So far, the main results stem from smooth convex variational problems. It is well-known, by the seminal works of De Giorgi and Nash, together with Schauder estimates, that minimizers are smooth. This is often called the Hilbert 19th problem. A conjecture, which we solved in two long papers, was that in the context of homogenization, one could prove the corresponding theory in the case of homogenization. In essence, this required higher-order regularity theory in the nonlinear setting. Moreover, we have obtained some of the first quantitative results in the context of homogenization and free boundary problems. In particular, we have shown, again in the context of homogenization, that the free boundaries of the obstacle problem have large-scale regularity in a quantified way. These types of problems have been open for a long time.
Finally, the most significant results in the project were obtained in the context of high-contrast homogenization. A quantitative theory has been open for a long time, and there were many conjecture by physicists. For example, before our work, superdiffusive scaling limits in random environments were understood only in an averaged (annealed) sense, and precise variance asymptotics with universal prefactors were conjectural. We have proved, in a quenched setting, that the variance grows as conjectured by physicists with an explicit constant. We further established a quenched invariance principle under the same scaling. Achieving such pathwise control was widely thought to be out of reach because incompressible drifts generate long-range, dynamically evolving correlations. Moreover, we introduced so-called coarse-grained ellipticity, a new observable that measures the effective ellipticity ratio of the operator after integrating out fluctuations below a given scale. By analyzing scale-by-scale behavior for this quantity, obtained through a direct analytic argument rather than the combinatorial or perturbative techniques typical in physics, we produced the first fully rigorous renormalization group framework.
