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Avenues in Probabilistic and Geometric Combinatorics

Final Report Summary - PROGEOCOM (Avenues in Probabilistic and Geometric Combinatorics)

The study of sharp threshold phenomenon and related isoperimetric questions and Fourier methods was the starting point of our research in probabilistic combinatorics. We proved a sharp threshold result for large alphabets and various applications for random graphs and hypergraphs and for social choice theory. We also found applications of the sharp threshold phenomenon to extremal combinatorial problems.

The study of noise in stochastic systems and related Fourier methods, turned out to be useful in many different areas - both theoretical and applied. On the theoretical side, we found some new correlation inequalities of Harris-type, and used them for giving partial results on a classical question of Chvatal in extremal combinatorics. We also studied noise sensitivity and stability of random graph properties.

We also studied noise sensitivity for systems of non interacting bosons and other intermediate-scale quantum systems that are designed to demonstrate superior computational powers. Our study leads to a theoretical argument for why quantum computers are not possible, an argument based on the mathematics of noise stability and noise sensitivity and its connections to the theory of computing. While controversial, exploring this avenue may have important implications to, mathematics, the theory of computing, and various areas of quantum physics.

We explored some avenues in geometric combinatorics, and also here isoperimetry was a central theme. We have made progress on understanding face numbers of embedded complexes, high dimensional analog for minors and induced graphs, and to applications of geometric and homological methods for extremal combinatorial problems. In this direction we mention the new theories of bipartite rigidity and bipartite minors, and the emerging connections between homological properties and the notion of chi-boundedness for graphs and hypergraphs. Finally, we were always fascinated by Tverberg's theorem in discrete geometry, and our project have led to the solution of an old conjecture on the sizes of parts for Tverberg's partition, and to representation-theoretic extensions as well as a matroidal extension of Tverberg's theorem itself.