Math 296, Special Student Seminar, Fall 2023

• Instructor: Viktor Ginzburg; office: McHenry 4124; email: ginzburg(at)ucsc.edu.

• Meetings: Tuesdays, 1:30-3:05pm (subject to change), starting 10/03; McHenry 1270.

• Description: The seminar will focus on applications of persistent homology methods to problems in geometry and analysis. The concept of persistent homology, aka persistence modules, originated in the field of Topological Data Analysis (TDA), lying at the very heart of the field. As the name suggests, the idea of TDA is to use topological methods to analyze data sets. For instance, persistent homology converts geometrical information, e.g., a data set, into a family of homology groups. Recently, a broad spectrum of extremely interesting applications of persistent homology have been developed in other areas of mathematics, ranging from differential geometry and topology to symplectic geometry and dynamics to pure analysis. In more applied areas, there are, as expected, applications to image recognition, data processing and math biology, to mention just a few. In this seminar we will start with an introduction to persistent homology and then explore some of these applications. Persistent homology and more generally TDA is a new, active and fast growing field and a familiarity with its basics will be of use in the future regardless of one's area of expertise or career path in mathematics or IT.

• Format: While the details are yet to be worked out, so far I plan the seminar to meet once a week, on Tuesdays, 1:30-3:05pm. (To be finialized.) The first meeting is on Tuesday, 10/03. We will also meet individually, as needed.

• Prerequisites: No prior knowledge of persistent homology or TDA is required or expected. Some background in manifolds and topology on the level of our Manifolds Sequence will definitely be helpful.

• Text: There will be no "official" textbook in this seminar. Below I will give and keep on updating some introductory and more specialized references.

• Introductory Reading:
  • Topological Persistence in Geometry and Analysis, L. Polterovich, D. Rosen, K. Samvelyan , J. Zhang; AMS, University Lecture Series, 74, 2020.

  • Computational Topology: An Introduction, H. Edelsbrunner and J.L. Harer; AMS, 2010.

  • Topology and Data, G. Carlsson, Bull. AMS 46(2009), 255-308.

• A concise and accessible introduction to homology: Algebraic Topology, A. Hatcher, Cambridge University Press, 2001.

• Some topic suggestions for talks and a further discussion:
  • Rips and Cech complexes -- a more in depth look and appplications to, e.g., manifold learning (John).

  • Multiparameter persistence (Sam).

  • Proof of the Isometry Theorem.

  • Wasserstein distances on the space of barcodes and stability; see, e.g., ArXiv 2006.16824 as a starting point.

  • Persistent homology transforms; see, e.g., ArXiv 1310.1030 and ArXiv 2208.14583.pdf and references therein.

  • Euler calculus; see, e.g., the survey ArXiv 1804.04740 .

  • Applied and computational aspects; see ArXiv 1601.01741 for example, but this is of course a huge subject.