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I would add that there are some specializations and generalizations of this notion that appeared since 1970, and that the survey of Anne Shepler and Sarah Witherspoon arxiv.org/abs/1404.6497 is a good place to look at to account for that.
@coco for what it's worth, Switzerland has (in Zurich) one of the Google's programming hubs. Several people I know well who had a past life of studying to become mathematicians found their happiness there. I cannot be certain that this is a good thing for you personally to try, but it is something worth considering. If you wish you can contact me by email, and I can get you in touch with someone I know, they can tell you a bit what to expect.
Perhaps you can clarify the wording of "related to ring theory". There are two possible understandings : 1) one can study lattices as something similar to associative (and commutative?) rings, thinking of the two operations as "addition" and "multiplication", or 2) one can study lattices that are meaningfully appear in ring theory (lattices of ideals in a ring, lattices of subvarieties in a variety of rings etc.) The comment of Todd Trimble is presumably about the first interpretation, while the immediate reaction of a typical ring theorist will be along the lines of the second interpretation.
@RobbieGoodwin indeed, the whole notion of "a pure math academic path" is difficult to comprehend for a non-mathematician, as almost every mathematician will have observed many times in their daily life. I would honestly suggest that you exercise some humility and avoid telling to mathematicians in what terms they should be talking about a serious real issue that arises in their profession.
@LSpice what you see as credential checking I view as pointing out that a person who is interesting in maths recreationally is very likely to not understand the pertinence of the question to a research mathematician.
@RobbieGoodwin MathOverflow is intended to be about what is relevant to researchers in maths. (Are you one?) I strongly believe that this question is relevant because researchers in maths who supervise PhD students encounter such situations in real life (for instance, the usual "career path" of academia requires making a lot of drastic decisions which some people cannot choose to make, as discussed in the OP). So this discussion forum has a critical mass of people whose expertise prepared them to give some meaningful answers to this particular question, which is the whole point.
The ordering of variables $a1,a2,a3,b1,b2,b3,c1,c2,c3,d1,d2,d3$ seems to be good. At least in Magma a Gröbner basis is computed in fractions of a second, once that order is given.
@YCor this might help a lot, thanks! (Due to many situations where I work with $S_n$-modules, I tend to ignore the usefulness of modular arithmetic even where it is useful!!)
Denote $x_i=a_i-a_{i+1}$. Then you are studying $$ \prod_{0\le i< j< n}\sin^2(x_i+x_{i+1}+\cdots+x_{j-1}), $$ and you conjecture that the maximum is attained when $x_0=\cdots=x_{n-1}=\frac{\pi}n$. This is a bit more symmetric conjecture, so perhaps something can be done. In any case, using this formulation, I checked the result for $n=3$ with almost no calculations, but you perhaps did that yourself already.
