We prove that the critical Maxwell–Klein–Gordon equation on ${\mathbb{R}}^{4+1}$ is globally well-posed for smooth initial data which are small in the energy norm. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.

We prove that there are only finitely many algebraically primitive Teichmüller curves in the minimal stratum in each prime genus at least 3. The proof is based on the study of certain special planes in the first cohomology of a translation surface which we call Hodge–Teichmüller planes. We also show that algebraically primitive Teichmüller curves are not dense in any connected component of any stratum in genus at least 3; the closure of the union of all such curves (in a fixed stratum) is equal to a finite union of affine invariant submanifolds with unlikely properties. Results of this type hold even without the assumption of algebraic primitivity. Combined with work of Nguyen and the second author, a corollary of our results is that there are at most finitely many nonarithmetic Teichmüller curves in $\mathcal{H}(4{)}^{\mathrm{hyp}}$.

We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental functions $f:\mathbb{C}\setminus \left\{0\right\}\to \mathbb{C}\setminus \left\{0\right\}$ that preserve the circle and whose singular set (apart from $0,\infty$) is finite and contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnold family of circle maps and its generalizations, and we solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão, and Vargas.

We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.

We consider the limiting behavior of discriminants, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety $X$ and linear systems on $X$. These are connected—we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological.