张拉网格位形空间几何

When is a planar rod configuration infinitesimally rigid?

We provide a way of determining the infinitesimal rigidity of rod configurations realizing a rank two incidence geometry in the Euclidean plane. We model each rod with a cone over its point set and prove that the resulting geometric realization of the incidence geometry is infinitesimally rigid in regular position if and only if the resulting cone graph is infinitesimally rigid in generic position. This is a generalization of the Molecular conjecture.

带状代数的几何 I：图构形空间的上同调

The geometry of zonotopal algebras I: cohomology of graphical configuration spaces

本文证明反图向量排列的内部带状代数同构于 Moseley–Proudfoot–Young 构形空间的上同调环，其整数形式对应整上同调环，并借助轨道调和环解释给出证明。

双键模型与动力学接触几何

The dimer model and dynamical incidence geometry

本文提出了一种双键模型的几何对应物，通过研究一致双回路配置，揭示了动力学接触几何中的新定理类，并探讨了参数化问题。

Schur四次曲线64条直线内部的Reye几何

The Reye geometry inside the 64 lines of the Schur quartic

本文识别出Schur四次曲线上Naskręcki--Pokora构型隐藏的经典Reye几何，确定64线关联几何的自同构群阶为4608，并典范扩展至176线排列。

On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry

We prove that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^2$ is greater than 7/4, then the set of {\ag three-point configurations determined by $E$ has positive three-dimensional measure}. We establish this by showing that {\ag a} natural measure on the set of {\ag such configurations} has {\ag Radon-Nikodym derivative} in $L^{\infty}$ if $\dH(E)> 7/4$, and the index 7/4 in this last result cannot, in general, be improved. This problem naturally leads to the study of a bilinear convolution operator, $$ B(f,g)(x)=\int \int f(x-u) g(x-v)\, dK(u,v),$$ where $K$ is surface meas…

On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lisoněk in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by Lisoněk has a distinguished structure of a coherent configuration of type $(2,2;3)$ and is embedded in two concentric spheres in $\mathbb R^8$. In this paper we study whether there exists any other similar embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set in $\mathbb R^n$, without assuming any restriction on the size of the set. We pr…

几何感知MCTS用于组合几何中的极值问题

Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry

提出几何感知蒙特卡洛树搜索框架，通过增量更新可行动作空间和利用几何对称性，高效解决组合几何中的极值问题，在多个问题上取得新最佳结果。

一类面分裂矩阵乘积中秩下降的几何结构

The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products

本文利用经典代数几何工具，刻画了由面分裂矩阵乘积构成的矩阵在k≤6时秩下降的几何条件，其中k=5以下由交叉比刻画，k=6由三次曲面理论描述。