26 August 2017:
Here is this week's progress:
1) Jason Erbele submitted an abstract for the Nov. 4-5 special session on applied category theory at UCR:
* Controllability and observability: diagrams and duality
Diagrams of systems appear in many different fields of study, and for good reason: they can dramatically simplify communication of and calculations with those systems. In many cases, large diagrams can be viewed as coming from piecing together smaller diagrams in ways that preserve important data, and complicated diagrams can be rewritten to produce simpler diagrams that represent the same behavior. Category theory provides a framework to reason with diagrams as mathematical objects that can be composed and transformed by rewrite rules. In particular, for linear, time independent control systems, the dual notions of controllability and observability can be expressed in terms of a the dual notions of epimorphism and monomorphism, as applied to certain composite diagrams. (Received August 25, 2017)
2) Nina Otter put a new paper on the arXiv with Hal Schenk and her two advisors, the mathematical biologist Heather Harrington and the homotopy theorist Ulrike Tillmann:
* Stratifying multiparameter persistent homology
> **Abstract.** A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.
3) Joseph Moeller has been visiting Metron Scientific Solutions in Reston since last Friday. He's talking to John Foley about our project, for example the use of "graphic monoids" for describing networks of commitments among a collection of agents.
were introduced by Lawvere. These are monoids obeying the identity xyx = xy, which means that if you try to commit to x, then to y, then to x it's the same as trying to commit to x and then trying to commit to y. You may or may not be able to make these commitments, which is why I say "try". But if you succeed in committing to x the first time, committing to it a second time doesn't change anything, even if you've made some other commitments in the meantime!
I'm hoping Joseph will get pulled into the more applied aspects of our project when he's there, since the cutting edge right now is using our math to get things done. But the applied aspects, done right, will probably involve a lot of brand new pure math, like what Joseph and John are doing right now with graphic monoids.
4) I wrote another blog article about our Metron project:
* Complex adaptive system design (part 4)
This is about the simplest example of a "network operad" and its simplest algebra.