有理多边形集平移铺砌的周期性与可判定性

Cluster Algebras and Symmetries of Regular Tilings

The classification of Grassmannian cluster algebras resembles that of regular polygonal tilings. We conjecture that this resemblance may indicate a deeper connection between these seemingly unrelated structures.

轴平行多边形集的平移铺砌

Translational tilings by axis-parallel polygonal sets

本文研究轴平行多边形集的平移铺砌，证明其拓扑极小铺砌为弱周期、存在准周期铺砌，且顶点坐标为实代数数时的平移铺砌问题可判定，推进了周期铺砌猜想相关研究。

多边形线镶嵌的完美匹配复形

Perfect Matching Complexes of Polygonal Line Tilings

本文研究多边形线镶嵌及$\left(2 \times n\right)$-网格图的完美匹配复形，利用离散Morse理论证明其要么可缩要么与球面的楔同伦等价，并刻画了无法扩展为完美匹配的所有匹配。

Families of convex tilings

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we …

多边形菱形平铺尖点处的Pearcey普适性

Pearcey universality at cusps of polygonal lozenge tiling

该研究证明一般单连通多边形菱形平铺在北极曲线尖点处的局部统计量收敛于Pearcey过程，通过与非相交伯努利随机游走的局部耦合及最优集中估计实现，验证了尖点边缘统计的普适性。

Decidability and Periodicity of Low Complexity Tilings

In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in $P$. A coloring $c$ is said to be of low complexity with respect to a rectangle if there exist $m,n\in\mathbb{N}$ and a set $P$ of $n\times m$ rectangular patterns such that $c$ is valid with respect to $P$ and $|P|\leq nm$. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, …

Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks

In this paper we study height fluctuations of random lozenge tilings of polygonal domains on the triangular lattice through nonintersecting Bernoulli random walks. For a large class of polygons which have exactly one horizontal upper boundary edge, we show that these random height functions converge to a Gaussian Free Field as predicted by Kenyon and Okounkov [28]. A key ingredient of our proof is a dynamical version of the discrete loop equations as introduced by Borodin, Guionnet and Gorin [5], which might be of independent interest.

边界镶嵌：密集的数字图像具有大的连通组件

Tiling with Boundaries: Dense digital images have large connected components

研究了数字图像中连通组件的密集排列问题，通过结构论和等周不等式确定了最大像素密度限制，并展示了结果的紧致性。

平移单调铺砌的不可判定性

Undecidability of translational monotilings

本文结合新开发技术与Aanderaa及Lewis的思想，证明了virtually $\mathbb{Z}^2$ 空间及更高维度 $\mathbb{Z}^d$（$d\geq 3$）中平移单调铺砌的算法不可判定性，解决了长期悬而未决的问题。

A formula for boundary correlations of the critical Ising model

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical $Z$-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.

金属均值王氏瓷砖 II：非周期计算机芯片的动力学

Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip

本文研究一族新的非周期王氏瓷砖集，其动力学与金属均值相关，并构造了到二维环面的因子映射，推广了黄金比例情形。