In this paper we prove the non-linear asymptotic stability of the five-dimensional Schwarzschild metric under biaxial vacuum perturbations. This is the statement that the evolution of $(SU (2) \times (U(1))$)-symmetric vacuum perturbations of initial data for the five-dimensional Schwarzschild metric finally converges in a suitable sense to a member of the Schwarzschild family. It constitutes the first result proving the existence of non-stationary vacuum black holes arising from asymptotically flat initial data dynamically approaching a stationary solution. In fact, we show quantitative rates of approach. The proof relies on vectorfield multiplier estimates, which are used in conjunction with a bootstrap argument to establish polynomial decay rates for the radiation on the perturbed spacetime. Despite being applied here in a five-dimensional context, the techniques are quite robust and may admit applications to various four-dimensional stability problems.

We investigate cosmological predictions on the early universe based on the noncommutative geometry (NCG) models of gravity coupled to matter. Using the renormalization group analysis for the standard model with right-handed neutrinos and Majorana mass terms, which is the particle physics content of the most recent NCG models, we analyze the behavior of the coefficients of the gravitational and cosmological terms in the Lagrangian derived from the asymptotic expansion of the spectral action functional of NCG. We find emergent Hoyle–Narlikar and conformal gravity at the see-saw scales and a running effective gravitational constant, which affects the propagation of gravitational waves and the evaporation law of primordial black holes and provides Linde models of negative gravity in the early universe. The same renormalization group analysis also governs the running of the effective cosmological constant of the model. The model also provides a Higgs-based slow-roll inflationary mechanism, for which one can explicitly compute the slow-roll parameters. The particle physics content allows for dark matter models based on sterile neutrinos with Majorana mass terms.

We show that certain superpotential and Kähler potential couplings of $\mathcal{N} = 1$ supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry. This applies to F-theory on a Calabi–Yau four-fold and three-fold compactifications of type II and heterotic strings with branes. The heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the four-fold computes non-perturbative corrections to mirror symmetry on the three-folds, including D-instanton corrections. We also propose a physical interpretation for the observation byWarner that relates the deformation spaces of certain matrix factorizations and the periods of non-compact four-folds that are ALE fibrations.

Using F-theory/heterotic duality, we describe a framework for analyzing non-geometric $T_2$-fibered heterotic compactifications to six- and four dimensions. Our results suggest that among $T_2$-fibered heterotic string vacua, the non-geometric compactifications are just as typical as the geometric ones. We also construct four-dimensional solutions that have novel type-IIB and M-theory dual descriptions. These duals are non-geometric with three- and four-form fluxes not of (2, 1) or (2, 2) Hodge type, respectively, and yet preserve at least $N = 1$ supersymmetry.