Multiparameter Persistent Homology (18w5140)

Arriving in Oaxaca, Mexico Sunday, August 5 and departing Friday August 10, 2018

Organizers

(University of Victoria)

Peter Bubenik (University of Florida, Mathematics)

(Princeton University)

Objectives

We propose a workshop to bring together specialists in multiparameter persistence with experts in neighboring fields who have an interest in applied topology but may not be familiar with the recent developments and main problems in multiparameter persistence. The workshop will provide a forum for these researchers to meet, exchange ideas, and explore possibilities for interdisciplinary collaboration on the development of novel practical tools and theory for topological data analysis.

In addition to specialists in multiparameter persistence, we plan to include in this workshop experts in statistics, computational commutative algebra, representation theory, and high performance computing; they will comprise at least 1/3 of the participants.

The primary topics explored will be as follows:

a) Statistical inference theory and methodology for emerging multiparameter persistence tools, e.g. computing bootstrap confidence regions for multiparameter persistence invariants.

b) Applications of distributed computing techniques to large-scale multiparameter persistence computation.

c) Invariants of multiparameter persistence modules. Which invariants are most useful for data analysis? What insights does representation theory offer?

d) Computing metrics between multiparameter persistence modules. Good metrics would have immediate application to machine learning (e.g. shape classification). A natural choice of metric is the interleaving distance [7,8], but its computation is a challenging problem in computational algebra, and remains open despite the efforts of several researchers. This problem is a generalization of the isomorphism testing problem, studied by Brooksbank, Luks [9,10].

e) Computing decompositions of multiparameter persistence modules into indecomposables. Partial steps towards this goal would have important applications to exploratory data analysis and visualisation. The problem is not understood by TDA community except in certain very simple settings, but general versions of it have been studied by computational algebraists [9,10,11].

f) Computing minimal presentations of multiparameter persistence modules. The multiparameter persistence modules studied by the TDA community are simply multigraded modules over a polynomial ring in several variables. While the problem of computing minimal presentations of such modules is well studied in commutative algebra [12,13], the development of optimized algorithms that suit the needs of TDA has not been explored, and the capabilities of existing algorithms for typical modules arising from data are not well understood.

Introductory talks will be given at the beginning of the workshop to bring people from different backgrounds up to speed, and all talks will be pitched at an accessible level. Experts from neighboring fields will be encouraged to give survey talks on topics outside of topological data analysis that are of interest to the TDA community. For example, we hope to have introductory talks on the bootstrap, Groebner bases, computation of minimal presentations, module isomorphism testing, and quiver representation theory. Parallel workgroups will be held in the afternoons, to allow smaller groups to discuss open questions and future research directions.

In summary, in this workshop, we plan to clarify the connections of multiparameter persistence with cognate fields; discuss fundamental open problems; generate interest in these problems among a diverse group of mathematicians; and set a research agenda for advancing this new subject. An additional goal is to bring together experts in these areas with talented young researchers who are motivated to advance topological data analysis.

List of references:

[1] Carlsson, Gunnar, and Afra Zomorodian. "The theory of multidimensional persistence." Discrete & Computational Geometry 42.1 (2009): 71-93.

[2] Lesnick, Michael, and Matthew Wright. "Interactive Visualization of 2-D Persistence Modules." arXiv preprint arXiv:1512.00180 (2015).

[3] Lesnick, Michael, and Matthew Wright. "Computing Bigraded Betti Numbers in Cubic Time." In preparation.

[4] Botnan, Magnus Bakke, and Michael Lesnick. "Algebraic Stability of Zigzag Persistence Modules." arXiv preprint arXiv:1604.00655 (2016).

[5] Bjerkevik, Håvard Bakke. "Stability of higher-dimensional interval decomposable persistence modules." arXiv preprint arXiv:1609.02086 (2016).

[6] Cochoy, Jérémy, and Steve Oudot. "Decomposition of exact pfd persistence bimodules." arXiv preprint arXiv:1605.09726 (2016).

[7] Chazal, Frédéric et al. "The structure and stability of persistence modules." arXiv preprint arXiv:1207.3674 (2012).

[8] Lesnick, Michael. "The theory of the interleaving distance on multidimensional persistence modules." Foundations of Computational Mathematics 15.3 (2015): 613-650.

[9]Brooksbank, Peter A, and Eugene M Luks. "Testing isomorphism of modules." Journal of Algebra 320.11 (2008): 4020-4029.

[10] Chistov, Alexander, Gábor Ivanyos, and Marek Karpinski. "Polynomial time algorithms for modules over finite dimensional algebras." Proceedings of the 1997 international symposium on Symbolic and algebraic computation 1 Jul. 1997: 68-74.

[11] Parker, Richard A. "The computer calculation of modular characters (the meat-axe)." Computational Group Theory, Academic Press, London (1984): 267-274.

[12] Kreuzer, Martin, and Lorenzo Robbiano. Computational commutative algebra 2. Springer Science & Business Media, 2005.

[13] La Scala, Roberto, and Michael Stillman. "Strategies for computing minimal free resolutions." Journal of Symbolic Computation 26.4 (1998): 409-431.