A reply to comments showing skepticism about how mathematical skills of someone like Tao could be relevant:
Last time I thought I would understood anything of Tao’s blog was around ~2019. Then he was working on curious stuff, like whether he could prove there can be finite-time blow-up singularities in Navier-Stokes fluid equations (coincidentally, solving the famous Millenium prize problem showing non-smooth solution) by constructing a fluid state that both obeys Navier-Stokes and also is Turing complete and … ugh, maybe I quote the man himself:
[...] one would somehow have to make the incompressible fluid obeying the Navier–Stokes equations exhibit enough of an ability to perform computation that one could programme a self-replicating state of the fluid that behaves in a manner similar to that described above, namely a long period of near equilibrium, followed by an abrupt reorganization of the state into a rescaled version of itself. However, I do not know of any feasible way to implement (even in principle) the necessary computational building blocks, such as logic gates, in the Navier–Stokes equations.
However, it appears possible to implement such computational ability in partial differential equations other than the Navier–Stokes equations. I have shown5 that the dynamics of a particle in a potential well can exhibit the behaviour of a universal Turing machine if the potential function is chosen appropriately. Moving closer to the Navier–Stokes equations, the dynamics of the Euler equations for inviscid incompressible fluids on a Riemannian manifold have also recently been shown6,7 to exhibit some signs of universality, although so far this has not been sufficient to actually create solutions that blow up in finite time.
The relation (if any, to proving stuff about computational agents alignment people are interested in) is probably spurious (I myself don’t follow either Tao’s work or alignment literature), but I am curious if he’d be interested in working on a formal system of self-replicating / self-improving / aligning computational agents, and (then) capable of finding something genuinely interesting.
A reply to comments showing skepticism about how mathematical skills of someone like Tao could be relevant:
Last time I thought I would understood anything of Tao’s blog was around ~2019. Then he was working on curious stuff, like whether he could prove there can be finite-time blow-up singularities in Navier-Stokes fluid equations (coincidentally, solving the famous Millenium prize problem showing non-smooth solution) by constructing a fluid state that both obeys Navier-Stokes and also is Turing complete and … ugh, maybe I quote the man himself:
The relation (if any, to proving stuff about computational agents alignment people are interested in) is probably spurious (I myself don’t follow either Tao’s work or alignment literature), but I am curious if he’d be interested in working on a formal system of self-replicating / self-improving / aligning computational agents, and (then) capable of finding something genuinely interesting.
minor clarifying edits.