Crossing numbers: Dead or alive. The crossing number of a graph G, denoted by cr(G), is the smallest number of edge crossings in any drawing of G in the plane. Clearly, cr(G)=0 if and only if G is planar. Garey and Johnson proved that determining cr(G) is an NP-complete problem. Over the past forty years, crossing numbers have proven to play a vital role in various fields, including discrete and computational geometry, number theory, and circuit layout (such as providing lower bounds on the chip area required for a graph's physical layout). Yet, many of the foundational questions that inspired research in this area remain open, and new mysteries continue to emerge. Quo vadis?

The syntomification of the integers (Z_p^syn) acts as a parameter space for p-adic cohomology theories. By the work of Drinfeld and others, it is known to carry a canonical formal group, and hence a map to the moduli stack of formal groups. Pulling back along this map gives rise to a certain line bundle equipped with a distinguished section "v_1". On the p-completed moduli stack of formal groups, the nonvanishing of v_1 defines the substack parametrizing "height-1" formal groups. Yet the preimage of this locus on Z_p^syn, when computed in p-adic formal stacks, is empty. There is a mismatch here because inverting v_1 on syntomic cohomology gives p-adic étale cohomology, but inverting it on the stack itself before passing to global sections gives nothing.

I will explain how passing to analytic stacks brings this missing locus into view. By realizing the p-completed moduli stack of formal groups as an analytic stack, we construct an algebro-geometric shadow of K(1)-localization.  Applied to syntomification, this produces an analytic stack whose dualizable quasi-coherent sheaves recover p-adic Galois representations, and more generally lisse étale sheaves on the generic fiber of a p-adic formal scheme. I will explain how a perfected version of this construction depends only on the generic fiber, providing a geometric counterpart to purity results for K(1)-localized K-theory. 

This is joint work in progress with Greg Andreychev.