Other Research Interests
Modes of Gaussian Mixtures
To many researcher’s surprise, n+1 isotropic Gaussian kernels in R^n can have n+2 modes. I call these additional modes “ghost modes.” More generally, there exist finite configurations of isotropic Gaussian kernels with superlinearly many modes. Moreover, even the most simple configuration exhibits an exponential number of critical points.
Curves in Space
Researchers across many fields are interested in the topological and geometric properties of shapes in Euclidean space. In a 2011 Journal paper, I bound the difference of lengths of curves in Euclidean space by a function of the total curvature and the Fréchet distance between the curves.
Heat Equation Homotopy
Persistent homology is a tool used to classify topological features that are present in data sets and functions. I am interested in using persistence to 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. Viewing each image as a real-valued function, I stack the corresponding persistence diagrams. This creates a vineyard of curves that connect the points in the diagrams. I am interested in using the vineyard arising from the heat equation homotopy to define a distance between two images.
Computer Science Education
I created virtual worlds using a 3D interactive animation environment, Alice. Alice 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.
Homotopy Classification Problems
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. In some cases, these classification problems are actually equivalent. I am interested in understanding these cases. In particular, I look at examples classifying groups by the centralizers and classifying the path components of function spaces.
Publications and Preprints
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“Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions” by Herbert Edelsbrunner, BTF, and Günter Rote. Conference: SoCG 2012 Journal (2013): Discrete and Computational Geometry [BibTeX]
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“The Difference of Lengths of Curves in R^n” by BTF. Journal: Acta Sci. Math. (Szeged), 2011. [BibTeX]
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Prelim: “Discovering Metrics and Scale Space”. 2010. Unpublished manuscript: [PDF] [BibTeX]
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Research Initiation Project: the Heat Equation Homotopy, 2009. axXiv:1002.1937 BibTeX
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“Homotopy Classification of the Components of the Space of Maps into an Aspherical Space: a Problem in the intersection of Group Theory and Topology.” University Scholar Senior Thesis. [BibTeX]
Presentations