In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension 4 with Einstein orbifolds as tangent flows at infinity. The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities that we introduce.

We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in SU(2) and mild decay at infinity are hyperkähler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's mu-functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized lambda-functional related to a notion of weighted mass.

We prove the instability of all conformally Kähler, Einstein 4-metrics which are not conformally flat. This settles the question of the stability all of the known compact and Ricci-flat gravitational instantons.

This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.

We provide the first example of continuous families of Poincaré-Einstein metrics developing cusps on the trivial topology $\mathbb{R}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański-Demiański. We additionally indicate how to construct similar examples on more complicated topologies.

We study weighted the notion of scalar curvature motivated by applications to Ricci flow in compact and AE manifolds. We define and study the weighted notion of ADM mass and Dirac operator extending among others classical eigenvalues estimates and Witten's formula. Ricci flow turns out to be the gradient flow of this weighted mass which also equals a weighted Dirichlet energy on spinors.

I study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. I more precisely show that spherical and hyperbolic 4-orbifolds (which are an Einstein metric spaces in a synthetic sense) with the simplest singularities cannot be Gromov-Hausdorff limits of smooth Einstein 4-metrics without relying on previous integrability assumptions. For this, I analyze the integrability of deformations of Ricci-flat ALE metrics. I also introduce preserved integral quantities based on the symmetries of Einstein metrics. I show that many previously identified obstructions to the desingularization of Einstein 4-metrics are equivalent to these quantities on Ricci-flat cones. In particular, all of the obstructions to desingularizations bubbling out Eguchi-Hanson metrics are recovered.

We study the stability and instability of ALE Ricci-flat metrics around which a Łojasiewicz inequality is satisfied for a variation of Perelman's functional adapted to the ALE situation. This functional was introduced by the authors in a recent work where it was moreover proven that it satisfies a good enough Łojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.

High-dimensional depth separation results for neural networks show that certain functions can be efficiently approximated by two-hidden-layer networks but not by one-hidden-layer ones in high-dimensions d. Existing results of this type mainly focus on functions with an underlying radial or one-dimensional structure, which are usually not encountered in practice. The first contribution of this paper is to extend such results to a more general class of functions, namely functions with piece-wise oscillatory structure, by building on the proof strategy of (Eldan and Shamir, 2016). A common theme in the proof of such results is the fact that one-hidden-layer fail to approximate high-energy functions whose Fourier representation is spread in the domain. On the other hand, existing approximation results of a function by one-hidden-layer neural networks rely on the function having a sparse Fourier representation. The choice of the domain also represents a source of gaps between upper and lower approximation bounds. Focusing on a fixed approximation domain, namely the (d-1)-dimensional sphere, we provide a characterization of both functions which are efficiently approximable by one-hidden-layer networks and of functions which are provably not, in terms of their Fourier expansion.

I motivate the following question: If an Einstein orbifold is a Gromov-Hausdorff limit of Einstein metrics bubbling out Eguchi-Hanson metrics, then, is it necessarily Kähler? It is done by identifying new obstructions to the desingularization of Einstein metrics which are specific to the compact situation. I then test these new obstructions for desingularization of a flat orbifold and show how obstructed this procedure is.

We introduce a functional, on ALE manifolds whose critical points are Ricci-flat and such that the Ricci flow is its gradient flow. The ADM-mass appears in its expression as a boundary term. We study its properties with respect to the Ricci flow and prove that it satisfies a weighted Łojasiewicz inequality. We moreover study the properties of the ADM-mass for ALE metrics with nonnegative scalar curvature near Ricci-flat metrics or on spin manifolds.