We initiate the study of simple games from the point of view of combinatorial topology. The starting premise is that the losing coalitions of a simple game can be identified with a simplicial complex. Various topological constructions and results from the theory of simplicial complexes then carry over to the setting of simple games. Examples are cone, join, and the Alexander dual, each of which have interpretations as familiar game-theoretic objects. We also provide some new topological results about simple games, most notably in applications of homology of simplicial complexes to weighted simple games. The exposition is introductory and largely self-contained, intended to inspire further work and point to what appears to be a wealth of potentially fruitful directions of investigation bridging game theory and topology.

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 simple games as well as weighted voting games where certain coalitions are considered impossible. Topological characterizations of various ideas from simple games are provided, as are the expressions for Banzhaf and Shapley-Shubik power indices for weighted games. 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.

Multiple Indicator Cluster Surveys (MICS), supported by UNICEF, are one of the most important global household survey programs that provide data on health and education of women and children. We analyze the Serbia 2014–2015 MICS dataset using topological data analysis which treats the data cloud as a topological space and extracts information about its intrinsic geometric properties. In particular, our analysis uses the Mapper algorithm, a dimension-reduction and clustering method which produces a graph from the data cloud. The resulting Mapper graph provides insight into various relationships between household wealth—as expressed by the wealth index, an important indicator extracted from the MICS data—and other parameters such as urban/rural setting, ownership of items, and prioritization of possessions. Among other uses, these findings can serve to inform policy by providing a hierarchy of essential amenities. They can also potentially be used to refine the wealth index or deepen our understanding of what it captures.