FoCM 2014 conference

Workshop B2 - Computational Topology and Geometry

December 15, 15:30 ~ 15:55 - Room A22

## Induced Matchings of Barcodes and the Algebraic Stability of Persistence

### Ulrich Bauer

### TU München, Germany - mail@ulrich-bauer.org

We define a simple, explicit map sending a morphism $f: M \to N$ 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.

Joint work with Michael Lesnick (IMA, Minneapolis, MN).