Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of $U$. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when $U$ is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram constructed from counts of infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as well as the positivity in the Laurent phenomenon, follow readily from the geometric description.

The Erdős--Faber--Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.

In this article we prove Crew's parabolicity conjecture of $F$-isocrystals. For this purpose, we introduce and study the notion of $\dagger$-hull of a sub-$F$-isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent $F$-isocrystals.

We classify $\mathrm{GL}(2,\mathbb{R})$ orbit closures of translation surfaces of rank at least half the genus plus $1$.

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface $(X,L)\in F_{2d}$. The main new notion is that of a recognizable divisor $R$, a choice which can be consistently extended to all central fibers of Kulikov models. We prove that any choice of recognizable divisor leads to a semitoroidal compactification of the period space, at least up to normalization. Finally, we prove that the rational curve divisor is recognizable for all degrees.

We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon >0$, with probability tending to one as $n\to \infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,\frac{1}{4}-\epsilon)$ other than those of $X$, and with the same multiplicities.

As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exists a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.

We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.

We will fix an error in our paper on the average Colmez conjecture by proving a slightly weaker statement than Theorem 2.7, which is sufficient for application to the main results.