Unusual formulae for the Euler characteristic

欧拉关于球面几何的工作：概述与评论

Euler's work on spherical geometry: An overview with comments

本文回顾欧拉关于球面几何的研究，涵盖其建立球面三角公式的方法、球面三角形面积计算及与相关领域的联系。

A formula for the Euler characteristics of even dimensional triangulated manifolds

An unusual formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.

Non-Abelian number theory and the structure of curves on surfaces

In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a certain length, either geometric or combinatorial. Some cases are rigurolosly determined and the Euler totient function emerges. More complicated orbits are computed to suggest an array of formulae continuing to involve the Euler totient function. We formulate six precise conjectures. These include a novel study of an "inverse" function of the Mirzakhani's asymp…

计算Seifert纤维化空间上的$SL(2,\mathbb{C})$联络数

Counting $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces

本研究从规范理论视角研究Seifert纤维化同调3-球面的$SL(2,\mathbb{C})$特征簇，通过引入陈-西蒙斯泛函的扰动证明局部化结果，得到稳定轨迹的欧拉示性数和庞加莱多项式公式，并建立欧拉示性数与对应奇点米尔诺数的关系。