Bradley J. Nelson

Contact

Email: firstnamelastname@protonmail.com
Location: 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 co-advised 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.

Curriculum Vitae

Announcements

• May 2022 New preprint with Lek-Heng Lim: What is an equivariant neural network?
• May 2022 New preprint on Parameterized Vietoris-Rips Filtrations via Covers
• January 2022 New preprint with Yuan Luo: Topology-Preserving Dimensionality Reduction via Interleaving Optimization
• Winter 2022 I am teaching a course on Scientific Computing with Python and a course on Topological Data Analysis.

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 Time-Aware Machine Intelligence. 2020-2022. With Lek-Heng 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. Bruel-Gabrielsson, 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: 10.3997/2214-4609.201901608.
• Boundary integral equation solution of high frequency scattering from obstacles in an unbounded linearly graded-index medium.
  A. H. Barnett, B. J. Nelson, and J. M. Mahoney.
  Journal of Computational Physics 297C:407-426, 2015.
• Chiral Polymerization in Open Systems From Chiral-Selective Reaction Rates.
  M. Gleiser, B. J. Nelson, and S. I. Walker.
  Origins of Life and Evolution of Biospheres 42:333-346, 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

• 02.18.2022 Vietoris-Rips Seminar
  Parameterized Vietoris-Rips 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 d-dimensional 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]

University of Chicago

• Winter 2022 (previous: Winter 2021)
  STAT 37411=CAAM 37411 - Topological Data Analysis.
  [Course Description] [GitHub] [Website]
  Graduate-Level 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 co-taught 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.

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