For any $2 \leqslant r \leqslant \infty , n \geqslant 2$, we prove the existence of an open set $U$ of $C^r$‑self‑mappings of any $n$‑manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $n$‑periodic points grows as fast as asked. This complements the works of Martens–de Melo–van Strien, Kaloshin, Bonatti–Díaz–Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness.

Furthermore, for any $1 \leqslant r \lt \infty$ and any $k \geqslant 0$, we prove the existence of an open set $\hat{U}$ of $k$-parameter families in $U$ such that for a generic $(f_p)^p \in \hat{U}$, for every $\lVert p \rVert \leqslant 1$, the map $f_p$ displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.

In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑$D_k$ case, it must have an $O(4)$‑multiplet.

We obtain an asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, relevant for random matrix theory. As a consequence we show that the density of states in the random normal matrix model has universal error function transition across smooth interfaces for the limiting eigenvalue density. The key ingredient in the proof of the asymptotic expansion is the construction of the orthogonal foliation flow: a smooth flow of closed curves foliating a neighborhood of the interface, where the weighted $L^2$-mass of the orthogonal polynomial concentrates. The defining property of the flow is that the orthogonal polynomial of a given degree with respect to the disintegrated weighted area measure on the curves is approximately stationary in the flow parameter. To compute the coefficient functions in the expansion, we develop an algorithm which collapses the orthogonality relations to conditions along the interface, which become inhomogeneous Toeplitz kernel equations.