We describe the Hall algebra ${\mathbf{H}}_X$ of an elliptic curve $X$ defined over a finite field and show that the group $SL(2,\mathbb{Z})$ of exact autoequivalences of the derived category $D^b(Coh(X\left)\right)$ acts on the Drinfeld double ${\mathbf{DH}}_X$ of ${\mathbf{H}}_X$ by algebra automorphisms. We study a certain natural subalgebra ${\mathbf{U}}_X$ of ${\mathbf{DH}}_X$ for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants $\mathbb{C}[x_1^{\pm 1},\dots ,y_1^{\pm 1},\dots {]}^{{\mathfrak{S}}_{\infty }}$, that is, the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.

We show that families of coverings of an algebraic curve where the associated Cayley–Schreier graphs form an expander family exhibit strong forms of geometric growth. We then give many arithmetic applications of this general result, obtained by combining it with finiteness statements for rational points of curves with large gonality. In particular, we derive a number of results concerning the variation of Galois representations in one-parameter families of abelian varieties.

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a nontrivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities $y^k=x^n$ whose links are the $(k,n)$ torus knots, and for the singularity $y^4=x^7-x^6+4x^{5}y+2x^3{y}^2$ whose link is the $(2,13)$ cable of the trefoil.

We show that if $r\ge 3$ and $\alpha$ is a faithful ${\mathbb{Z}}^r$-Cartan action on a torus ${\mathbb{T}}^d$ by automorphisms, then any closed subset of $({\mathbb{T}}^d{)}^2$ which is invariant and topologically transitive under the diagonal ${\mathbb{Z}}^r$-action by $\alpha$ is homogeneous, in the sense that it is either the full torus $({\mathbb{T}}^d{)}^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some $d$-dimensional invariant subtorus. A counterexample is constructed for the rank $2$ case.

Suppose that $g\ge 3$, that $n\ge 0$, and that $\ell \ge 1$. The main result is that if $E$ is a smooth variety that dominates a codimension $1$ subvariety $D$ of ${\mathcal{M}}_{g,n}\left[\ell \right]$, the moduli space of $n$-pointed, genus $g$, smooth, projective curves with a level $\ell$ structure, then the closure of the image of the monodromy representation ${\pi }_1(E,e_o)\to {\mathrm{Sp}}_g\left(\mathrm{Ẑ}\right)$ has finite index in ${\mathrm{Sp}}_g\left(\mathrm{Ẑ}\right)$. A similar result is proved for codimension $1$ families of principally polarized abelian varieties.

We study the orbits of a polynomial $f\in \mathbb{C}\left[X\right]$, namely, the sets $\{\alpha ,f(\alpha ),f(f\left(\alpha \right)),\dots \}$ with $\alpha \in \mathbb{C}$. We prove that if two nonlinear complex polynomials $f,g$ have orbits with infinite intersection, then $f$ and $g$ have a common iterate. More generally, we describe the intersection of any line in ${\mathbb{C}}^d$ with a $d$-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.