Statement for Karl - Dieter Crisman

My current research is primarily in the mathematics of voting. This discipline lies at the interface of mathematics, political science, and economics, though it has connections to any field which could involve aggregation of preferences, such as statistics or psychology. It also is a proven, fertile field for undergraduate research. The math of voting, in its classical form, asks questions about what is and isn’t possible with various voting methods; these include the usual plurality vote, a runoff system such as in television ‘reality shows’ or primaries, or weighted voting systems such as the Borda Count and those used in collegiate sports polls. The seminal theorems are intriguing even to the non-expert. Arrow proved in the early 50’s (essentially) that, for three or more candidates, there does not exist a voting system which fulfills the most basic criteria of fairness (symmetry, really) and simultaneously agrees with all pairwise comparisons between candidates. Twenty years later, Gibbard and Satterthwaite proved that there also does not exist such a system which cannot be manipulated by insincere voting (for an example from the most recent presidential campaign, such as an Obama supporter supporting Clinton in the primary as more ‘electable’). Before describing current work, we need a little notation. The basic concept is that of a voting profile, which is a collection of voters along with their preferences. For instance, the set {4, 3, 1, 0, 0, 0} is often thought of as representing a profile where four voters prefer A B C, three support A C B, and one has the preference order C A B. If we think of a profile as a vector, we can think of the set of all profiles as a vector space. It turns out that there are geometric decompositions of this profile space which are highly profitable in analyzing different methods and their relative strengths (most notably done by D. Saari), and it further turns out that one can re-encode these ideas algebraically in terms of representation theory (e.g., as done by M. Orrison and his students); the portion of a profile space which is subject to the fewest paradoxes is called the Basic or Borda portion. My work tries to answer new questions from these algebraic/geometric points of view, including the following: