There are some postings at MathJobs.org that specify Topological Data Analysis. These include:
Postdoc:
https://www.mathjobs.org/jobs/jobs/12518 Florida State University; department interests include TDA
https://www.mathjobs.org/jobs/jobs/12021 Several possible sites, including Florida State University where department interests include TDA
Faculty:
https://www.mathjobs.org/jobs/jobs/12350 Hong Kong University, tenured or tenure track
https://www.mathjobs.org/jobs/jobs/12666 Northeastern University, open rank
https://www.mathjobs.org/jobs/jobs/12155 University of Luxembourg, expected to create a new research group
https://www.mathjobs.org/jobs/jobs/12493 Florida State University, tenure track; department interests include TDA
https://www.mathjobs.org/jobs/jobs/12102 North Carolina State University (Raleigh), tenure track