1. Entropic methods

Entropy FreimanRuzsa lower bounds
Theorem. Let $X$ be any $\mathbb{F}_2^n$valued random variable such that $d[X,X]\leq \log K$, where $d$ is the entropic Ruzsa distance. Then there is a subspace $H\subset \mathbb{F}_2^n$ such that $d[X,U_H]\leq 11\log K$, where $U_H$ is uniform on $H$.Problem 1.1.
[Terence Tao] Can we give lower bounds for $d[X,U_H]$? 
Entropy proof of Sidorenko
There is a proof system for graph homomorphism inequalities using flag algebras.Problem 1.2.
[Yufei Zhao] Define a proof system for graph homomorphism inequalities using entropy.
Can all known cases of Sidorenko be entropized?
Can the proof that even girth graphs are locally Sidorenko be entropized? 
Inverse theorems with polynomial bounds
Given $f:\mathbb{F}_2^n\to \mathbb{F}_2^m$ with $\mathbb{P}_{x+y=z+w}(f(x)+f(y)=f(z)+f(w))\geq \epsilon$, then Polynomial FreimanRuzsa implies that $f$ correlates with a linear function.
Set $D_hf(x)=f(x+h)f(x)$. If $\mathbb{P}_{x,h_1,h_2,h_3}(D_{h_1}D_{h_2}D_{h_3}f(x)=0)\geq \epsilon$, then PFR implies that $f$ correlates with a quadratic function.Problem 1.3.
[Freddie Manners] Is there a direct entropic proof of the above results with polynomial bounds?
Cite this as: AimPL: Highdimensional phenomena in discrete analysis, available at http://aimpl.org/highdimdiscrete.