Random triangles on flat tori

Random eigenfunctions on flat tori: universality for the number of intersections

We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues with a given smooth curve in flat tori are universal under various families of randomness.

全等六边形对平坦环面的铺砌

Tilings of Flat Tori by Congruent Hexagons

本文在所有顶点度数为3的条件下，分类三类一般六边形及两种特殊情形的平面铺砌，并据此刻画平坦环面的六边形铺砌与模空间。

The structure of $I_4$-free and triangle-free binary matroids

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most $2$.