Almost-tiling the plane by ellipses

Periodicity and ergodicity in the trihexagonal tiling

We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular form: the regions have infinite area and consist of the plane with a periodic family of triangles removed. We also completely describe initial conditions for periodic and drift-periodic light rays.

Rational Angles and Tilings of the Sphere by Congruent Quadrilaterals

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a3b). Parallel to a complete classification by Cheung, Luk and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings.

强几乎有限性

Strong almost finiteness

本文提出强几乎有限性概念，证明其等价于 Property A 加 Følner 性质，并推广了顺从 Cayley 图的经典结果，进而构造出一类可由 Elliott 不变量分类的 C*-代数。