Geometric and Topological Combinatorics in Economics, Game theory, & Optimization

These 20 lectures tell the story of an intellectual trip that starts in areas of mathematics, geometry and topology, that are known for their pure nature, but the trip ends with central concerns in discrete applied mathematics and mathematical economics. The ultimate goal is to revisit some beautiful problems in game theory, and optimization. Indeed, geometric and topological theorems have been used, among many others places, for understanding optimality criteria of mathematical optimization, equilibrium theorems for games, and in algorithms for fair-division and voting. Although some of the theorems presented are well-over hundred-years-old we try to visit new generalizations and extensions that promise future new applications. We imagine the typical reader is a graduate student, who may not have seen this part of mathematics, but we hope you will find the trip pleasurable. Let us mention five motivating applications examples.