Martti Karvonen’s scientific contributions

How does one build a robust and general theory of temporal data? To address this question, we first draw inspiration from the theory of time-varying graphs. This theory has received considerable attention recently given the huge, growing number of data sets generated by underlying dynamics. Examples include human communication, collaboration, economic, biological, chemical networks, and epidemiological networks. We distill the lessons learned from temporal graph theory into the following set of desiderata for any mature theory of temporal data: 1. Categories of Temporal Data: Any theory of temporal data should define not only time-varying data, but also appropriate morphisms thereof. 2. Cumulative and Persistent Perspectives: In contrast to being a mere sequence, temporal data should explicitly record whether it is to be viewed cumulatively or persistently. Furthermore there should be methods of conversion between these two viewpoints. 3. Systematic 'Temporalization': Any theory of temporal data should come equipped with systematic ways of obtaining temporal analogues of notions relating to static data. 4. Object Agnosticism: Theories of temporal data should be object agnostic and applicable to any kinds of data originating from given underlying dynamics. 5. Sampling: Since temporal data naturally arises from some underlying dynamical system, any theory of temporal data should be seamlessly interoperable with theories of dynamical systems. In this paper we lay the foundations of a categorical theory for temporal data that satisfies the above list of desiderata.

What is a time-varying graph, or a time-varying topological space and more generally what does it mean for a mathematical structure to vary over time? Here we introduce categories of narratives: powerful tools for studying temporal graphs and other time-varying data structures. Narratives are sheaves on posets of intervals of time which specify snapshots of a temporal object as well as relationships between snapshots over the course of any given interval of time. This approach offers two significant advantages. First, when restricted to the base category of graphs, the theory is consistent with the well-established theory of temporal graphs, enabling the reproduction of results in this field. Second, the theory is general enough to extend results to a wide range of categories used in data analysis, such as groups, topological spaces, databases, Petri nets, simplicial complexes and many more. The approach overcomes the challenge of relating narratives of different types to each other and preserves the structure over time in a compositional sense. Furthermore our approach allows for the systematic relation of different kinds of narratives. In summary, this theory provides a consistent and general framework for analyzing dynamic systems, offering an essential tool for mathematicians and data scientists alike.