实根猜想在五维及更高维的球面上失效

Subword complexes via triangulations of root polytopes

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $β$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polyto…

格的局部覆盖最优性：Leech格与根格E8

Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8

我们表明，水蛭格给出了一个球覆盖，这是局部最不稠密的格覆盖。我们证明了一个类似的结果是假的根格E8。为此，我们构造了一个不太稠密的覆盖格，其Delone细分与E8的Delone细分具有共同的细化。新的格产生一个球面覆盖，其密度比以前最著名的格A8* 小12%以上。目前，Leech格是第一个也是唯一已知的具有非单纯Delone细分的局部最优格覆盖的例子。我们在此特别回答迪克森在1968年提出的一个问题。通过证明Leech格是刚性的，我们的答案在某种意义上甚至是最强的。

多边界三角剖分的显式enumeration

Explicit enumeration of triangulations with multiple boundaries

我们列举了有根三角剖分的一个领域与多个洞的边缘的总数和每个边界组件的长度。证明依赖于一个组合恒等式，由于W. T。图特

图的真正可靠性根源

On the Real Reliability Roots of Graphs

本文研究图的可靠性多项式的真实根，证明几乎所有图都有非实根，并在特定区间内根密集。

Kleinian 球填充、反射群与算术性

Kleinian sphere packings, reflection groups, and arithmeticity

本文研究Kleinian球填充等相关问题，解决了高维晶体球填充存在性难题，提出几何加倍构造方法并探究真整填充，技术依托洛伦兹二次型等相关计算。

All Terminal Reliability Roots of Smallest Modulus

Given a connected graph $G$ whose vertices are perfectly reliable and whose edges each fail independently with probability $q\in[0,1],$ the \textit{(all-terminal) reliability} of $G$ is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in $q$). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem -- among all reliability polynomials of graphs on $n$ vertices, which has a root o…