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Avenues in Probabilistic and Geometric Combinatorics (PROGEOCOM)
Start date: May 1, 2013, End date: Apr 30, 2018 PROJECT  FINISHED 

We consider problems in geometric and probabilistic combinatorics and discuss some applications to and connections with other areas.One underlying theme of our proposal is discrete isoperimetric relations.On the probabilistic side we discuss applications of Fourier analysis of Boolean functions to the study of threshold behavior of random graphs and other stochastic models, and propose ten directions for this emerging theory. One crucial problem is the study of near equality cases of Harper's isoperimetric inequality.On the geometric side we discuss the relation between the number of (k-1)-dimensional faces and the number of k-dimensional faces for complexes that can be embedded in 2k-dimensions. We also consider metrical and algorithmical problems on graphs of polytopes and Helly-type theorems.
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