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

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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.