Many extremal problems in harmonic analysis can be viewed as problems of optimization of profit over controlled random walks. As such they have an associated Monge–Amp`re equation, whose solution (called the Bellman function) gives full information over the initial extremal problem.

Paata Ivanisvili and myself observed that one can move this technique one step further if one notices that a wide class of harmonic analysis problems provides us with concrete Bellman function that have the following interesting property. Namely, if a specially selected Legendre transform is applied to this Bellman function, then a new function appear, and it now solves a “dual” extremal problem on any discrete cube independent of dimension (and so also gives a Gaussian isoperimetric inequality independent of dimension). I will give 4 lectures on the subject complemented by the lecture of Paata Ivanisvili on Wednesday. His lecture is devoted to application of this duality to Ehrhard type inequalities. My lectures will be 1) “The heating of the Beurling–Ahlfors transform”; 2) “Improving Beckner–Sobolev inequality on Hamming cube and with Gaussian measure”; 3) “Extremal problems for martingale square function and their duals on Hamming cube”; 4) “Burkholder function and quasi-convexity”.