ResearchGraduate Resarch ProjectsPersistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification Fall 2022 - Spring 2023 I worked with Professor Ekaterina and Professor Guowei Wei on this project. We developed a semi-supervised graph based method for data classification. We propose an algebraic topology-based semi-supervised method called persistent Laplacian-enhanced graph MBO (PL-MBO) by integrating persistent spectral graph theory with the classical Merriman-Bence- Osher (MBO) scheme. Specifically, we use a filtration procedure to generate a sequence of chain complexes and associated families of simplicial complexes, from which we construct a family of persistent Laplacians. Overall, it is a very efficient procedure that requires much less labeled data to perform well compared to many ML techniques, and it can be adapted for both small and large datasets. We evaluate the performance of the proposed method on data classification, and the results indicate that the proposed technique outperforms other existing semi-supervised algorithms.
Undergraduate Research ProjectsHonors College Thesis: Twisted Central Configurations of the Eight-Body problem The University of Southern Mississippi, Hattiesburg, MS I did an honors project advised by Professor Zhifu Xie . I studied a special type of central configuration: twisted central configurations of the eight-body problem. We showed that when the two square configurations have a common centroid, the configuration can form a central configuration only if the ratio of the size of the two squares falls into one of three intervals. Moreover there are some numerical evidences that there are exactly three nested central configurations for each given mass ratio m1/m5.
Wright W. and Annie Rea Cross Research: Central configurations in the planar 6-body problem forming two equilateral triangles The University of Southern Mississippi, Hattiesburg, MS I was fortunate to get Wright W. and Annie Rea Cross Mathematics Undergraduate Research Scholarship. As a cross scholar, I worked with Professor Zhifu Xie and Hamas Tahir on this project. In this project, six bodies are located on two equilateral triangles 123 and 456. Assume that both triangles are symmetrical about the line connecting m3 and m4. Within these configurations, the six body configuration is not a central configuration if the triangle 123 is above or below the triangle 456. It is also not a central configuration if more than two of the six bodies are collinear. When the two equilateral triangle configurations have a common centroid, masses on each equilateral triangles must be same respectively and the configuration can form a central configuration only if the ratio of the lengths of the sides between 123 and 456 falls into one of five intervals. Moreover there are some numerical evidences that, first there are exactly two nested central configurations but there may be one, two, or three twisted nested CCs for a given mass ratio; and second, there exists central configurations other than same centroid. |