I've not been able to make use of the Sturmfels/Cox stuff. Nor really any of the supposedly applied mathematics texts I've read. So I switched to just trying to understand simple mathematical objects that don't have the marketing behind them. Started with baez week52 a couple years ago and various things built on that.
Many years before I made that switch I did pick up Eisenbud’s commutative algebra with a view to algebraic geometry. Even not understanding most of it, it's worth an econometric look even just for the appendices. Econometricians *always* use ℤ, never ℝ, and plus the ring embedding idea and his view of matrices are just different.
Plus commutative algebra is about curved surfaces, not straight planes. Which we clearly deal with.
However I don't believe I approached algebraic geometry correctly the first time. What helped me now is Mumford’s
Curves and their Jacobians, Arnol'd plane curves and caustics, Bill Fulton, Miles Reid, HF Baker, Durfee double points, … people who have an actually concrete curve with some weirdness to it. And toward that end, Emmy Murphy's IAS lecture on mirror symmetry for the trefoil mentioned the Koras-Russell variety, which I thought would be good to do some calculations with. Chapter 2 of the Macaulay book is again Eisenbud’s, and it's mercifully concrete.
As far as career (thread category), the idea is just to separate intellectual interest from stuff that's actually useful. (e.g. FIX protocol has no intellectual interest, the Ikosahedron does)
No one will pay you to play with Floer theory, and making an image recognizer work on a phone isn't interesting.
