image-left     I am a math PhD candidate at Michigan State University working with Guo-Wei Wei to develop tools in Topological Data Analysis (TDA) for computational biology.

I earned my M.A. in Mathematics, B.S. in Computer Science, and B.S. in Mathematics at The University of Alabama.

ORCID:0000-0001-7810-7770

Research

Overview

Persistent Topological Laplacians (PTLs) generalize persistent homology, like the one in this paper by Rui Wang. They also generalize the graph Laplacian of spectral graph theory to higher-order simplicial complexes (and other structures, such as path complexes). The eigenvalues (spectra) of a PTL reveals persistent homology and geometric qualities of the complex. Computing the PTL and its eigenvalues is hard.

As part of the Spring 2025 WinCompTop+AATRN Tutorial-a-thon, I made a YouTube tutorial that introduces Persistent Laplacians.

I am currently interested in:

  1. The behavior of persistent (and non-Persistent) topological Laplacians defined in different contexts, such as directed flag (clique) complexes3, the chain complex arrising from Khovanov homology5, and with cellular sheaves.
  2. Ways to efficiently compute the persistent Laplacian and its eigenvalues. Persistent Laplacian computations are much slower in practice than persistent homology. I am working on a Python and C++ library for fast and user-friendly computations.
  3. Applying persistent Laplacians to problems in molecular biology, by way of machine learning methods4, 6.
  4. Applying persistent homology of asymmetric/directed networks (directed flag/clique complexes and Dowker complexes) to study dynamical systems arrising from climate science.

Papers and preprints

  1. Dong Chen, Gengzhuo Liu, Hongyan Du, Benjamin Jones, Junjie Wee, Rui Wang, Jiahui Chen, Jana Shen, and Guo-Wei Wei,"Drug Resistance Predictions Based on a Directed Flag Transformer," 2024. Submitted.
  2. Benjamin Jones and Guo-Wei Wei, "Khovanov Laplacian and Khovanov Dirac for Knots and Links," arXiv:2411.18841 [math.GT], 2024. Submitted.
  3. Mushal Zia, Benjamin Jones, Hongsong Feng, Guo-Wei Wei, "Persistent Directed Flag Laplacian (PDFL)-Based Machine Learning for Protein-Ligand Binding Affinity Prediction," 2024. Accepted to Journal of Chemical Theory and Computation. (arxiv) (doi)
  4. Benjamin Jones and Guo-Wei Wei, "Persistent Directed Flag Laplacian," Foundations of Data Science, 2024. (arxiv) (doi)
  5. S. Ahmed Ullah, X. Yang, Ben Jones, S. Zhao, W. Geng, and G. Wei, "Bridging Eulerian and Lagrangian Poisson-Boltzmann solvers by ESES," Journal of Computational Chemistry. 2023, 1. (doi)
  6. Benjamin Jones, S. Ahmed Ullah, S. Wang, and S. Zhao, "Adaptive pseudo-time methods for the Poisson-Boltzmann equation with Eulerian solvent excluded surface", Communications in Information & Systems, 21(1), 85-123, (2021). (arxiv) (doi)

Contributed Talks

  1. Graduate Student Geometry and Topology Conference 2025, Indiana University Bloomington. "Persistent and combinatorial Laplacians for topological data analysis and their introduction to knot theory," April 2025.
  2. Women in Computational Topology (WinCompTop) + Applied Algebraic Topology Research Network (AATRN) Spring 2025 Tutorial-a-thon, YouTube tutorial. "Persistent Laplacians: What they are, why you should care, and how to compute them," February 2025.
  3. Joint Mathematics Meetings (JMM), Seattle, WA. AMS Special Session on Topological Data Analysis: Theory and Applications. "Efficient Computation of Persistent Laplacians," January 2025.
  4. Joint Mathematics Meetings (JMM), Seattle, WA. MRC Climate Science at the Interface Between Topological Data Analysis and Dynamical Systems Theory. "Dynamics-Aware Filtrations," January 2025.
  5. SIAM Conference on Mathematics of Data Science (MDS24), Atlanta, GA. Minisymposium on Exploring the Intersection of Topological and Geometric Data Analysis with Biological Applications. "Persistent Directed Flag Laplacian," October 2024.
  6. 4th Workshop on Computational Persistence (ComPer), Graz University of Technology (by zoom). "Efficient Computation of Persistent Laplacians," September 2024.
  7. Mathematical Biosciences Workshop, Penn State University. "Persistent Directed Flag Laplacian," August 2024.

Seminar Talks

  1. MSU TDA Seminar. April 2025.
  2. MSU Student Geometry and Topology Seminar. "Combinatorial and Persistent Laplacians: from Graphs to TDA," October 2024.
  3. MSU Operator Algebras Reading Seminar. "The story of how Vaughan Jones used operator algebras to spark a revolution in topology," March 2024.
  4. University of Alabama Applied Math Seminar, Tuscaloosa, AL. "Adaptive Pseudo-Time Methods for the Poisson-Boltzmann Equation with Eulerian Solvent Excluded Surface," December 2020.

Theses

Ph.D. (in progress): B. Jones. “Aspects of Applied Geometric and Algebraic Topologies.” Doctoral thesis at Michigan State University. Committee: Guo-Wei Wei (chair), Mark Iwen, Ekaterina Merkurjev, and Elizabeth Munch.

Master’s: B. Jones. “Adaptive pseudo-time methods for the Poisson-Boltzmann equation with Eulerian solvent excluded surface,” (2021). Master’s thesis at The University of Alabama.
Committee: Shan Zhao (chair), Xiaoyan Hong, Mojdeh Rasoulzadeh, and Wei Zhu.

Contact information

benjamindanieljones@gmail.com

jones657@msu.edu