I don't seem to see as much discoveries in other areas like number theory, calculus, algebra etc...
Is this a bias of what gets covered on HN, or are this type of geometrical problems the currently most active field of mathematics?
In case you didn't notice: the article is about a discovery in number theory. The primary mathematician in this story (search the article for "Stange is a number theorist…"), the ones referenced and quoted ("Elena Fuchs, a number theorist", James Rickards, Peter Sarnak, Alex Kontorovich, Jeffrey C. Lagarias, …) are all number theorists, and the paper itself (https://arxiv.org/abs/2307.02749) was posted under math.NT.
Apollonian circles are geometry, but the conjecture is about the integers that show up as the curvatures of packings, and specifically about the "certain numerical buckets" they happen to fall into. Of course mathematics is ultimately a connected whole; e.g. Jean Bourgain mentioned in the article would not be considered primarily a number theorist.
[And of course there's a bias in what gets covered: researchers work in all areas and it's far from true that "this type of geometrical problems the currently most active field of mathematics", but the ones that can be turned into a good story (and geometry is easier to explain / show) are more likely to get picked up by media like Quanta; and some of them are more likely to be posted to HN and to be upvoted. And some of them are likely to be interpreted as about geometry anyway!]
Analysis is an extremely active field, PDEs have almost endless amounts of open research questions. Usually they are not very flashy and can be very non-geometrical.
Algebra is extremely hard to write about for a general audience. Trying to communicate any result which does not have a simple visual interpretation seems like a nightmare.
Numerics are often things where advances are hard to relay to an audience, as it usually is about incremental improvements instead of breakthroughs.
I'd say that HN posts a lot of quanta articles, and quanta has a "bias" towards results that can be explained to a semi-lay audience. You really don't want to know enough about modular elliptic curves to understand Wiles' proof of Fermat's conjecture. But sometimes number theory proofs come up here too.
Or unless by calculus he means analysis, which is really active. Especially things like PDEs.
And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.
But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers.
Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations.
It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths... I mean, this used to be THE subject in most engineering programs round the world. These days it's more like bootstrapping the intuition.
If you want to calculate a numerical solution to a novel analytic problem you absolutely need a good understanding of analysis. Being able to plug things into Matlab is only useful if Matlab implements a good solver for that problem.
>It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths.
Indeed. We need 5+ years of analysis now.
And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
What point are you trying to make?
Yep, what you want is a good and predictable approximation.
> What point are you trying to make?
In practice most Matlab solutions you are describing are awful, in terms of performance and in terms of accuracy due to ignorance about underlying theory.
What if you want to make a numerical method for something you can't look up the recipe for?
Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.
> The point is that 3-5 years of analysis might be an overkill
Rarely have I met an engineer and said, wow that guy knows TOO much about math. What do you think they should substitute?
But the amount of time available is fundamentally limited and it has to be balanced.
E.g., in my personal circumstances probability theory, statistics, deeper proof understanding would all be useful. I had all of these at varying depths.
But it's only analysis that we were getting for years and years and years...
And I see why and how this is such an important subject historically. It's just not that big anymore relative to other things.