The CUNY Data Science and Applied Topology Reading Group is joint between the Mathematics and Computer Science programmes. We meet Fridays 11.45 -- 12.45 in GC 4419. You can contact us at cunygc@appliedtopology.nyc.

Our plan is to primarily read and discuss seminal papers in data science, in applied topology and in topological data analysis. Each seminar one participant takes the responsibility to present a paper and prepare items for discussion. We expect occasionally to be able to invite external speakers.

## Schedule

Current schedule can be found here.

We will be sending out announcements through a mailing list; you can subscribe here.

## Organizers

- Mikael Vejdemo-Johansson, Computer Science Programme, CUNY Graduate Center; Department of Mathematics, CUNY College of Staten Island
- Azita Mayeli, Mathematics Programme, CUNY Graduate Center; Department of Mathematics, CUNY Queensborough Community College

## Suggested papers

We have compiled a list of papers that might be interesting to present.

# Schedule

### Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $\displaystyle F_2\xrightarrow{\partial_2} F_1 \xrightarrow{\partial_1} F_0$ such that $M\cong \ker{\partial_1}/\im{\partial_2}$. It runs in time $O(\sum_i |F_i|^3)$ and requires $O(\sum_i |F_i|^2)$ memory, where $|F_i|$ denotes the size of a basis of $F_i$. We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.