Goodwillie Calculus and Applied Topology

我的建议是关于持久同调和古德威利微积分。******假设我们在空间中有一些点S。我们可以把它们看作是数据集中的样本。我们可以尝试通过计算传统的统计量来理解它们，比如它们的平均值。然而，如果这些点都是从一个圆中选择的，我们怎么能看到这个圆呢？这类问题就是持久性同源性的设计目的。******这些点本身就是离散的点，完全不相连。我们可以慢慢地让它们变大，把它们变成像水滴一样的球形斑点。一开始它们都是分开的，但当我们把它们变大时，它们就会开始结合在一起。如果点安排得足够好，因为我们从一个圆开始，一个（厚）圆最终会出现。如果我们继续把斑点变大，最终圆圈的中心会被填满，最后我们只剩下一个大的斑点。******持久同源性可以用来理解这个圆出现然后消失的方式。特别是，如果当一个点周围的每个斑点都达到大小时出现一个圆圈，当每个斑点都达到大小时消失，那么我们得到一个“条”，开始于和结束于。条的长度表示当圆的半径增加时，圆持续的时间。对一般数据集S这样做可能会产生许多不同的柱状图。它们一起被称为数据集S的条形码，并告诉我们数据集的几何形状。持续时间较长的条形图（即如果-是大的）代表了更重要的几何特征。我们也可以用这种方法来理解数据集的高维几何特征。******这种查看数据的方式已经存在了20多年，并且一直在不断发展。我们建议研究持久同源的一些具体方面。首先，我们将研究条形码之间的度量（或距离）。这可以用来比较两个数据集的几何形状。S中的点也可以随时间变化，这就给出了两个参数的持久性。出现的结构要复杂得多，我们也会研究它们。******方案中的古德威利微积分部分也与几何有关。如果我们考虑机器人手臂的可能位置，台球桌上球的可能位置，或者星系中所有恒星的可能位置，我们就得到了流形。我们试着通过对流形应用特定的结构（称为函子）来理解它，看看我们得到了什么。问题是函子通常很复杂，所以我们把它们分解成所谓的泰勒塔。这类似于在微积分中取一个函数并用它的泰勒多项式代替它。一般的函数可能很复杂，但多项式却容易理解得多。我们最感兴趣的是可能出现的塔，换句话说，在这种情况下多项式是什么

My proposal is concerned with persistent homology and Goodwillie calculus.******Imagine we have a bunch of points S in space. We can think of them as samples from a data set. We can try to understand them by computing traditional statistical quantities like their mean (or average). However if the points were all selected from a circle, how can we see the circle? This type of question is what persistent homology was designed to answer. ******The points themselves are just discrete dots, completely disconnected. We can slowly make them bigger, turning them into spherical blobs like water droplets. At first they will all stay separate, but as we make them bigger they will start to join together. If the points are arranged nicely enough, since we started with points from a circle, a (thick) circle will eventually appear. If we keep making the blobs bigger eventually the center of the circle will be filled in and we end up with just one big blob. ******Persistent homology can be used to understand the way this circle appears and then disappears. In particular if a circle appears when each blob around a point is size and disappears when each blob is size , then we get a “bar” starting at and ending at . The length of the bar says how long the circle lasts as the blobs increase in radius. Doing this for a general data set S can give rise to many different bars. Together these are called the bar code of the data set S, and tell us about the geometry of the data set. Bars that persist for longer (ie if - is large) represent more significant geometric features. We can also use this method to understand higher dimensional geometric features of data sets.******This way of looking at data has been around for over 20 years, and has been undergoing continuous development. We propose to study some specific aspects of persistent homology. First we will study metrics (or distances) between bar codes. This can be used to compare the geometry of two data sets. The points in S can also be changing in time, and this gives two parameter persistence. The structures that arise are much more complicated and we will also study them. ******The Goodwillie calculus part of the proposal is also concerned with geometry. If we consider say the possible positions of a robot arm, the possible positions of the balls on a pool table, or the possible positions of all the stars in the galaxy, we get something called a manifold. We try to understand the manifold by applying certain constructions to it (called functors) and seeing what we get. The problem is that the functors are often complicated, and so we resolve them into their so called Taylor tower. This is analogous to taking a function in calculus and replacing it by its Taylor polynomials. A general function can be very complicated, but polynomials are much easier to understand. We are mostly interested in what possible towers can occur, in other words what are the polynomials in this context.**