Paul L Bendich, Research Professor

I am a mathematician whose main research focus lies in adapting theory from ostensibly pure areas of mathematics, such as topology, geometry, and abstract algebra, into tools that can be broadly used in many data-centered applications.

My initial training was in a recently-emerging field called topological data analysis (TDA). I have been responsible for several essential and widely-used elements of its theoretical toolkit, with a particular focus on building TDA methodology for use on stratified spaces. Some of this work involves the creation of efficient algorithms, but much of it centers around theorem-proof mathematics, using proof techniques not only from algebraic topology, but also from computational geometry, from probability, and from abstract algebra.

Recently, I have done foundational work on TDA applications in several areas, including to neuroscience, to multi-target tracking, to multi-modal data fusion, and to a probabilistic theory of database merging. I am also becoming involved in efforts to integrate TDA within deep learning theory and practice.

I typically teach courses that connect mathematical principles to machine learning, including upper-level undergraduate courses in topological data analysis and more general high-dimensional data analysis, as well as a sophomore level course (joint between pratt and math) that serves as a broad introduction to machine learning and data analysis concepts.

Contact Info:

Office Location:	121 Physcis Bldg, Durham, NC 27708
Email Address:
Web Page:	http://www.paulbendich.com

Teaching (Fall 2024):

• MATH 412.01, TOPOLOGICAL DATA ANALYSIS Synopsis

  Gross Hall 318, TuTh 08:30 AM-09:45 AM

  (also cross-listed as COMPSCI 434.01)

• MATH 713.01, TOPOLOGICAL DATA ANALYSIS Synopsis

  Gross Hall 318, TuTh 08:30 AM-09:45 AM

Office Hours:

Monday, 11 AM - Noon, Math 210

Friday, 11:45 - 1 PM Gross Hall 327

Education:

Ph.D.

Duke University

2008

Specialties:

Topology
Applied Math

Research Interests:

I work in computational topology, which for me means adapting and using tools from algebraic topology in order to study noisy and high-dimensional datasets arising from a variety of scientific applications. My thesis research involved the analysis of datasets for which the number of degrees of freedom varies across the parameter space. The main tools are local homology and intersection homology, suitably redefined in this fuzzy multi-scale context. I am also working on building connections between computational topology and various statistical data analysis algorithms, such as clustering or manifold learning, as well as building connections between computational topology and diffusion geometry.

Undergraduate Research Supervised

• Marshall Ratliff (2014/08-present)
  Data RTG Topology for Music and Brains Program, and Research Independent Study (Cover Trees for Jazz Data) 
• Derrick Nowak (2014/08-present)
  Data RTG Topology for Music and Brains Program 
• Carmen Cox (2014/08-present)
  Data RTG Topology for Music and Brains Program 
• Alex Pieloch (2014/08-present)
  Data RTG Topology for Music and Brains Program 
• Aaron Park (2014/08-present)
  co-mentored with Ezra Miller 
• Bingxi Lin (May, 2013 - July, 2013)
  Data RTG REU program on Multi-Scale Topology for Signals and Images 
• Michael Ogez (May, 2013 - July, 2013)
  Data RTG REU program on Multi-Scale Topology for Signals and Images 
• Ben Dreyzen (May, 2013 - July, 2013)
  Data RTG REU program on Multi-Scale Topology for Signals and Images 
• Bryan Jacobson (2012 - 2014)  

Recent Publications   (More Publications)

1. Solomon, YE; Bendich, P, Convolutional persistence transforms, Journal of Applied and Computational Topology (January, 2024) [doi]  [abs]
2. Catanzaro, MJ; Dharna, A; Hineman, J; Polly, JB; McGoff, K; Smith, AD; Bendich, P, Topological Decompositions Enhance Efficiency of Reinforcement Learning, IEEE Aerospace Conference Proceedings (January, 2024) [doi]  [abs]
3. Jin, Y; McDaniel, R; Tatro, NJ; Catanzaro, MJ; Smith, AD; Bendich, P; Dwyer, MB; Fletcher, PT, Implications of data topology for deep generative models, Frontiers in Computer Science, vol. 6 (January, 2024) [doi]  [abs]
4. Koplik, G; Borggren, N; Voisin, S; Angeloro, G; Hineman, J; Johnson, T; Bendich, P, Topological Simplification of Signals for Inference and Approximate Reconstruction, IEEE Aerospace Conference Proceedings, vol. 2023-March (January, 2023), ISBN 9781665490320 [doi]  [abs]
5. Smith, AD; Angeloro, G; Catanzaro, MJ; Patel, N; Bendich, P, Topological Parallax: A Geometric Specification for Deep Perception Models, Advances in Neural Information Processing Systems, vol. 36 (January, 2023)  [abs]

Recent Grant Support

• Geometric and Topological Methods for Multi-Modal Data Analysis and Fusion, Air Force Office of Scientific Research, FA9550-18-1-0266, 2018/06-2023/06.      

Conferences Organized

• conference. December 06, 2011, conference, December 2011