日常喝咖啡，可以使我们感悟到奇妙的拓扑学原理。我们均匀搅拌咖啡，咖啡静止后会有一个分子回到初始位置，这是经典的布劳威尔不动点定理，我们用代数拓扑的方法加以证明；咖啡拉花经过搅拌后变得愈发复杂，但是咖啡拉花的抽象模式保持不变，这反映了曲面同胚映射的动力学特性，可以用泰希米勒空间和布劳威尔不动点定理加以证明。

Drinking coffee in daily life can help us to observe surprising and beautiful topological principles. Suppose we gently stir the coffee, when the coffee becomes still，there must be a coffee molecule that returns to the initial position. This phenomenon can be described by the classical Brouwer fixed point theorem, which can be proven using algebraic topology method. Gently swirl the pitcher, the latte art pattern becomes more and more complex, but the abstract pattern will remain the same. This reflects the dynamics of surface homeomorphisms, and can be proven using Teichmuller space theory and Brouwer fixed point theorem.