The paper introduces a method to solve inverse problems for hyperbolic systems where the leading-order terms are nonlinear. We apply the method to the coupled Einstein-scalar field equations and study the question of whether the structure of space-time can be determined by making active measurements near the world line of an observer. We show that such measurements determine the topological, differential, and conformal structure of the space-time in the optimal chronological diamond-type set containing the world line. In the case when the unknown part of the space-time is vacuum, we can also determine the metric itself. We exploit the nonlinearity of the equation to obtain a rich set of propagating singularities, produced by a nonlinear interaction of singularities that propagate initially as for linear wave equations. This nonlinear effect is then used as a tool to solve the inverse problem for the nonlinear system. The method works even in cases where the corresponding inverse problems for linear equations remain open, and it can potentially be applied to a large class of inverse problems for nonlinear hyperbolic equations encountered in practical imaging problems.

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic 0, prove almost all cases in positive characteristic, and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. The first, regeneration, is a process opposite to specialization, which preserves the geometric genus and does not require the class of the curve to extend. The second, called the marked point trick, allows for a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic 0.

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.