Persistent homology is a method for probing topological properties of point clouds and function. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be “topological noise.” In our work, We attempt to bring statistical ideas to persistent homology. In particular, we have derived confidence intervals that allow us to separate topological signal from topological noise.
It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n >=2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpinan and Williams exhibited n+1 isotropic Gaussian kernels in R^n with n+2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points and that the resilience of the extra mode grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.
We bound the difference in length of two curves in terms of their total curvatures and the Fréchet distance. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner, and it generalizes a result by Fáry and Chakerian.
Researchers across many fields are interested in the topological and geometric properties of shapes in Euclidean space. We describe three open questions related to metrics and scale space. First, we present a result that bounds the difference of lengths of curves in Euclidean space by a function of the total curvature and the Fréchet distance between the curves. Although this result improves upon another inequality that additionally depends on the dimension in which the curves are embedded, it is still unknown if the new inequality is tight. Second, we explore the deep structure, or scale space, of images. The scale space of an image is a family of related images obtained through convolution with the Gaussian kernel. We ask how to extend this idea to the study of a more general class of functions. Finally, we introduce the vineyard--a collection of curves in R^3--associated with a homotopy. In particular, we use the heat equation homotopy, which arises from scale space, to measure the distance between two functions. We ask how to quantify the relationship between this metric and other metrics between functions.
Persistence homology is a tool used to classify 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. In this project, I am concerned with using persistence to characterize the difference between two functions f, g : S R, where S is a topological space embedded in R^n. Furthermore, I assume that we have a homotopy between the functions f and g. By stacking the persistence diagrams, we create a vineyard of curves that connect the points in the diagram for f with the points in the points in the diagram for g. The choice of homotopy will affect the resulting vineyard. I investigated the changing behavior of the persistence diagrams when the heat equation is used to create a homotopy between the functions f and g.
A basic problem in mathematics is the classification problem. In group theory we have the Classification Problem for Groups: Given a collection G of groups, classify the groups G in G up to isomorphism. And in homotopy theory, we have the Homotopy Classification Problem for Spaces: Given a collection T of topological spaces, classify the spaces X in T up to homotopy equivalence. We show that in some cases, these classification problems are actually equivalent. In addition, we give some examples classifying groups by the centralizers and classifying the path components of function spaces.
Computer Science Education
adviser: Dr. Stephen Cooper
I created virtual worlds using a 3D interactive animation environment, Alice. This environment had previously been used as a program visualization tool for introductory computer science classes. The worlds I created enable use of this tool at the intermediate programming level by introducing the concepts of lists and arrays. This research has provoked interest in expanding the use of Alice to higher level CS courses.