WWDC26：2026 年 6 月 8 日至 12 日 (太平洋时间)

In this paper we prove, that none of the three combinatorial 8- manifolds on 15 vertices constructed by Brehm and Kuhnel, each of which is a cohomology quaternionic projective plane, can be combinatorially embedded in the Euclidean 12-space, though they have tight polyhedral embeddings in Euclidean 14-space. This extends a similar method already known for the nonembeddings of real and complex projective planes.

It is shown that the percolation model of hydrogen-bonded crystals, which is a 6-vertex model with bond defects, is completely equivalent with an 8-vertex model in an external electric field. Using this equivalence we solve exactly a particular 6-vertex model with bond defects. The general solution for the Bethe-like lattice is also analyzed.

We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.