General Relativity (GR), Einstein’s theory of gravity, predicts that under certain conditions the evolution of spacetime inevitably breaks down in finite time, giving rise to singularities—regions where curvature becomes unbounded and the classical description of physics ceases to apply. Such singularities appear at the Big Bang, inside black holes, and in many cosmological and collapse models. The Hawking–Penrose incompleteness theorems ensure that singularities arise generically, but they provide almost no information about their structure, stability, or dynamical formation. As a result, some fundamental questions remain unresolved:
– How do singularities actually form?
– What are their precise geometric and analytic properties?
– Are they stable under perturbations, or do small changes lead to drastically different behavior?
– Does determinism hold in GR, as conjectured by the Strong Cosmic Censorship hypothesis?
Over the last decades, partial progress has been made—e.g. on symmetric or homogeneous models, or in special matter models—but the full understanding of singularity formation in generic, non-symmetric solutions remains one of the most challenging open problems in mathematical physics. Recent breakthroughs have illuminated particular regimes (e.g. subcritical Big Bang formation, Cauchy horizon stability), revealing that singular behaviors are far richer and more subtle than previously expected. Yet core scenarios predicted by physics, such as the oscillatory BKL (Mixmaster) singularity, remain completely unverified in the general vacuum case.
This project is motivated by the need to cross this frontier by developing new mathematical techniques—analytic, geometric, and PDE-based—to attack these problems in settings far closer to the fully general Einstein equations.
Overall Objectives
The project addresses two fundamental singularity scenarios:
Big Bang singularities in vacuum
Construct the first non-symmetric solutions exhibiting oscillatory or spiky behavior, thereby testing the BKL conjecture beyond homogeneous or symmetric models.
Develop a general method to construct singularities with discontinuous asymptotic profiles (spikes).
Identify and dynamically characterize the single functional degree of freedom responsible for instabilities, proving codimension stability of Kasner-like singularities and delimiting the stable manifold within vacuum dynamics.
Spacelike singularities inside black holes
Show that spacelike singularities arise generically in one-ended black holes beyond spherical symmetry for realistic matter models (massless scalar field, Oppenheimer–Snyder dust).
Establish stable blow-up near such singularities and understand how null and spacelike singular components coexist in dynamical interiors.
Expected Impact
The project is expected to reshape our understanding of singularity formation in GR and address long-standing conjectures in the field:
The first evidence of oscillatory Big Bang dynamics without symmetry would provide a crucial step toward validating or refining the BKL conjecture.
A general construction of spike singularities could force a reformulation of the notion of “generic” singularity and the expected locality of asymptotic dynamics.
Codimension stability results would clarify which singularities are dynamically relevant and provide the first rigorous identification of the mechanism driving oscillatory instabilities.
New results on black hole interiors would show that spacelike singularities are not artifacts of symmetry but persist under perturbations—advancing our understanding of Strong Cosmic Censorship and the global structure of black holes.
Overall, the project aims to deliver foundational advances in the mathematical analysis of the Einstein equations, significantly deepening our comprehension of the most extreme and fundamental phenomena predicted by General Relativity.
