On rich lenses in planar arrangements of circles and related problems

由小弦支撑且与自然实代数函数相容的圆周排布

Arrangements of circles supported by small chords and compatible with natural real algebraic functions

本文提出一类本质上不同的圆周排布，研究其诱导实代数映射与函数的拓扑性质、组合学及全局结构应用。

Morse-Bott 函数的小圆排列

Arrangements of small circles for Morse-Bott functions

本文提出一种通过归纳添加小圆构造圆排列的系统方法，将其与Morse-Bott函数及Reeb图联系起来，研究排列中区域坍缩图的局部变化。

On the Average Complexity of the $k$-Level

Let ${\cal L}$ be an arrangement of $n$ lines in the Euclidean plane. The \emph{$k$-level} of ${\cal L}$ consists of all vertices $v$ of the arrangement which have exactly $k$ lines of ${\cal L}$ passing below $v$. The complexity (the maximum size) of the $k$-level in a line arrangement has been widely studied. In 1998 Dey proved an upper bound of $O(n\cdot (k+1)^{1/3})$. Due to the correspondence between lines in the plane and great-circles on the sphere, the asymptotic bounds carry over to arrangements of great-circles on the sphere, where the $k$-level denotes the vertices at distance at mo…

Optimal packings of congruent circles on a square flat torus

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super resolution of images." We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.

圆的排列、由其围成的区域及标号的 Poincaré-Reeb 图

Arrangements of circles, the regions surrounded by them and labeled Poincaré-Reeb graphs

本文研究圆的排列及其围成的区域，通过在 Poincaré-Reeb 图的顶点和边上引入标号来编码圆的信息，并讨论了添加圆引起的新类型图变化，发现该区域是自然实代数映射的像。

两两相交的伪圆形成的二角形的最大数量

The maximum number of digons formed by pairwise crossing pseudocircles

本研究将Grünbaum关于两两相交伪圆排列中二角形最大数量为2n−2的猜想，从柱面排列、几何圆情形推广至任意简单两两相交伪圆排列，给出了该猜想的完整证明，解决了悬置五十余年的公开问题。

半径可变的圆堆积的刚性

Rigidity of Circle Packings with Flexible Radii

针对表示平面图且半径受等式与不等式约束的圆堆积，本文提出一阶刚性的对偶充分条件，并讨论超越一阶的刚性研究难点。

Cutting Algebraic Curves into Pseudo-segments and Applications

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then …

Well-distributed great circles on S^2

Let $C_1, \dots, C_n$ denote the $1/n-$neighborhood of $n$ great circles on $\mathbb{S}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$ \sum_{i, j = 1 \atop i \neq j}^{n}{|C_i \cap C_j|^s} \gtrsim_s \begin{cases} n^{2 - 2s} \qquad &\mbox{if}~0 \leq s < 2 \\ n^{-2} \log{n} \qquad &\mbox{if}~s = 2\\ n^{1- 3s/2} \qquad &\mbox{if}~s > 2. \end{cases} .$$ For $s=1$ there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in $\mathbb{R}^2$ the lower bound is $\gtrsim \log{n}$) and there are strong connection…

Topological realizations of line arrangements

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of $2$-spheres embedded, in the symplectic, smooth, or topological (locally flat) categories, in the complex projective plane. We find obstructions to realizability in the topological category, which a…

双射的加法三元组，或环面半皇后问题

Additive triples of bijections, or the toroidal semiqueens problem

本文用改造的Hardy–Littlewood圆法证明了模n群上双射加法三元组数量的渐近公式，统一解决了环面半皇后、正交态射及循环拉丁方横截线等计数问题。