After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler–Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log canonical pairs of log general type. We then proceed to prove the Miyaoka–Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the abundance theorem for threefolds, which is independent of positivity results for cotangent sheaves established by Miyaoka.

A theorem of Anderson and Bando, Kasue and Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov–Hausdorff sense, one has to add singular spaces, called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces.

This raises some natural issues, in particular: Can all Einstein orbifolds be Gromov–Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one?

In this first paper, we prove that Einstein manifolds sufficiently close, in the Gromov–Hausdorff sense, to an orbifold are actually close to a gluing of model spaces in suitable weighted Hölder spaces. The proof consists in controlling the metric in the neck regions thanks to the construction of optimal coordinates.

This refined convergence is the cornerstone of our subsequent work on the degeneration of Einstein metrics or, equivalently, on the desingularization of Einstein orbifolds, in which we show that all Einstein metrics Gromov–Hausdorff close to an Einstein orbifold are the result of a gluing-perturbation procedure. This procedure turns out to be generally obstructed, and this provides the first obstructions to a Gromov–Hausdorff desingularization of Einstein orbifolds.

In this second article, we prove that any desingularization in the Gromov–Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluing-perturbation procedure that we develop. This builds on our first paper, where we proved that a Gromov–Hausdorff convergence implies a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

The description of Einstein metrics as the result of a gluing-perturbation procedure sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we extend the obstruction to the desingularization of Einstein orbifolds found by Biquard, and prove that it holds for any desingularization by trees of quotients of gravitational instantons only assuming a mere Gromov–Hausdorff convergence instead of specific weighted Hölder spaces. This is conjecturally the general case, and can at least be ensured by topological assumptions such as a spin structure on the degenerating manifolds. We also identify an obstruction to desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE spaces.

We prove several rigidity theorems related to and including Lytchak’s problem. The focus is on Alexandrov spaces with $\text{curv}\ge 1$, nonempty boundary and maximal radius $\frac{\pi }{2}$. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that, when the boundary is either geometrically or topologically spherical, it is possible to obtain strong rigidity results. In contrast to this, one can show that with general lower curvature bounds and strictly convex boundary only cones can have maximal radius. We also mention some connections between our problems and the positive mass conjectures.

We study the quantum Witten–Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi (2020) as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $𝜖$ and a quantum parameter $\hslash$. When $\hslash =0$, this series restricts to the Witten–Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves.

We establish a link between the $𝜖=0$ part of the quantum Witten–Kontsevich series and one-part double Hurwitz numbers. These numbers count the number of nonequivalent holomorphic maps from a Riemann surface of genus $g$ to ${\mathbb{P}}^1$ with a complete ramification over $0$, a prescribed ramification profile over $\infty$ and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil (2005) proved that these numbers have the property of being polynomial in the orders of ramification over $\infty$. We prove that the coefficients of these polynomials are the coefficients of the quantum Witten–Kontsevich series.

We also present some partial results about the full quantum Witten–Kontsevich power series.

We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.

The Stolz–Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. We extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant K–theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.

We develop a theory of convex ancient mean curvature flow in slab regions, with Grim hyperplanes playing a role analogous to that of half-spaces in the theory of convex bodies.

We first construct a large new class of examples. These solutions emerge from circumscribed polytopes at time minus infinity and decompose into corresponding configurations of “asymptotic translators”. This confirms a well-known conjecture attributed to Hamilton; see also Huisken and Sinestrari (2015). We construct examples in all dimensions $n\ge 2$, which include both compact and noncompact examples, and both symmetric and asymmetric examples, as well as a large family of eternal examples that do not evolve by translation. The latter resolve a conjecture of White (2003) in the negative.

We also obtain a partial classification of convex ancient solutions in slab regions via a detailed analysis of their asymptotics. Roughly speaking, we show that such solutions decompose at time minus infinity into a canonical configuration of Grim hyperplanes. An analogous decomposition holds at time plus infinity for eternal solutions. There are many further consequences of this analysis. One is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the midplane of their slab.