We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure-preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for ${\mathrm{II}}_1$ factors.

We prove N. Takahashi’s conjecture determining the contribution of each contact point in genus-0 maximal contact Gromov–Witten theory of ${\mathbb{P}}^2$ relative to a smooth cubic E. This is a new example of a question in Gromov–Witten theory that can be fully solved despite the presence of contracted components and multiple covers. The proof relies on a tropical computation of the Gromov–Witten invariants and on the interpretation of the tropical picture as describing wall-crossing in the derived category of coherent sheaves on ${\mathbb{P}}^2$.

The same techniques allow us to prove a new sheaves/Gromov–Witten correspondence, relating Betti numbers of moduli spaces of one-dimensional Gieseker semistable sheaves on ${\mathbb{P}}^2$, or equivalently, refined genus-0 Gopakumar–Vafa invariants of local ${\mathbb{P}}^2$, with higher-genus maximal contact Gromov–Witten theory of $({\mathbb{P}}^2,E)$. The correspondence involves the nontrivial change of variables $y=e^{i\hslash }$, where y is the refined/cohomological variable on the sheaf side, and ℏ is the genus variable on the Gromov–Witten side. We explain how this correspondence can be heuristically motivated by a combination of mirror symmetry and hyper-Kähler rotation.

It is widely expected that generic black holes have a nonempty but weakly singular Cauchy horizon, due to mass inflation. Indeed this has been proven by the author in the spherical collapse of a charged scalar field, under decay assumptions of the field in the black exterior which are conjectured to be generic. A natural question then arises: can this weakly singular Cauchy horizon close off the space-time, or does the weak null singularity necessarily “break down,” giving way to a different type of singularity? The main result of this paper is to prove that the Cauchy horizon cannot ever “close off” the space-time. As a consequence, the weak null singularity breaks down and transitions to a stronger singularity for which the area-radius r extends to 0.