Balanced and Unbalanced Collections

Tuesday, November 11, 2014 - 10:30am - 10:55am
Keller 3-180
Louis Billera (Cornell University)
A collection of nonempty subsets of the set {1,...,n} is said to be balanced if the convex hull of the indicator functions of these sets in the n-cube meets the diagonal. The collection is unbalanced otherwise.

Balanced collections were defined 50 years ago by Lloyd Shapley (who was awarded a Nobel Prize in Economics in 2012) in his study of cores of cooperative games. Minimal balanced collections played an important role in determining when such games arise from economic trading models. One can view minimal balanced collections as generalized partitions.

Maximal unbalanced collections arose recently in physics in the study of thermal field theory, a combination of quantum field theory and statistical mechanics. They are also closely related to the study of threshold Boolean functions, threshold collections and voting games. We consider a hyperplane arrangement whose regions correspond to unbalanced collections.

I will say a bit about the applications of balanced and unbalanced collections and give relations between the various questions they ask. The talk will be more a survey than a recitation of new results, although some new approaches will be described. In particular, there are many questions here that modern geometric enumerative combinatorics ought to be able to answer.
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