This text describes the content of the Takagi lectures given by the author in Kyoto in 2017. The lectures present some aspects of the theory of sharp thresholds for boolean functions and its application to the study of phase transitions in statistical physics.
Phase Transitions on Nonamenable Graphs
Counting without sampling. New algorithms for enumeration problems using statistical physics
Boolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input
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