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Joe Moeller (Mar 29 2024 at 18:53):I've accepted a postdoc position at CalTech. I'll be working with Aaron Ames on using category theory to gain a deeper understanding of stability and control of dynamical systems.
Evan Patterson (Mar 29 2024 at 18:55):Congrats Joe!
Joe Moeller (Mar 29 2024 at 19:06):Thanks! I'm super excited to work on really interesting ideas and at such a cool place. Cherry on top is that I'm back home in SoCal.
Matteo Capucci (he/him) (Mar 29 2024 at 19:38):That's cool to hear! I'm curious what the approach is.
Joe Moeller (Mar 29 2024 at 20:55):I think we should have a paper out within a reasonable amount of time. I'll sorta randomly throw "end of summer" out there. So the approach will be revealed then!
Niles Johnson (Mar 30 2024 at 12:32):Congrats!! Sounds interesting!
Joe Moeller (Oct 11 2024 at 00:29):John and Todd and I finally finished the sequel to our Schur functors paper:
2-rig extensions and the splitting principle
John Baez, Joe Moeller, Todd Trimble
Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on K-theory. Here we categorify the splitting principle and generalize it to the context of 2-rigs. A 2-rig is a kind of categorified "ring without negatives", such as a category of vector bundles with ⊕ as addition and ⊗ as multiplication. Technically, we define a 2-rig to be a Cauchy complete k-linear symmetric monoidal category where k has characteristic zero. We conjecture that for any suitably finite-dimensional object r of a 2-rig R, there is a 2-rig map E:R→R′ such that E(r) splits as a direct sum of finitely many "subline objects" and E has various good properties: it is faithful, conservative, essentially injective, and the induced map of Grothendieck rings K(E):K(R)→K(R′) is injective. We prove this conjecture for the free 2-rig on one object, namely the category of Schur functors, whose Grothendieck ring is the free λ-ring on one generator, also known as the ring of symmetric functions. We use this task as an excuse to develop the representation theory of affine categories - that is, categories enriched in affine schemes - using the theory of 2-rigs.
Joe Moeller (Oct 11 2024 at 00:31):My pseudo-promised deadline of "end of summer" for my first paper with Aaron Ames has probably passed, though you couldn't tell from the weather. We are almost done. The basic theory is complete, we're just working on some more examples to include.
John Baez (Oct 11 2024 at 01:03):Please finish soon so the weather here cools down! :sunglasses:
Joe Moeller (Feb 25 2025 at 03:13):Ok, here’s part one!
https://arxiv.org/abs/2502.15276
Categorical Lyapunov Theory I: Stability of Flows
Aaron D. Ames, Joe Moeller, Paulo Tabuada
Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for “doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for establishing the stability of flows. To illustrate these constructions, we show how classical notions of stability, e.g., for continuous and discrete time dynamical systems, are captured by this categorical framework for Lyapunov theory. Finally, to demonstrate the extensibility of our framework, we illustrate how enriched categories, e.g., Lawvere metric spaces, yield settings for stability enabling one to “do Lyapunov theory'' in enriched categories.
Joe Moeller (Feb 25 2025 at 03:15):What ended up happening is we kept adding and adding to the paper until it became a monster. This part one is ironically what we had done at the end of summer :upside_down:
Joe Moeller (Feb 25 2025 at 19:39):I'm also speaking about this project in the Topos Institute's online colloquium on Thursday:
https://topos.institute/events/topos-colloquium/
Joe Moeller (Mar 05 2025 at 22:06):Here's my talk at the Topos Institute's online colloquium:
https://www.youtube.com/live/dI033-TjymA?si=9C153AvSuKTYiwzy