RUI: Triangulations, Set Intersections, Fair Division, and Voting

Harvey Mudd College, Claremont CA

Investigators

Abstract

RUI: Triangulations, Set Intersections, Fair Division, and Voting (DMS - 1002938) The investigator's prior work has introduced methods from combinatorial topology and discrete geometry to the study of fair division questions and voting problems. The current project will support the the development of the mathematics behinds these tools and the solution of several combinatorial questions that have been motivated by his prior work, including: (1) the study of triangulations of cubes and simplotopes, (2) the further development of combinatorial fixed point theorems and constructive solutions, and (3) the development of set intersection theorems and associated applications in social choice theory and fair division. This project will also support the active participation of undergraduates in this research. Informally speaking, a "fair division" problem asks: how can we divide a set of goods among parties in such a way that each can be satisfied according to some notion of fairness. Social choice theory asks: how does a society make a group choice (e.g., in an election) in a way that best aggregates the preferences of all the individuals? Questions of fairness and social choice are of interest to political scientists, economists, and game theorists, and motivate interesting mathematical questions. The space of preferences and the preference sets of each person are often naturally geometric sets (e.g., convex, connected, polyhedral), and the desired solution is often at the intersection of such sets. This project aims to prove mathematical theorems (e.g., about set intersections and triangulations of polyhedra) that have direct bearing on important problems in the social sciences involving voting and fairness.