Building on previous work, this paper extends the modeling of political structures from simplicial complexes to hypergraphs. This allows the analysis of more complex political dynamics where agents who are willing to form coalitions contain subsets that would not necessarily form coalitions themselves. We extend topological constructions such as wedge, cone, and collapse from simplicial complexes to hypergraphs and use them to study mergers, mediators, and power delegation in political structures. Concepts such as agent viability and system stability are generalized to the hypergraph context, alongside the introduction of the notion of local viability. Additionally, we use embedded homology of hypergraphs to analyze power concentration within political systems. Along the way, we introduce some new notions within the hypergraph framework that are of independent interest.

We use simplicial complexes to model weighted voting games where certain coalitions are considered unlikely or impossible. Expressions for Banzhaf and Shapley-Shubik power indices for such games in terms of the topology of simplicial complexes are provided. We calculate the indices in several examples of weighted voting games with unfeasible coalitions, including the U.S. Electoral College and the Parliament of Bosnia-Herzegovina.

Institute for Mathematics and Democracy The Institute for Mathematics and Democracy (IMD)1 was created to help cast light on the important role that mathematics plays in our political systems. Founded in 2019, it is housed at Wellesley College, a liberal arts institution with a strong interest in interdisciplinary work. The mission of IMD is to promote a deeper understanding of the mathematics that underlies many socioeconomic forces and political processes so that citizens can make informed political decisions and effect meaningful change. It supports research, teaching, and outreach in mathematics and democracy, and brings together educators and activists who recognize the importance of political quantitative literacy. It acts as a facilitator, a repository, and nexus to bring research and pedagogy of disparate individuals who synthesize mathematics and democracy in their research and teaching. The timeliness of these efforts is apparent as IMD is finding a wide and growing audience among students, researchers, educators, activists, and the public at large. Over thirty mathematicians now serve as scholars and affiliates for IMD, many of whom have pursued or supervised research, run summer programs, or created curricula at the intersection of politics and mathematics. The institute has also supported over twenty student researchers, almost half of them from underrepresented groups. The activities of IMD are rooted and motivated by mathematics, but they lie at the unique interdisciplinary interface of mathematics, statistics, political science, economics, and history. Not only do they fill an evident educational and research gap, but they serve a social purpose: political

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of high-dimensional string links. We conjecture that graphing is injective on all rational homotopy classes.

Abstract This article gives an overview of the applications of algebraic topology methods in signal processing. We explain how the notions and invariants such as (co)chain complexes and (co)homology of simplicial complexes can be used to gain insight into higher-order interactions of signals. The discussion begins with some basic ideas in classical circuits, continues with signals over graphs and simplicial complexes, and culminates with an overview of sheaf theory and the connections between sheaf cohomology and signal processing.

We apply topological data analysis, specifically the Mapper algorithm, to the U.S. COVID-19 data. The resulting Mapper graphs provide visualizations of the pandemic that are more complete than those supplied by other, more standard methods. They encode a variety of geometric features of the data cloud created from geographic information, time progression, and the number of COVID-19 cases. They reflect the development of the pandemic across all of the U.S. and capture the growth rates as well as the regional prominence of hot-spots. The Mapper graphs allow for easy comparisons across time and space and have the potential of becoming a useful predictive tool for the spread of the coronavirus.