Research

Topological Data Analysis
The last 15 years has seen the explosion of new applications of parts of mathematics that were largely consider pure and esoteric up to that point by the scientific community. Our group focuses on applying topological methods to data, using the theory of persistent homology, created by Herbert Edelsbrunner and collaborators. We develop new applications of results from topology, recent work includes sliding windows with persistence for finding periodic patterns, persistent local homology for finding local point cloud shape, and persistent obstruction theory for finding consistency in databases, system internals, networks, etc.
The last 15 years has seen the explosion of new applications of parts of mathematics that were largely consider pure and esoteric up to that point by the scientific community. Our group focuses on applying topological methods to data, using the theory of persistent homology, created by Herbert Edelsbrunner and collaborators. We develop new applications of results from topology, recent work includes sliding windows with persistence for finding periodic patterns, persistent local homology for finding local point cloud shape, and persistent obstruction theory for finding consistency in databases, system internals, networks, etc.
Biological Clocks
Much recent research concerns the structure of networks and their application to biological systems. Less is known about the dynamics on those networks and how these describe the functioning of those systems. We focus on Boolean Networks, Dynamical Systems and Stochastic Networks and are particularly interested in understanding the relationship between these as models for gene regulation.
Much recent research concerns the structure of networks and their application to biological systems. Less is known about the dynamics on those networks and how these describe the functioning of those systems. We focus on Boolean Networks, Dynamical Systems and Stochastic Networks and are particularly interested in understanding the relationship between these as models for gene regulation.
Yeast Cell Cycle
The duplication and division cycle of a cell is a complex event, one that has been studied for a long time. Budding yeast, Saccharomyces cerevisiae, is a widely studied model of this process. We work with Prof. Steve Haase of the Duke Biology Department to understand how networks of gene transcription and cyclin/Cdk complexes coordinate to control cell cycle events. In particular, we are providing mathematical approaches to questions of network structure and dynamics. The mathematical methods we use have been developed in our projects in computational topology and network dynamics and are suggesting new experiments that will unveil critical network structure.
The duplication and division cycle of a cell is a complex event, one that has been studied for a long time. Budding yeast, Saccharomyces cerevisiae, is a widely studied model of this process. We work with Prof. Steve Haase of the Duke Biology Department to understand how networks of gene transcription and cyclin/Cdk complexes coordinate to control cell cycle events. In particular, we are providing mathematical approaches to questions of network structure and dynamics. The mathematical methods we use have been developed in our projects in computational topology and network dynamics and are suggesting new experiments that will unveil critical network structure.
Geometric Models
Managing and understanding data requires models that capture critical structure and properly deal with noise in order to avoid misclassification and over-fitting. Traditionally this has meant the use of statistical models. We use geometric models that learn their structure directly from the data themselves.
Managing and understanding data requires models that capture critical structure and properly deal with noise in order to avoid misclassification and over-fitting. Traditionally this has meant the use of statistical models. We use geometric models that learn their structure directly from the data themselves.
Statistics on Persistence Diagrams
Since persistence is a critical tool for capturing the intrinsic shape of data, it is important that one be able to do statistical analyses on persistence diagrams. Questions such as “what is the expected persistence diagram of a dataset of this type?”, “how much do shapes vary for this kind of data”, etc. can now be answered in statistical terms but computing means and variances of persistence diagrams.
Since persistence is a critical tool for capturing the intrinsic shape of data, it is important that one be able to do statistical analyses on persistence diagrams. Questions such as “what is the expected persistence diagram of a dataset of this type?”, “how much do shapes vary for this kind of data”, etc. can now be answered in statistical terms but computing means and variances of persistence diagrams.
Machine Learning with Persistence
TDA methods take point clouds, distance matrices and other kinds of inputs and return persistence diagrams, which are collections of points in the plane that represent shape features and their size measurement (persistence). We recently developed the capability (included in [ref]) to use these diagrams as features in machine learning.
TDA methods take point clouds, distance matrices and other kinds of inputs and return persistence diagrams, which are collections of points in the plane that represent shape features and their size measurement (persistence). We recently developed the capability (included in [ref]) to use these diagrams as features in machine learning.