Induced Matchings and the Algebraic Stability of Persistence Barcodes

We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\ker f$ and $\mathop{\mathrm{coker}} f$. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational Geometry. Includes a proof that no definition of induced matching can be fully functorial (Proposition 5.10), and an extension of our single-morphism characterization of the interleaving relation to multidimensional persistence modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde

The Reality of Moral Imperatives in Liberal Religion

This paper uses a classic one-liner attributed to Dostoyoevski’s Ivan Karamozov, Without God everything is permitted, to explore some differences between what I term traditional and liberal religion. The expansive connotations and implications of Ivan’s words are grounded in the historic association of wrongfulness and punishment, and in a reaction against the late modern challenge to the inexorability of that association, whether in liberal religion or in secular moral thought. The paper argues that, with its full import understood, Ivan’s claim begs critical questions of the meaning and source of compulsion and choice, and of knowledge and belief regarding the specific content of religiously grounded moral norms. Liberal religion views knowledge of the Will of God in ways that pervasively tend to emphasize the place of human discernment of that Will, finding clarity in ambiguity and complexity, in a focus on process as well as outcome, in an openness to questioning as well as honoring tradition, and in a willingness to go beyond the propositional aspects of a text in seeking its meaning. In all these ways, it may seem to dilute, even to dissolve, the imperative quality of moral norms. One’s attraction to or wariness of a liberal approach is to a significant extent grounded in non-theological preferences, and reflexive condemnation of liberal religion because of these differences fails to engage with the question whether they are better calculated to serve the task of achieving moral knowledge. While it would therefore aid the process of coming to grips with that question for traditional religion to resist the tendency to dismissal, caricature and polarization, adherents to liberal religion need to take on a greater responsibility for articulating their own theological position more fully. I undertake briefly to take my own advice

Teaching and Learning in Community: Staff-Student Learning Partnerships As Part of a College Education

This paper offers descriptive analyses of two staff-student educational partnership programs of the Teaching and Learning Initiative (TLI) at Bryn Mawr College. The focal programs partner college employees with undergraduate students in unique, reciprocal learning partnerships and student-mentored introductory staff computing courses. While community engagement traditionally focuses attention beyond the campus and identifies off-campus community members as beneficiaries of college students’ efforts, these programs focus on students’ relationships with people whose labor sustains the campus in egalitarian, collaborative, educational experiences. In focusing this argument on the educational benefits of such experiences to students, I explore the connections to liberal education. I also argue that intra-campus community engagement enhances students’ understandings and capacities to challenge limiting hierarchies and divisions. I further argue that this kind of engagement enables students to learn within and across diversity, while developing as people and leaders of campus-based civic initiatives