Department of Mathematics
 Search | Help | Login | pdf version | printable version

Math @ Duke





.......................

.......................


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
Office Phone:  (919) 660-2811
Email Address: send me a message
Web Page:  http://www.paulbendich.com

Teaching (Spring 2024):

  • MATH 238L.001, DATA ANALYSIS AND DECISION SCI Synopsis
    Gross Hall 318, TuTh 08:30 AM-09:45 AM
  • MATH 238L.01L, DATA ANALYSIS AND DECISION SCI Synopsis
    Gross Hall 105, Tu 01:25 PM-03:10 PM
Office Hours:

Monday, 11 AM - Noon, Math 210

Friday, 11:45 - 1 PM Gross Hall 327
Education:

Ph.D.Duke University2008
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. 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]
  3. Solomon, E; Wagner, A; Bendich, P, FROM GEOMETRY TO TOPOLOGY: INVERSE THEOREMS FOR DISTRIBUTED PERSISTENCE, Journal of Computational Geometry, vol. 14 no. 2 Special Issue (January, 2023), pp. 172-196 [doi]  [abs]
  4. Voisin, S; Hineman, J; Polly, JB; Koplik, G; Ball, K; Bendich, P; D‘Addezio, J; Jacobs, GA; Özgökmen, T, Topological Feature Tracking for Submesoscale Eddies, Geophysical Research Letters, vol. 49 no. 20 (October, 2022) [doi]  [abs]
  5. Solomon, E; Wagner, A; Bendich, P, From Geometry to Topology: Inverse Theorems for Distributed Persistence, Leibniz International Proceedings in Informatics, Lipics, vol. 224 (June, 2022), ISBN 9783959772273 [doi]  [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

 

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320