Let $\mathfrak{g}$ be a semisimple Lie algebra, let $\mathfrak{t}$ be its Cartan subalgebra, and let W be the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel–Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus {\mathfrak{t}}^{\ast })∕W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimodules over rational and trigonometric Cherednik algebras.

We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the r-uniform Erdős–Rényi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erdős–Rényi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.

A Riemannian cone $(C,g_C)$ is by definition a warped product $C={\mathbb{R}}^{+}\times L$ with metric $g_C=d{r}^2\oplus r^2{g}_L$, where $(L,g_L)$ is a compact Riemannian manifold without boundary. We say that C is a Calabi–Yau cone if $g_C$ is a Ricci-flat Kähler metric and if C admits a $g_C$-parallel holomorphic volume form; this is equivalent to the cross-section $(L,g_L)$ being a Sasaki–Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi–Yau manifolds asymptotic to some given Calabi–Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer’s classification of ALE hyper-Kähler 4-manifolds without twistor theory.