James Fairbanks’s research while affiliated with University of Florida and other places

It is folklore that tree-width is monotone under taking subgraphs (i.e. injective graph homomorphisms) and contractions (certain kinds of surjective graph homomorphisms). However, although tree-width is obviously not monotone under any surjective graph homomorphism, it is not clear whether contractions are canonically the only class of surjections with respect to which it is monotone. We prove that this is indeed the case: we show that - up to isomorphism - contractions are the only surjective graph homomorphisms that preserve tree decompositions and the shape of the decomposition tree. Furthermore, our results provide a framework for answering questions of this sort for many other kinds of combinatorial data structures (such as directed multigraphs, hypergraphs, Petri nets, circular port graphs, half-edge graphs, databases, simplicial complexes etc.) for which natural analogues of tree decompositions can be defined.

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.

In many sciences one often builds large systems out of smaller constituent parts. Mathematically, to study these systems, one can attach data to the component pieces via a functor F. This is of great practical use if F admits a compositional structure which is compatible with that of the system under study (i.e. if the local data defined on the pieces can be combined into global data). However, sometimes this does not occur. Thus one can ask: (1) Does F fail to be compositional? (2) If so, can this failure be quantified? and (3) Are there general tools to fix failures of compositionality? The kind of compositionality we study in this paper is one in which one never fails to combine local data into global data. This is formalized via the understudied notion of what we call a lavish presheaf: one that satisfies the existence requirement of the sheaf condition, but not uniqueness. Adapting \v{C}ech cohomology to presheaves, we show that a presheaf has trivial zeroth presheaf-\v{C}ech cohomology if and only if it is lavish. In this light, cohomology is a measure of the failure of compositionality. The key contribution of this paper is to show that, in some instances, cohomology can itself display compositional structure. Formally, we show that, given any Abelian presheaf F : C^op --> A and any Grothendieck pretopology J, if F is flasque and separated, then the zeroth cohomology functor H^0(-,F) : C^op --> A is lavish. This follows from observation that, for separated presheaves, H^0(-,F) can be written as a cokernel of the unit of the adjunction given by sheafification. This last fact is of independent interest since it shows that cohomology is a measure of ``distance'' between separated presheaves and their closest sheaves (their sheafifications). On the other hand, the fact that H^0(-,F) is a lavish presheaf has unexpected algorithmic consequences.

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.

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.

Classical planning representation languages based on first-order logic have preliminarily been used to model and solve robotic task planning problems. Wider adoption of these representation languages, however, is hindered by the limitations present when managing implicit world changes with concise action models. To address this problem, we propose an alternative approach to representing and managing updates to world states during planning. Based on the category-theoretic concepts of C-sets and double-pushout rewriting (DPO), our proposed representation can effectively handle structured knowledge about world states that support domain abstractions at all levels. It formalizes the semantics of predicates according to a user-provided ontology and preserves the semantics when transitioning between world states. This method provides a formal semantics for using knowledge graphs and relational databases to model world states and updates in planning. In this paper, we conceptually compare our category-theoretic representation with the classical planning representation. We show that our proposed representation has advantages over the classical representation in terms of handling implicit preconditions and effects, and provides a more structured framework in which to model and solve planning problems.