@HarryGindi @MikeMiller: There are many applications of homotopical ideas in analysis. Conley index is a great example, this is a beefed up version of Morse theory. In Morse theory one looks at the changes in homotopy type of a manifold when passing critical values. A big tool is the gradient flow. Conley theory shows that you basically can do this for many dynamical systems. But now the topology changes around equilibria, or more generally invariant sets.

Another thing that comes to mind is the h-principle. This idea should be very attractive: To find solutions to (usually geometric) Pde's one proves two things: First: A formal solution of the PDE exists and secondly: a formal solution can be wiggled to a real solution. The first part is pure obstruction theory, i.e. homotopy theory.

I would also like to say that many theorem's in topology are of a dynamical systems nature. For example the existence of vector fields without zeros on manifolds with zero euler characteristic can be interpreted dynamically.

The list goes on, but I'll stop here.

@ThomasRot @ThomasRot I expected tbat, but I meant more like how homotopical algebra is homotopical. Like, is there a 'floppier' version of analysis that makes sense in a homotopy-coherent way.

For complex analysis, you can basically make do with Lurie's stuff because analytic things are so rigid

But real analysis seems like it would be all messed up

Like, do the local pictures on real smooth Lurie stacks (however you define rhem) give you PDEs like usual

If not, are they like PDE up to some kind of homotopy?

I don't know enough about that

@HarryGindi yes there is. although like everything else on twitter it can get mixed into a lot of different things. I'm on twitter, along with Emily Riehl, Eugenia Cheng, John Baez, several of John Baez's students, a lot of undergrad math majors, and some other people who read but don't post much (e.g. Charles Rezk).

@JonathanBeardsley It's better if I don't fall to temptation and be a jerk on twitter in the presence of colleagues... I'll keep those kinds of interactions for my cave-dwelling friends 😊