The combinatorial type of an elliptic $K3$ surface with a zero section is the pair of the $\mathit{ADE}$-type of the singular fibers and the torsion part of the Mordell–Weil group. We determine the set of connected components of the moduli of elliptic $K3$ surfaces with fixed combinatorial type. Our method relies on the theory of Miranda and Morrison on the structure of a genus of even indefinite lattices and on computer-aided calculations of $p$-adic quadratic forms.

We develop a new method to construct explicit regular minimal surfaces in Euclidean space that are defined on the entire complex plane with controlled geometry. More precisely, we show that for a large class of planar curves $\left(x\right(t),y(t\left)\right)$, we can find a third coordinate $z\left(t\right)$ and normal fields $n\left(t\right)$ along the space curve $c\left(t\right)=\left(x\right(t),y(t),z(t\left)\right)$ so that the Björling formula applied to $c\left(t\right)$ and $n\left(t\right)$ can be explicitly evaluated. We give many examples.

We construct, for $m\ge 6$ and $2n\le m$, closed manifolds $M^m$ with finite nonzero $\phi (M^m,S^n$), where $\phi (M,N)$ denotes the minimum number of critical points of a smooth map $M\to N$. We also give some explicit families of examples for even $m\ge 6$ and $n=3$, taking advantage of the Lie group structure on $S^3$. Moreover, there are infinitely many such examples with $\phi (M^m,S^n)=1$. Eventually, we compute the signature of the manifolds $M^{2n}$ occurring for even $n$.

We extend the theory of Ihara zeta functions to noncompact arithmetic quotients of Bruhat–Tits trees. This new zeta function turns out to be a rational function despite the infinite-dimensional setting. In general, it has zeros and poles in contrast to the compact case. The determinant formulas of Bass and Ihara hold if we define the determinant as the limit of all finite principal minors. From this analysis we derive a prime geodesic theorem, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.

We consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass, and welded band-pass moves. Interrelationships between these moves are discussed, and, for each of these moves, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a byproduct, we obtain that all of the local moves mentioned are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.