Which conjectures have been solved (or partially solved) using entropy methods?

Question

I am studying how entropy-based arguments (in the sense of Shannon, Boltzmann, or Perelman's geometric entropy) can be used to prove or guide the resolution of major mathematical conjectures.

Some examples include:

1. Poincaré and Geometrization Conjectures , solved by Perelman using a monotone entropy functional in Ricci flow
   Perelman, The entropy formula for the Ricci flow and its geometric applications

2. Shepp–Olkin Entropy Concavity Conjecture , proved by Hillion & Johnson (2015)
   Hillion & Johnson, Proof of the Shepp–Olkin entropy concavity conjecture

3. Gaussian optimizer (minimum output entropy) Conjecture , proved for quantum Gaussian channels
   Giovannetti–Holevo–Garcia-Patrón, Gaussian optimizer conjecture for quantum channels

4. Sidorenko’s Conjecture — partially resolved using information-theoretic inequalities
   Li & Szegedy, On the log-convexity of graph homomorphism functions

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Question:
Are there other important conjectures (solved or open) whose proofs or partial results crucially rely on entropy or information-theoretic principles?

Answer 1

Von Neumann asked around 1940 if two bilateral Bernoulli shifts are always measure theoretic isomorphic. Kolmogorov introduced the notion of entropy in the theory of dynamical systems in 1958 and showed that it is invariant by isomorphism thus giving a negative answer to the problem raised by Von Neumann. Ornstein showed in 1972 that two bilateral Bernoulli shifts are isomorphic if and only if they have the same entropy.

Answer 2

Marton's conjecture, a central problem in additive combinatorics, was proven recently by Gowers, Green, Manners, and Tao using entropy methods. This is part of a very fruitful line of using entropy methods to study additive combinatorics; see this article by Green, Manners, and Tao for some explanation of how the two fields relate.

One way to phrase their result is as follows: if $A\subset(\mathbb Z/2\mathbb Z)^n$ satisfies $\lvert\{a+b:a,b\in A\}\rvert\leq K\lvert A\rvert$, then there exists a subgroup $V\leq(\mathbb Z/2\mathbb Z)^n$ satisfying $\lvert V\rvert\leq\lvert A\rvert$ such that $A$ intersects at most $2K^{12}$ cosets of $V$.