Preexisting Papers

[jpfairbanks]

We need to populate the recent publications page with recent papers. We need the authors list, abstract, link to pdf, downloaded pdf, and a figure for the preview. This can be a figure from the paper that you think hypes it up.

• @benjaminmerlinbumpus,
  • the sheaf theory preprint
  • ACT extended abstract
• @tylerhanks, AlgebraicControl papers
  • CDC 2023, Hanks et al
  • CDC 2023, She at al
  • JLAMP Brown et al
  • ICGT Brown et al
  • ACT abstract
• @lukem12345
  • Morris et al JOCS
  • Patterson et al Diagrammatic eqns
  • ACT abstract

[jpfairbanks]

Here is a summary of papers that I wrote for someone with annotations. Not all of the annotations should go on the website.

· (Royal Society) ACT for compositional modeling of disease dynamics. No control on there, but a lot of good compositionality for dynamics. Really proud of the slice categories approach to imposing structure onto the ways you can combine systems https://royalsocietypublishing.org/doi/10.1098/rsta.2021.0309

· (Compositionality) Compositional Modeling of Biological Systems relating combinatorial (graph-based) models of system structure to dynamical systems that happen on those structures. We introduce a category of Linearly Parameterized Dynamical Systems that could be used for Control. We looked in the context of Ecological/Biological systems and proved that Lotka-Volterra Dynamics are functorial for Graphs and Signed Graphs. So that if you build a big ecosystem by glueing subsystems, the space of dynamical systems on the populations is built by gluing the dynamics https://arxiv.org/abs/2301.01445

· (CDC2023) Characterizing Compositionality of LQR from the Categorical Perspective. Basically an impossibility result for when LQR of a composite will be the LQR of the components. It was our first step towards applying ACT in control https://arxiv.org/abs/2305.01811
· (ACC2024) Modeling Model Predictive Control: A Category Theoretic Framework for Multistage Control Problems https://arxiv.org/abs/2305.03820

· First Order Optimization with Operads. In this paper we relate convex optimization problems to their gradient flows, and use the fact that Euler’s method is functorial for resource sharing open dynamical systems to show that gradient descent is functorial for open smooth optimization problems. More significant theorems relate undirected wiring diagrams to distributed optimization algorithms via the gradient flow dynamical system. Basically every UWD gives you a distributed optimization algorithm based on message passing. This construction recovers existing methods like Uzawa’s and Dual Decomposition. The extended example is Minimum Cost Network Flow, which is about connectivity in discrete topology (in that graphs/networks are discrete spaces). https://arxiv.org/abs/2403.05711

· Gradient Descent is a Hypergraph functor. Relates Multitask learning to Gradient Flows and the paper above. (submitted last night)

· Temporal Data Structures with Sheaves and Cosheaves. In this paper we develop a data model for temporal data which can be used to study samples from dynamical systems. This shows off the sheaf skills. When we put this on Arxiv, we got great feedback from Justin Curry, who is a Topological Data Analysis guy. He was really impressed by the adjunction relating the persistent data and cumulative data perspectives https://arxiv.org/abs/2402.00206v1

· Sheaf theory for Computational Complexity. Here we invented an method that yields a fixed parameter tractable algorithm for any computational problem that can described as “does this sheaf have a global section?” A lot of graph algorithms fall into this class as long as you choose your notion of “space” correctly. We developed a class of Grothendieck Topologies that relate to things like tree width decompositions, which are the standard tool in FPT graph algorithms. https://arxiv.org/abs/2302.05575

· Decapodes is our flagship product on multiphysics modeling. By next Spring we will have extended it to multiphysics on multiple spatial domains which will use serious Algebraic Geometry and Topology concepts such as diagrams in the category of sheaves over a space and discrete simplicial manifolds: https://arxiv.org/abs/2401.17432, https://arxiv.org/abs/2204.01843