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

RUI：三角剖分，集合交叉点，公平划分和投票（DMS - 1002938）研究者之前的工作已经将组合拓扑和离散几何的方法引入到公平划分问题和投票问题的研究中。目前的项目将支持这些工具背后的数学发展，以及他之前工作所激发的几个组合问题的解决方案，包括：(1)立方体和单纯形的三角剖分研究，(2)组合不动点定理和构造解的进一步发展，以及(3)集相交定理的发展及其在社会选择理论和公平划分中的相关应用。本项目还将支持本科生积极参与本研究。非正式地说，“公平分配”问题问的是：我们如何在各方之间分配一组商品，使每个人都能根据某种公平的概念得到满足。社会选择理论提出的问题是：一个社会如何做出群体选择（例如，在选举中），以最好地汇集所有个人的偏好？公平和社会选择问题是政治学家、经济学家和博弈论家感兴趣的问题，并激发了有趣的数学问题。每个人的偏好空间和偏好集通常是自然的几何集合（例如，凸的、连通的、多面体的），期望的解决方案通常在这些集合的交叉点。该项目旨在证明与社会科学中涉及投票和公平的重要问题有直接关系的数学定理（例如，关于多面体的集合交叉点和三角形）。

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.