A few specific research directions that I am currently interested in include (1) general-purpose methods for computing persistent relative
homology and applications to data science, (2) algorithm unrolling techniques for implementing deep neural networks, (3) the Laplace-Beltrami
operator and its applications to geometric deep learning and (4) formalization of the manifold hypothesis. My broad interests are listed below.
Mathematics
Topology, Geometry, Homological algebra
Applied Mathematics
Numerical linear algebra, Computational geometry
Algorithms & Data Science
Topological data analysis, (Geometric) machine learning, Complexity theory
Projects
Persistent Relative Homology for Data Science
I am currently working as the sole author of an open-source software module to implement a general-purpose method to compute
persistent relative homology with an emphasis on applications in TDA and data science. The implementation is written in Python and
utilizes low-level Rust via the Open Applied Topology (OAT)
project. The code utilizes modern data structures for matrix algebra which compress sparse matrices and lazily generate their
rows and columns on an as-needed basis. An example is included below.
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Undergraduate Capstones
Exact and Stable Computation of Persistent Relative Homology
Working under Drs. Lori Ziegelmeier and Gregory Henselman-Petrusek (PNNL), I
developed a novel algorithm for the direct computation of persistent relative homology. Utilizing a recently presented
matrix factorization scheme, our method can compute the barcode decomposition
and persistence module of a pair of topological spaces equipped with arbitrary filtrations. The only restriction on these filtrations
is that the inclusion of spaces is preserved at each level. This is the most general formulation of Persistent Relative homology, and
our work provides a novel and simplified framework for studying it and its variants via U-match decomposition. I completed an undergraduate
honors thesis related to this work, which can be viewed at the links below.
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Evolving Mobile Agents in Conway’s Game of Life
This project was completed as my computer science capstone during the Fall of 2023, when I took an introductory
course in Artificial Intelligence with my academic advisor Susan Fox.
Using the popular cellular automaton Conway's Game of Life,
we investigated to capacity for agents (or cell tilings) within the game to exhibit emergent or intelligent behavior.
The explicit goal was to write a genetic algorithm which, given a randomly generated popualtion of cell tilings, could
be used to evolve subsequent generations of cell tilings which are capable of directed growth and movement. Our
genetic algorithm is implemented in Python and required the NumPy library and elementary linear algebra in order
to assess the behavior and fitness of the population. The project also includes a front-end animation built with
Matplotlib. Check out the links below to see the project in more detail.
Numerical Techniques for the Wave Equation
This project was completed as my mathematics capstone during the Fall of 2023, when I took a math modeling
course with Will Mitchell. This
was primarily an investigation of numerical techniques for approximating solutions to partial
differential equations, and in particular finite differences and the Fourier spectral method. I studied each of
these techniques using the wave equation, and implemented both. Check out the links below to see my work in
more detail.
