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I was invited to give an introductory talk on “Higher structures in mathematics and physics” at a meeting of a general mathematics audience with an emphasis on differential equations.
I am still fine tuning my “slides” (a sparsely typeset nLab stream, really), but here is what I have so far
Comments are welcome.
with an emphasis on differential equations.
Let me try you out with a naive question from a possible audience member:
If you’re so keen to move from equality to isomorphism to equivalence, why are you still speaking about differential equations? Or is it that the meaning of ’equal’ and so ’equation’ is to be changed?
Good point. I’ll add a comment on that to the notes tomorrow when I am more awake again.
But the answer is in Higher Prequantum Geometry (schreiber) (and with full details in an upcoming writeup with Igor Khavkine): A differential equation is an equalizer of two maps out of a jet bundle in the Cahier topos sliced over some base $\Sigma$, and it is formally integrable precisely if this inherits the structure of a coalgebra over the jet comonad. Now the homotopified version of this statement verbatim applies to the $\infty$-Cahiers topos. The resulting “differential equivalence” or maybe “differential homotopy” is then the homotopy equalizer between two morphisms out of a jet $\infty$-bundle of an $\infty$-bundle, and this is formally integrable precisely if it is an $\infty$-coalgebra over the jet $\infty$-comonad.
We won funds for organizing a Durham-Symposium on “Higher Strutcures in M-Theory” this August. No funds left now that speakers have been invited, unfortunately, but everyone who happens to come by is welcome.
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