Pi Mu Epsilon Lecture Series

The Pi Mu Epsilon lecture series is made possible through the support of the UNL Mathematics Department and the Nebraska Alpha Chapter of Pi Mu Epsilon.

For more information contact Tri Lai.

Pi Mu Epsilon Lecture 2025
Ismar Volic, Wellesley College 

Friday, March 7, 2025 
4:00 pm, 115 Avery Hall 

The Shape of Democracy: Topological Methods in Social Choice Theory 

In recent years, mathematicians have applied sophisticated techniques to shed new light on practices like voting, districting, and apportionment of legislative seats, providing novel points of view on which of them work well and which do not. Topology has been remarkably successful in this endeavor. One of the reasons is that special kinds of combinatorial geometric objects called simplicial complexes are a natural tool for modeling interactions in political systems and revealing their deeper underlying structures. I will discuss how simplicial complexes can be used to study political coalitions, with some basic topological constructions then easily translating into political situations such as merging of parties, introduction of mediators, or delegation of power. The topological point of view also supplies a refined take on game-theoretic notions like the Banzhaf and Shapley-Shubik power indices of agents in a political system. I will also present a generalization to hypergraphs which captures an even richer collection of political dynamics concepts. Time permitting, a recasting of some classical results from social choice theory in topological and category-theoretic terms will also be mentioned. No prior knowledge of topology, geometry, or political science is required.