RUI: Combinatorial fixed point theorems, polytopes, and preference sets

RUI：组合不动点定理，多面体和偏好集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数学经济学中某些“公平分割”问题的研究中。目前的提案将支持这些方法背后的数学的发展，以及由他之前的工作所激发的几个组合问题的解决方案，包括：(1)对多边形的最小三角化和最小简单覆盖的研究，(2)组合不动点定理和相关简单算法的进一步发展，以及(3)对社会选择问题有影响的凸性定理的发展。本研究的广泛影响包括其在群体决策、公平分配和投票问题上的跨学科应用，以及本科生在本研究中的积极参与和培训。通俗地说，“公平分配”问题是指找到在玩家之间分配一系列商品的方法，使每个人都能根据某种公平概念得到满足。经济学家、政治学家和博弈论家都对这个话题感兴趣，它经常引发有趣的数学问题。这类问题的可能解决方案通常是一个高维的多面体（polytope），人们通常可以通过将多面体分解成许多块（四面体），并根据玩家的偏好在多面体周围走动来找到解决方案。因此，利用三个领域（组合学、拓扑学和离散几何）的思想以及相关的经典凸性定理，如何从四面体构建多面体的数学将直接应用于涉及人类策略和决策制定的社会科学中的重要问题。

RUI: Combinatorial Fixed Point Theorems, Polytopes, and Preference SetsThe investigator's prior work has introduced methods from combinatorial topology and discrete geometry to the study of certain "fair division" questions in mathematical economics. The current proposal will support the development of the mathematics behind these methods and the solution of several combinatorial questions that have been motivated by his prior work, including: (1) the study of minimal triangulations and minimal simplicial covers of polytopes, (2) the further development of combinatorial fixed point theorems and associated simplicial algorithms, and (3) the development of convexity theorems that have bearing on social choice problems. Broader impacts of the proposed work include its interdisciplinary applications to group decision making, fair division, and voting problems, and the active participation and training of undergraduates in the proposed research.Informally speaking, a "fair division" problem is concerned with finding methods for dividing a set of goods among players in such a way that everyone can be satisfied according to some notion of fairness. This topic is of interest to economists, political scientists, and game theorists, and it often motivates interesting mathematical questions. The set of possible solutions in such problems is often a polyhedron of high dimension (a polytope), and one can usually find a solution by breaking the polytope into many pieces (tetrahedra), and walking around the polytope in a way that is guided by player preferences. Thus the mathematics of how polytopes can be built up from its tetrahedral building blocks using ideas from three areas (combinatorics, topology, and discretegeometry) as well as related classical convexity theorems will have direct application to important problems in the social sciences involving human strategy and decision making.