Holmstrom Some ideas II

Local Langlands

See emails from Strauch for TMF, TAF connections with Local Langlands

Algebraic homotopy groups

Is it possible to compute motivic homotopy groups of for example spheres or other basic varieties? In general, compute [X,Y].

Computer-assisted homotopy theory

Can one represent finite spectra in a computer, and compute things like smash product, homotopy groups, etc. Same question for motivic spectra, for example over Q or over finite fields.

Some people who have published on such things: Julio Rubio and Francis Sergeraert could compute for example $H_5(S^4 \vee S^5)$ already in the 80s. Also, Gonzalez-Diaz and Real have done something on the Steenrod algebra. Here is more useful info on Sergeraert

Can we combine this with computational algebraic geometry, and be able to treat simplicial varieties?

A different idea: Can we use “density arguments”, for example, given a cohomology group/cycle group: if we can produce a list of linearly independent elements, and then generate “truly random” other elements: if these always happen to be a linear combination of our list, it is “almost certain” that this list is a basis. Hence finite generation.

nLab page on Some ideas II