Let G be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let Γ be a lattice in G, with $\mathrm{\pi }:G\to G∕\mathrm{\Gamma }$ the quotient map. For a semialgebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $\mathrm{\pi }(X)$ in the compact nilmanifold $G∕\mathrm{\Gamma }$.

Our theorem describes $\mathrm{cl}(\mathrm{\pi }(X))$ in terms of finitely many families of cosets of real algebraic subgroups of G. The underlying families are extracted from X, independently of Γ. We also prove an equidistribution result in the case of curves.

Let $d\ge 3$ be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant $\mathfrak{d}>0$ such that

$$\mathbb{P}(A\text{is singular in}\mathbb{R})\le n^{-\mathfrak{d}},$$

for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field ${\mathbb{F}}_p$. The proof combines a local central limit theorem and a large deviation estimate.

We use the “higher Hida theory” recently introduced by the second author to p-adically interpolate periods of nonholomorphic automorphic forms for ${\mathrm{GSp}}_4$, contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree-4 (spin) L-function of automorphic representations of ${\mathrm{GSp}}_4$, and the degree-8 L-function of ${\mathrm{GSp}}_4\times {\mathrm{GL}}_2$.

We find a geometric interpretation of the ${\mathbb{A}}_{q,t}$ algebra, the algebra which appeared in the previous work of Erik Carlsson and the author on the proof of the shuffle conjecture. This allows us to construct a representation of “the positive part” of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven, and Xin is then shown to be a corollary of this relation.

For any manifold with polynomial volume growth, we show that the dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau’s 1974 conjecture about polynomial growth harmonic functions holds.