My research focuses on creating and adapting theoretical and algorithmic tools for data analysis via applied topology and applied algebraic geometry. The unifying theme for both is topological data analysis (TDA).
Statistical TDA and Persistence Landscapes
Considerable theoretical research over the past 15 years has adapted computationally intractable tools from topology, particularly algebraic topology, into forms useful for data analysis. Persistent homology, one of the major tools resulting from that research, offers many advantages to analysts. Persistent homology algorithms typically take a shape as input, and output a persistence diagram which encodes the size and number of holes in the shape. To use statistical and machine learning methods, a persistence diagram must be transformed into a real-valued vector in an appropriate way so that averages, sums, and differences may be taken.
The persistence landscape introduced by Peter Bubenik in 2015 provides a robust method of making this transformation. Together with Leo Betthauser and Peter Bubenik, I have introduced a new graded persistence diagram, and shown that it is in a strong sense equivalent to the persistence landscape. This result yields an elegant connection between persistence landscapes and the combinatorial Möbius inversion used to construct persistence diagrams.
The graded diagram perspective on landscapes shows that landscapes are a refinement of usual persistence diagrams. It also suggests multiparameter versions of graded diagrams and more refined stability guarantees relating the distance between landscapes to the distance between persistence diagrams.
Sampling Real Algebraic Varieties for TDA
The published version of this project is available at https://ieeexplore.ieee.org/abstract/document/8999343 and an extended preprint is available at https://arxiv.org/abs/1802.07716. Slides from a presentation are available here. My MSc thesis on the subject is here.
Real algebraic varieties are the sets of solutions to systems of polynomial equalities. They arise naturally in applications where modelling a physical system is subject to distance and angle constraints. Real varieties are in general one of the classes of spaces with self-intersections which are simple to describe (via a variety’s defining polynomial equations). To apply persistent homology to real algebraic varieties we need a dense sample of points from the variety as input. The most common specification of real varieties, however, is by listing a variety’s defining equations.
Together with Emilie Dufresne, Heather Harrington, and Jon Hauenstein, I have developed an adaptive algorithm to generate dense point samples from algebraic varieties while keeping the size of the point sample low. An implemented version of the algorithm is available as the Python package tdasampling: https://github.com/P-Edwards/tdasampling.
Topological Data Analysis for Actin Networks
Actin proteins in cells are involved in a large range of cellular and bodily functions like cell movement and muscle contraction. Individual actin monomers link together in chains to form semi-rigid fibers, and these fibers exhibit medium range geometric structure that acts as a type of scaffolding for the cell. Incorrect regulation of actin in cells can lead to disease in humans, so current research looks to quantify the way different actin regulatory mechanism affect network structure.