Projects

These research projects are advised or led by Dr. Brittany Terese Fasy, Dr. David L. Millman, and Dr. Binhai Zhu. The focus of the research conducted by the club CompTAG is applied topological data analysis, and computational geometry. Our goal is to apply theoretical knowledge on the subjects to real data sets in new and meaningful ways. This page is meant to be a way for those interested in our work to be introduced to the many projects we work on.

In this project we are extending work on reconstructing shapes using persistence diagrams. We are currently working on extending our results from this our paper "Learning Simplicial Complexes from Persistence Diagrams" that was published at CCCG18.

In this project we use topological features of data collected on roads and maps to learn about how people move in different infrastructures. We also use this information to see if the current infrastructure is well designed or otherwise. This work is often done in collaboration with Dr. Carola Wenk at Tulane University, and has many sub-projects that are within the scope of analyzing data on road networks.

Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a set of other persistence diagrams is linear in the number of diagrams or do not offer performance guarantees. In this research project, we apply concepts from locality-sensitive hashing to support approximate nearest neighbor search in the space of persistence diagrams.

The tools of topological data analysis (TDA), such as persistence diagrams, provide new descriptors of data sets, with the potential to uncover new insights into data. However, persistence diagrams do not naturally lend themselves to statistical goals, such as inferring certain population characteristics, because their complicated structure makes common algebraic operations--such as addition, division, and multiplication-- challenging (e.g., the mean might not be unique). In our research at the intersection of topology and statistics, we define new topological desriptors, and prove statistical properties of these descriptors.

Finding the minimum area swept out by continuously deforming a closed curve in the plane to a point is an area of active research. One application is measuring the similarity between two sets of GPS coordinates. We use properties of the curves to develop efficient algorithms to find the minimum homotopy area.

Facility location is a traditional geometric optimization problem, the objective being optimizing certain geometric criteria to build optimal (or sub-optimal) locations for the facilities. However, in many modern applications, significant constraints could be placed on the facilities (e.g., a noisy facility should be far away from the residential areas). We studied some of these problems recently, e.g., Hitting a Set of Line Segments with One or Two Discrete Centers, CCCG 2018, by Xiaozhou He, et al.

The long-term goal of this project is to develop quantitative methodology for detecting geometric and topological features in point clouds extracted from (histology) images. Of particular relevance, this project considers the setting of prostate cancer classification, which is based on a pathologist grading of histology slides using the Gleason grading system. These pathology slides are a source of biomedical big data that are increasingly available as archived material. Developing these quantitative methods will be a significant advance towards a (semi-)automated quantification of prostate cancer aggressiveness. This award supports an interdisciplinary team of investigators in computational mathematics, computer science, biomedical engineering, and pathology to develop mathematical and computational tools based on topological descriptors and machine learning in order to distinguish between different morphological types of prostate cancer.

Biofilms Resources and Information Database (BRaID) is a centralized database and analysis tool for the diverse types of biofilm-related data such as metagenomics, images, and trajectories, etc. BRaID will give biofilm researchers easier access to large amounts of data and analytical tools.

The TDA package in R is an important tool for the implementation of key functions used in topological data analysis. However, the TDA package could be made more effective. Currently, we are working to simplify use of the package and enhance its viability in the future by improving and expanding upon its documentation. The end goal of this is to limit gaps between mathematical knowledge and computational experience, to make the package more accessible to a greater number of people. Furthermore, contributions to the package in the future will also be made simpler with increased documentation. Along with improving the package's documentation, we are also working to create the infrastructure to better test code functionality before it is contributed to the package. Lastly, we are working on creating a survey to request user feedback and to help understand how the TDA package can be further improved going forward.