Title: Entropy, Counting, and Shearer’s Inequalities
Abstract:
The Shannon entropy has proven to be a powerful tool for addressing problems in extremal combinatorics. After a brief introduction to the Shannon entropy of discrete random variables, we will present several examples where the entropy method demonstrates its utility in this field. A significant portion of the talk will focus on Shearer's inequalities in information theory, which have notable applications in combinatorics, particularly in areas such as graph theory, Boolean functions, and lattice problems. This talk explores several formulations of these inequalities and delves into some of their intriguing applications and generalizations (including contributions by the speaker), highlighting their utility in solving combinatorial problems across these domains.