I have a donut. Its boundary is a two-dimensional surface embedded in three-dimensional space, and surely is homeomorphic to a torus. If we fix a Riemannian metric on the space, it induces a two-dimensional Riemannian metric on the donut. Forgetting the scaling, we get a conformal structure on the surface. If we fix an orientation, this defines a complex structure.
Suppose my donut is modeled in cylindrical coordinates $(r,z,\theta)$ by the equation $(r-1)^2 + z^2=\rho^2$. What is the $j$-invariant as a function of $\rho$?
Find an actual, real donut, and approximate it by a surface of this form. Estimate $\rho$ and thereby estimate its $j$-invariant.
Determine how your donut differs from this model, and estimate in which direction that might change its $j$-invariant. Or solve a different model that better fits your donut.
I would be happy to see progress on any of these vitally important problems, which you will no doubt agree have many fruitful applications to mathematics and our daily life.