持久图与热方程同伦
Persistence Diagrams and the Heat Equation Homotopy
摘要：
持久性同源性是一种用于测量数据集和函数中存在的拓扑特征的工具。当我们遍历子级数据集或感兴趣的函数时，持久性将这些特征的出生和死亡配对。我关心的是用持久性来刻画两个函数f，g：M -> R之间的差异，其中M是一个拓扑空间。此外，我通过将热方程应用于差函数g-f来制定从g到f的同伦。通过堆叠与这个同伦相关的持久性图，我们创建了一个曲线葡萄园，将f的图中的点与g的图中的点连接起来。我看了这些图，其中M是正方形，球体，环面，还有克莱因瓶。查看这四个拓扑，我们注意到随着持久性图随时间变化的趋势（和差异）。
英文摘要：
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. I am concerned with using persistence to characterize the difference between two functions f, g : M -> R, where M is a topological space. Furthermore, I formulate a homotopy from g to f by applying the heat equation to the difference function g-f. By stacking the persistence diagrams associated with this homotopy, we create a vineyard of curves that connect the points in the diagram for f with the points in the diagram for g. I look at the diagrams where M is a square, a sphere, a torus, and a Klein bottle. Looking at these four topologies, we notice trends (and differences) as the persistence diagrams change with respect to time.
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