I agree -- abstract algebra is elementary and some aspects of it can (and should), be taught in high-school. AA is a domain of maths which people won't even encounter if not studying a STEM field and such crash course could serve as a good outreach/vulgarisation material.
I really liked the materials in the link and think they are suitable for talented high-schoolers. Cox-Little-O'Shea is a fantastic book I studied as an undergrad and learned a lot from it, I wish someone would expose me to it earlier in life.
There's plenty of good math software for algebra which is not very popular (SageMath comes to mind as the most commonly used) and even more code that is simply inside knowledge. The problem is that people in the academia are not being paid for writing software but for publishing articles, even though the community value of a good package outweighs many papers. For example, good luck finding something that will compute non-commutative Groebner bases :)
During my maths phd time in France I really enjoyed so-called "colles". It was a 1h long 1 on 1 conversation with the student where (s)he was given a complex problem (way too difficult to simply solve in an hour) and a sequence of hints and smaller, easier tasks, leading to the solution. I would often help the interviewee when stuck/correct computations, let them follow the wrong path to make them realize why it was wrong and generally allow for unsupervised exploration of the task. Even though quite often students were not able to get to the end, or even close, I could get a decent idea about their knowledge and understanding of the matter. When I first did it, I instantly thought that's a good way to do a job interview -- being prepared and interactive, have a conversation instead of one way question-answer traffic.
The caveat: I needed to really think through the problems beforehand, understand other possible solutions, traps and potential dead ends (ie. spend time on it).
When a few years later I was asked to do some recruiting at my company (DS/R&D positions, not SE), first thing I did was to prepare a few sets of interconnected problems, to gauge the person's knowledge and how does (s)he think when encountering a new problem with all necessary tools at hand. The problems were difficult but I never expected anyone to actually solve them.
I did dome technical interviews as a candidate, being asked about random math puzzles/algos one can google in 30 sec and which are already implemented in standard libraries -- even though I usually could solve most of them I sincerely wish such interviewers would ef off and stop wasting my time. We are grown-ups, I already spent my fair share of time solving elementary riddles during high school math competitions and I'd like to be treated like a serious professional. During my years as a ML Engineer/dev I never EVER needed to implement a single tree/graph/whatever algorithm from scratch. Also, many adults have a life, family, other full-time job and grinding leetcode is not something anyone should be expected to do.
To be honest, quite a few times I doubted that the interviewer would know how to solve a similar problem without having the solution checked in company's interview problems database. Also, time pressure and stress are a buzzkiller.
As for multiple interview rounds -- a non-starter. Recently, while exploring the market, a top-tier betting company asked me to do a take home, 4h technical interview and then another long take-home (unpaid, of course). I told them that they are ridiculous and asked to never contact me again. I'm at liberty to do so as I have a job I'm happy with and zero need to actually change it, but if someone has been laid off I can see people get grinded to death -- both mentally and physically -- by such interview processes.
I attended a conference talk of a FB AI-engineer talking about her paper with backprop equations so obviously wrong my eyes hurt, and incorrect definitions of objects. It did not stop her from participating (btw. this is always unclear -- who did what) in state-of-the art research in object detection.
PhD is overrated in the deep learning context. It is more about forging the intellectual resilience and ability to pursue ideas for months/years than learning useful things/tricks/theorems.
Finished a PhD in mathematics and wanted to 1) leave academia 2) move back to my country of origin.
Picking up some programming was pretty straightforward, writing code is half (or even less) as frustrating as doing abstract research and pays 3x better here. The choice was easy.
Nvidia GPU drivers and CUDA installations on Linux. Got better recently but I've spent many nights fixing these installs, wrestling with nouveau etc.
Compilation of deep learning libraries from source on Windows -- don't do it.
Tikz -- LaTeX drawing "language". The images produced can be stuninngly beatiful, but the manual has 1200+ pages and it takes forever to become productive. If only journals accepted hand-drawn images...
Linear algebra IS about linear transformations and vector spaces.
The thing is that the field over which the space is defined can be quite arbitrary (finite, infinite, not algebraically closed etc.) which has immense consequences on the behavior of such objects.
When one drops the assumption on finite number of dimensions, the story becomes wild (and is known as functional analysis, beautiful and extremely useful branch of mathematics).
My experience from a decade of doing professional maths is that there are no shortcuts. You only learn maths by doing hundreds (thousands) of exercises, both mundane and more exctiting.
Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it all felt trivial.
There's no shortcut to understanding maths, just a lot of time spent in solitude trying to make sense of all these abstract concepts (and they DO make sense).
A similar, though less sophisticated, scam has been around in France for over a decade.
They put ads with professional photos on websites with rental offers (the price is good, almost too good to be true but you're desperate for a place to live and think that maybe, just maybe, you got very lucky), talk to you in polite and agreeable manner, offer a visit and then demand a downpayment before the actual visit occurs.
The red flag was that they demanded a scan of ID before the meeting (the landlord cannot ask for it beforehand) and of course the downpayment request. I searched online after that and found out it's been around for a looong time.
Be careful who you're paying!
Pretty funny to see Simpson's name showing up here. I have kept two souvenirs from my academic life: the original manuscript of my thesis and Simpson's very eloquent and overwhelmingly positive review of it.
Regarding the topic, I'd say that mathematics, as a whole, is a set of intuitions and ideas, rather than strict proofs. I've spent a fair amount of time working my way through some of Kontsevich's ideas and attended many of his talks -- he rarely (if ever) gives any proofs, my educated guess is that he proposes some extremely profound ideas and his collaborators grind through the details (not always). And it still works just fine.
I remember Kenji Fukaya saying once, talking about a PDE-heavy theorem: "We are trying to keep the details of the proof under 500 pages and it's not sa easy". The main issue here is that people who don't do professional, academic research in mathematics are unaware of the complexity of modern maths. It has evolved immensely during last 50 years, the theories are just layers and layers of foundational work one has to assimilate before getting any work done. Rigorous verification takes years and no one in the academia is being offered a job for proof-reading of existing papers.
One should also remember that referees of the papers are not being paid, it's considered "work for the community" and it's hard to blame them for not reading the papers in detail.
I really liked the materials in the link and think they are suitable for talented high-schoolers. Cox-Little-O'Shea is a fantastic book I studied as an undergrad and learned a lot from it, I wish someone would expose me to it earlier in life.
There's plenty of good math software for algebra which is not very popular (SageMath comes to mind as the most commonly used) and even more code that is simply inside knowledge. The problem is that people in the academia are not being paid for writing software but for publishing articles, even though the community value of a good package outweighs many papers. For example, good luck finding something that will compute non-commutative Groebner bases :)