Bradley J. Nelson
Contact
Email: firstnamelastname@protonmail.comLocation: New York, New York
GitHub: bnels
Google Scholar
I am now employed outside of academia  you can contact me at firstnamelastname@protonmail.com (with appropriate substitutions).
I am a William H. Kruskal Instructor in the Department of Statistics at the University of Chicago. I completed my PhD at the Institute for Computational and Mathematical Engineering (ICME) at Stanford Univesity in June 2020, advised by Gunnar Carlsson and coadvised by Jonathan Taylor.
My primary research interests are in applied and computational algebraic topology, particularly topological data analysis. I am interested in the interface between applied/computational topology and other areas of mathematics, data science, and computing.
News
Announcements

May 2022 New preprint with LekHeng Lim: What is an equivariant neural network? 
May 2022 New preprint on Parameterized VietorisRips Filtrations via Covers 
January 2022 New preprint with Yuan Luo: TopologyPreserving Dimensionality Reduction via Interleaving Optimization 
Winter 2022 I am teaching a course on Scientific Computing with Python and a course on Topological Data Analysis.
Research
Overview
My primary research focus is on topological data analysis. This is a relatively young branch of applied mathematics, rooted in the more classical subject of algebraic topology. The general goal is to learn something about data using various topological invariants, and a popular tool is persistent homology.
More broadly, I am interested in applied mathematics, machine learning, and data science.
Recent/Current Projects
 DARPA HR00112190040: Topolgical Control for TimeAware Machine Intelligence. 20202022. With LekHeng Lim. Developing and applying tools from topological data analysis to neural networks for temporal reasoning.

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint.
G. Carlsson, A. Dwaraknath, B. J. Nelson.
[arXiv:1911.10693] [Code on GitHub] 
Accelerating Iterated Persistent Homology Computations with Warm Starts.
Y. Luo, B. J. Nelson
[arXiv:2108.05022] Code in BATS. 
Topological Regularization for Dense Prediction.
D. Fu, B. J. Nelson
[arXiv:2111.10984] Code forthcoming.
Publications

Clustering of Electromagnetic Showers and Particle Interactions with Graph Neural Networks in Liquid Argon Time Projection Chambers Data.
F. Drielsma, et al.
Phys Rev. D 104, 072004 (2021) [Link] 
A Topology Layer for Machine Learning.
R. BruelGabrielsson, B. J. Nelson, A. Dwaraknath, P. Skraba, L. J. Guibas, G. Carlsson.
AISTATS 2020. [arXiv:1905.12200] [Code on GitHub] 
Texture Based Classification of Seismic Image Patches Using Topological Data Analysis.
R. Sarkar, B. J. Nelson.
81st EAGE Conference and Exhibition 2019. DOI: 10.3997/22144609.201901608. 
Boundary integral equation solution of high frequency scattering from obstacles in an unbounded linearly gradedindex medium.
A. H. Barnett, B. J. Nelson, and J. M. Mahoney.
Journal of Computational Physics 297C:407426, 2015. 
Chiral Polymerization in Open Systems From ChiralSelective Reaction Rates.
M. Gleiser, B. J. Nelson, and S. I. Walker.
Origins of Life and Evolution of Biospheres 42:333346, 2012.
PhD Dissertation

Parameterized Topological Data Analysis. Bradley J. Nelson.
Ph.D. dissertation. Stanford University. June 2020.
[link][defense slides]
Older, Unpublished Projects

Sparse canonical correlation analysis.
X. Suo, V. Minden, B. Nelson, R. Tibshirani, M. Saunders.
arXiv:1705.10865
Talks and Presentations

02.18.2022 VietorisRips Seminar
Parameterized VietorisRips Filtrations via Covers. [slides] 
03.01.2021 SIAM CSE, Minisymposium on Emerging Directions in Computational Topolgy
Parallel Computation of Zigzag Homology using Matrix Factorizations. [abstract] [slides] 
01.08.2021 Joint Mathematics Meetings, AMS Special Session on Applied Topology
A fibration model for ddimensional image patches. [abstract] [slides] 
11.05.2020 CAAM Colloquium, University of Chicago
Topological Data Analysis, Linear Algebra, and Optimization. [announcement] [slides] 
05.03.2019 Stanford Exploration Project Seminar
Stanford, CA [slides] 
10.01.2018 AI at SLAC seminar
Stanford Linear Accelerator (Menlo Park, CA) 
08.11.2017 Applied Algebraic Topology 2017
Hokkaido University (Sapporo, Japan) [slides] 
08.01.2017 SIAM Algebraic Geometry 2017
Atlanta, GA [slides]
Teaching
University of Chicago

Winter 2022 (previous: Winter 2021)
STAT 37411=CAAM 37411  Topological Data Analysis.
[Course Description] [GitHub] [Website]
GraduateLevel topics course on Topological Data Analysis. 
Winter 2022 (previous: Fall 2020)
STAT 37830=CAAM 37830  Scientific Computing with Python.
[Course Description] [GitHub] [Website] [Reader]
For students getting started with scientific computing. Covers the Python language and its numerical libraries, as well as use of computing resources.
Stanford University

Fall 2018 CME 193  Introduction to Scientific Python.
[Explore Courses] [GitHub] [Website]
Introduction to Python and its popular scientific computing libraries. 
Spring 2018 CME 258  Libraries for Numerical Linear Algebra and Optimization.
[Explore Courses] [GitHub]
Designed and cotaught with Ron Estrin. Introduction to different libraries for numerical linear algebra and optimization. 
Winter 2018 CME 257  Advanced Topics in Scientific Computing with Julia.
[Explore Courses] [GitHub]
Updated for Julia v0.6.2, and added a new lecture on metaprogramming. 
Fall 2015 CME 257  Advanced Topics in Scientific Computing with Julia.
[Explore Courses] [GitHub]
This is a new course that I created to introduce the Julia language to students interested in using it for development/research.
Miscellaneous

Questions about mathematical modeling or algorithms?
Come by Computational Consulting.
A service offered by ICME graduate students.