To any k-dimensional subspace of ℚn one can naturally associate a point in the Grassmannian Grn,k(ℝ) and two shapes of lattices of rank k and n−k respectively. These lattices originate by intersecting the k-dimensional subspace with the lattice ℤn. Using unipotent dynamics we prove simultaneous equidistribution of all of these objects under a congruence conditions when (k,n)≠(2,4).

Socio-diversity, the variety of human opinions, ideas, behaviors and styles, has profound implications for social systems. While it fuels innovation, productivity, and collective intelligence, it can also complicate communication and erode trust. So what mechanisms can influence it? This paper studies how fundamental characteristics of social networks can support or hinder socio-diversity. It employs models of cultural evolution, mathematical analysis, and numerical simulations. We find that pronounced inequalities in the distribution of connections obstruct socio-diversity. In contrast, the prevalence of close-knit communities, a scarcity of long-range connections, and a significant tie density tend to promote it. These results open new perspectives for understanding how to change social networks to sustain more socio-diversity and, thereby, societal innovation, collective intelligence, and productivity.