We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. ${\bf 162}$ (2005), 439--485], and new hardness gaps for Unique-Games.

The Grassmann graph ${\sf Gr}_{\sf{global}}$ contains induced subgraphs ${\sf Gr}_{\sf{local}}$ that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is $o(1)$ inside all subgraphs ${\sf Gr}_{\sf{local}}$ whose order is $O(1)$ lower than that of ${\sf Gr}_{\sf{global}}$. We prove that pseudorandom sets have expansion $1-o(1)$, greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically regular, and simple index $0$ curves in dimensions greater than four are generically super-rigid, implying, e.g., that the Gromov-Witten invariants of Calabi-Yau $3$-folds reduce to sums of local invariants for finite sets of embedded curves. We also establish partial results on super-rigidity in dimension four and regularity of branched covers, and briefly discuss the outlook for bifurcation analysis. The proofs are based on a general stratification result for moduli spaces of multiple covers, framed in terms of a representation-theoretic splitting of Cauchy-Riemann operators with symmetries.

In this work we initiate the mathematical study of naked singularities for the Einstein vacuum equations in $3+1$ dimensions by constructing solutions which correspond to the exterior region of a naked singularity. A key element is our introduction of a new type of self-similarity for the Einstein vacuum equations. Connected to this is a new geometric twisting phenomenon which plays the leading role in singularity formation.

Prior to this work, the only known examples of naked singularities were the solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar-field system, as well as other solutions explored numerically for either the spherically symmetric Einstein equations coupled to suitable matter models or for the Einstein equations in higher dimensions.

A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Ampère equation that had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.