Can we, with arbitrary precision, calculate what's the place on earth that's closest to the sun on a given date? If so, what's the math like? Say, if I wanted to calculate this within 1m, 10cm or 1cm precision, what exactly should I take into account?
Calculation of the subsolar point is relatively straightforward. The actual point moves at nearly 500m/s in a circle, as the Earth spins. The linked website does the calculation in a few lines of python using a free library, The value is probably correct to within a few 10s of kilometres.
But you are looking for a different order of accuracy: If you (for some reason) need 1cm accuracy, you would need to take into account the actual, non-spherical shape of the Earth.
The Earth is not a sphere, it can be better modelled as an oblate ellipsoid. But for 1cm accuracy, you would need to consider all of the mountains and hills. And you would need a very accurate time, not just a date. That is not a practical calculation. I can think of no practical application.