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

芮：三角剖分、集合交集、公平分割和投票(DMS-1002938)研究者以前的工作介绍了从组合拓扑学和离散几何到公平分割问题和投票问题的研究方法。目前的项目将支持这些工具背后的数学发展和几个组合问题的解决，这些问题是由他以前的工作所激励的，包括：(1)立方体和单形的三角剖分的研究，(2)组合不动点定理和构造性解的进一步发展，以及(3)集合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。这个项目还将支持本科生积极参与这项研究。从非正式的角度讲，一个“公平分配”的问题是：如何在各方之间分配一套商品，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：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、经济学家和博弈论者感兴趣的问题，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)，而期望的解往往在这些集合的交点处。这个项目的目的是证明数学定理(例如，关于集合相交和多面体的三角剖分)，这些定理与社会科学中涉及投票和公平的重要问题有直接关系。

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.