We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.

Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called-M DOZZ formula, for the three point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e., to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first nontrivial rigorous integrability result on Gaussian Multiplicative Chaos measures.

We show that a second Neumann eigenfunction $u$ of a Euclidean triangle has at most one (non-vertex) critical point $p$, and if $p$ exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of $u$ are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the ``hot spots" conjecture for triangles in the plane.

The Allen--Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang--Wei) of the Allen--Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen--Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen--Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques--Neves in 3-dimensions regarding min-max constructions of minimal surfaces.

Allen--Cahn min-max constructions were recently carried out by Guaraco and Gaspar--Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie--Marques--Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every $p = 1, 2, 3, \ldots$, a two-sided embedded minimal surface with Morse index $p$ and area $\sim p^{\frac{1}{3}}$, as conjectured by Marques--Neves.