This article analyzes the interplay between symplectic geometry in dimension $4$ and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds.

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there are some new phenomena in the finite case which deserve closer study.

We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Deligne and Rapoport on the reduction of the modular curve $X(p)$.

We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding Kac-Moody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or $2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$, $E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves.

Let $f$ be a function such that for every $\varepsilon > 0,\ n^{\log n} \leq f(n) \leq n^{\varepsilon n}$ holds if $n$ is sufficiently large. Suppose that $\log f(n) / \log n$ is nondecreasing. Using sequences of finite alternating groups, for every such $f$ we construct a $4$-generator group $\Gamma$ such that $s_n(\Gamma)$, the number of subgroups of index at most $n$ in $\Gamma$, grows like $f(n)$.

This essentially completes the investigation of the ``spectrum'' of possible subgroup growth types and settles several questions posed by Lubotzky, Mann, and Segal.

As a by-product we obtain continuously many nonisomorphic $4$-generator residually finite groups with isomorphic profinite completions.

Our construction also sheds some light on a problem of Grothendieck [Gr]; we obtain an abundance of pairs of finitely generated residually finite groups $\Gamma_0 \lessthan \Gamma$, such that the natural map $\widehat \imath : \widehat\Gamma_0 \to \widehat\Gamma$ between profinite completions is an isomorphism, but $\Gamma_0 \not \cong \Gamma$.