Advances in topological data analysis (TDA) require efficient algorithm designs that can extract algebraic structures hidden in the data. This talk centers around the theme that special geometric/combinatorial constructions underlying the algebra can facilitate designing efficient algorithms in TDA. As examples of this premise, we present three (four time permitting) cases: (i) how a geometric viewpoint on zigzag persistence in terms of the Mayer-Vietoris pyramid helped designing a fast algorithm for computing zigzag persistence from an input zigzag filtration, (ii) how the special structure of two-dimensional grid ($\mathbb{Z}^2$) helped designing an efficient algorithm for computing the generalized rank (rank of the limit-to-colimit map) for $2$-parameter persistence, (iii) how combinatorics helped designing efficient algorithms for multiparameter persistence, and (iii) how combinatorial multivectors representing dynamical systems helped designing efficient algorithms for computing algebraic summaries such as Connection Matrices and Conley-Morse barcodes.

I will discuss methods for analyzing complex datasets based on adding geometric/topological/measure-theoretic attributes to merge trees and Reeb graphs generated from the data. This will be a two-part talk/tutorial, where I intend to simultaneously explain the underlying theory and illustrate it via interactive code examples. This will be based on a few papers which approach these ideas from different perspectives, which are joint work with Justin Curry, Haibin Hang, Washington Mio, Osman Okutan, and Florian Russold. Interactive notebooks associated to the talk will be available here: https://github.com/trneedham/Graph-Based-Methods-In-TDA.

Open-source software (OSS) development is a central topic at the intersection of IS and organizational research. Research in this area has consistently shown that OSS development processes are dynamic, non-deterministic, and emergent. In this presentation, I introduce a new method and a new theory for studying how OSS development processes change over time. The new topological data analysis (TDA) approach, Temporal Mapper, allows us to theorize the OSS development process as a complex dynamical system. Specifically, it enables us to visualize and conceptualize the emergence of recurrent patterns of action in the OSS development process. Due to a lack of data and methods, early work on organizations as complex dynamical systems was largely metaphorical. Recent advancements in topological data analysis (TDA), such as Temporal Mapper, make it possible to conduct rigorous empirical studies, especially in areas like OSS where detailed data are available. These methodological advancements create new opportunities to revisit earlier scholarly interests in viewing organizations as complex dynamical systems.