MSRI-UP 2015: Geometric Combinatorics Motivated by the Social Sciences
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In only the last 75 years or so has mathematics enjoyed a symbiotic relationship with the social sciences. On one hand, mathematics can be used to model questions in the social sciences; on the other hand, the social sciences motivate new mathematical questions. Game theory was born out of a question to model human interaction and decision-making.
As an example, the space of preferences is often a geometric space whose combinatorial structure encodes player preferences in interesting ways. It many 'fair division' problems (e.g., how to divide a cake fairly among several people) the space of preferences is often a convex polytope, and in a voting theory the space of preferences (the political spectrum) is commonly modeled as a line but could be something else as well.
In this REU we will consider mathematical problems motivated by 'fair division' questions and voting theory, using a mix of techniques from combinatorics, convex geometry, and analysis. Students who have had a course in which they have had to write proofs and also linear algebra or discrete mathematics are eligible to apply. It can be helpful, though not necessary, to have some exposure to economics, game theory, or a course in programming.