Georgia Institute of Technology College of Sciences

The KLS conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) and present recent progress resulting in the current best bound, as well as a tight bound for the log-Sobolev constant (both with Yin Tat Lee). The conjecture has led to several techniques of general interest.