"RUI: Combinatorial fixed point theorems, polytopes, and fair division"
The 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 from economic problems, including: (1) the development of new combinatorial fixed point theorems, (2) associated simplicial algorithms, and (3) the study of minimal triangulations and minimal simplicial covers of polytopes. Broader impacts of the proposed work include its interdisciplinary application to negotiation support for group decision making, 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 discrete geometry) will have direct application to important problems in the social sciences involving human strategy and decision making.
This award is being jointly funded by the Division of Mathematical Sciences and the Directorate of Social, Behavioral, and Economic Science as part of the Mathematics Sciences Priority Area.
