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Ray Adams (Apr 15 2026 at 02:09):Hello folks. I am returning to academia after 31 years. I retired from a career in quantitative finance 5 years ago. While fighting my way through a depressive episode, I "filled in the blanks" of my education as an applied mathematician to secure knowledge of various pure math topics.
I have fallen in love with category theory and HoTT. I was pursuing a greater understanding when the most unexpected thing happened. My advisor from 31 years ago (who is now 90) is writing a textbook for junior students to interest then in the theory of distributed static/dynamic material designs in optimal control (a subject within the calculus of variations).
So, I am working on a DSL in python to bring his text to life and I will edit his text. Longer term, I would love to apply category theory to his science (FYI - my advisor is Konstantin Lurie). This science is absolutely adjacent to mathematical physics. (I am wondering if @John Baez might be interested.
In any case, I would love to hear from this community about applications of CT to analysis and related subjects.
Thanks for reading my post. - Ray
John Baez (Apr 15 2026 at 02:31):Hi! I don't know anything about optimal material design, but there should be a categorical formulation for certain kinds of statics, which describes how we build things out of pieces, and how they respond to strains. In the linear regime it might be analogous to the theory of linear electrical resistors, which was worked out categorically here:
But we focused on the linear case just to keep things simple and focus on the new categorical ideas; there should be nonlinear generalizations, and indeed some have been studied. There's also this:
fosco (Apr 15 2026 at 05:55):Have a look here https://www.researchgate.net/publication/230774973_Synthetic_Calculus_of_Variations
it's not an easy find -but that chapter is now subsumed, I believe, by chapter 4 of https://www.cambridge.org/core/books/abs/synthetic-differential-topology/calculus-of-variations-in-sdg/045A2436E9F06D5CA96DC1737A3E3E32 which is easier to find via the usual methods.
Ray Adams (Apr 15 2026 at 11:33):Thank you both so much for providing me with a runway for takeoff.
John Baez (Apr 15 2026 at 22:14):I'd say that the synthetic approach to calculus (and calculus of variations) is somewhat orthogonal to the compositional approach to open systems which I'm always advocating. That is, you can do one without the other - though it might be best to do both. You can certainly learn a lot about one without understanding the other!
A brief caricature: the synthetic approach to calculus introduces infinitesimals via objects like "the infinitesimal arrow" - a thing that physicists often draw, but takes work to formalize:
The compositional approach to open systems says that we should study physical systems that have a "boundary" at which they can potentially interact with other systems... and we should study the process of gluing together systems along their common boundary.
Ray Adams (Apr 15 2026 at 23:45):Thanks again for your response John. I am out of my depth currently in multiple dimensions after 31 years away from academia and merely a year of new study.
I will say that Dr. Lurie's work is remarkable as a whole. But, his most remarkable contribution to the field is space-time composite materials. There are concrete boundaries in these combined dimensions that must be reconciled and the physics are very complex. Naively (as in years out from the ability to make competent statements), your compositional approach sounds natural to the nature of these solutions.
That said, I have collected your references and those suggested by fosco. I will work through them all. I really appreciate both of your responses.