you might look at how jordan canonical form works in the case of 2x2 matrices over Z. that is mostly about existence of C, but if you look at the proofs they may hint at where lack of uniqueness can arise.
I feel dumb. I have a 9 month review coming up for my PhD and, while I get the concepts behind them, I keep forgetting things.
I've had a lot of time off for my health.
It's scary stuff.
During my 1st attempt at a PhD, I had loads of time off for my health, then, too. I'm concerned that the same thing'll happen again here.
I think the amount of insights and eureka-moments I've had so far are thin on the ground compared to last time too.
I just haven't done as many exercises.
It's a beautiful area to study, much more structured than combinatorial group theory. But there's less experiments I can do on a computer when I'm stuck.
Linear algebraic groups, I mean.
I could rant for a while . . .
I still have COVID.
Also, I feel like I'm past it. I'm 32. I haven't got my PhD yet. I have only one published article. That puts me behind in academia, if I'm to have a career.
If I fail this attempt or get an MPhil, I'll try elsewhere. My psychiatrist said he knows people on their third PhD attempt. He encouraged me.
Sometimes it just doesn't work out, y'know?
@geocalc33 Hi :)
Mathematics can be cruel.
I'm not sure I even get what an algebraic variety is, as sad as it sounds, and it's fundamental!
Then again, algebraic geometry is notoriously difficult.
Do you agree on your end? Lemma: if a theory has a model of cardinality $m$ then it has one with cardinality $\geq$ than $m$. Ex: there is no a theory $K$ whose models have finite domain.
Mental health problems put huge weight on shoulders of any person having it, so it is understandably. I would be proud for myself if I worked 9+ hours on math in my past (like you in your undergrad years).
Ok. If such a theory existed, then we could have some model whose domain is infinite. If the statement is correct (the book had at least one typo, I even posted a quesiton here about one), and I understood it -- that it has _only_ models with finite domain. But I am suspicious that it is not what is meant.
