naturally by considering partial translation algebras associated with subspaces of trees.