We address two pressing questions in the theory of the Korteweg–de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in ${\mathbb{R}}^3$ with entropy at most 2 and to all closed hypersurfaces in ${\mathbb{R}}^4$ with entropy at most $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. When combined with recent work of Daniels-Holgate, this strengthens Bernstein and Wang’s low-entropy Schoenflies-type theorem by relaxing the entropy bound to $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt, Simon, and Smale).

We introduce and analyze free energy landscapes defined by associating to any point inside the sphere a free energy calculated on a thin spherical band around it, using many orthogonal replicas. This allows us to prove for general spherical models several results related to the Thouless–Anderson–Palmer (TAP) approach originally introduced in the 1970s for the Sherrington–Kirkpatrick model. We establish a TAP representation for the free energy, valid for any overlap value which can be sampled as many times as we wish in an appropriate sense. We call such overlaps multi-samplable. The correction to the Hamiltonian in the TAP representation arises in our analysis as the free energy of a certain model on an overlap dependent band. For the largest multi-samplable overlap it coincides with the Onsager reaction term from physics. For smaller multi-samplable overlaps the formula we obtain is new. We also derive the corresponding TAP equations for critical points. We prove the above without appealing to the celebrated Parisi formula or the ultrametricity property.

We prove that any overlap value in the support of the Parisi measure is multi-samplable. For generic models, we further show that the set of multi-samplable overlaps coincides with a certain set that arises in the characterization for the Parisi measure by Talagrand. Moreover, using the Parisi measure of the model on the sphere, we express that of the aforementioned model on the band related to TAP correction. The ultrametric tree of pure states can be embedded in the interior of the sphere in a natural way. For this embedding, we show that the points on the tree uniformly maximize the free energies we define. From this we conclude that the Hamiltonian at each point on the tree is approximately maximal over the sphere of same radius, and that points on the tree approximately solve the TAP equations for critical points.

We prove the spherical density conjecture of Sarnak for the principal subgroup of ${\mathrm{SL}}_n(\mathbb{Z})$ of square-free level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.