My research project
Spatial variation of collective efficacy
What if residents of a neighbourhood could influence crime rates by their behaviour? This is what postulates the theory of collective efficacy. Collective efficacy is the conviction shared by a group of people that they can work together to successfully complete a specific task. The idea is that the difference in neighbourhoods’ inner structure leads to spatial variation in crime rates. Many models exist that show the negative link between collective efficacy and crime but the literature in studying the formation of patterns is still limited. I work on a novel discrete model of collective efficacy that allows for a mathematical investigation of neighbourhood and resource effects on the formation of collective efficacy and transitions between different regions of collective efficacy. I also study its continuum version and explore more deeply the variation of collective efficacy depending on the neighbourhood definition.
We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and the first rational Čech cohomology is d-dimensional. We construct examples of such “homological Pisot” substitutions whose tiling flows do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.