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A lot of recent math discoveries seem to be geometrical: circle packing, infinitely tiling patterns, neighbor coloring theorem, ... I don't seem to see as much

A lot of recent math discoveries seem to be geometrical: circle packing, infinitely tiling patterns, neighbor coloring theorem, ... 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?

travisjungroth 3y ago
All of those new discoveries involve running code. More people are learning to code, more libraries are being written, compute costs are going down. It’s hard to break ground on a “sit and think” problem, especially one that is approachable enough that it’s also understandable news when you solve it.
- syndicatedjelly 3y ago
  Here's your answer as to why these kinds of discoveries make it to the front page of HN.
  - commonlisp94 3y ago
    The math literacy here is pretty poor when people imagine mathematicians to be solving college "calculus and algebra" problems.
dboreham 3y ago
Aren't all those things the same ultimately?
- deleted 3y ago
  [deleted]
vkazanov 3y ago
Well, algebra and calculus have been around for a while. 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.
- constantcrying 3y ago
  Nothing is more important for numerics than a veey good grasp of analysis.
  - vkazanov 3y ago
    Yup. 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.
    - commonlisp94 3y ago
      > Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
      - vkazanov 3y ago
        Understanding that something is fundamentally an approximation is important. Users of these packages are well aware of the fact. What point are you trying to make?
        commonlisp94 3y ago
        > fundamentally an approximation is important. 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.
    - 3y ago
gowld 3y ago
Langlands Program, one if many examples, illustrates (ha!) that nearly all math is geometrical. Math doesn't care about silly human distinctions like "Algebra", "Analysis", and "Geometry"
klyrs 3y ago
Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra... but the open questions there tend to be quite esoteric. That said, we did recently see a novel approach to solving quadratic equations ( https://www.sciencealert.com/math-genius-finally-discovers-easy-way-to-solve-quadratic-equations-after-4-000-years https://www.sciencealert.com/math-genius-finally-discovers-e... ). 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.
- constantcrying 3y ago
  >Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra Or unless by calculus he means analysis, which is really active. Especially things like PDEs.
QuesnayJr 3y ago
I think other fields it's gotten much harder to explain the results. For example, it was a big deal in number theory a few years ago when 10 authors proved a weak form of the Shimura-Taniyama conjecture for number fields with complex multiplication. (The simplest case of Shimura-Taniyama implies Fermat's last theorem.) It would take an entire course to explain just the statement of the theorem.
svat 3y ago
> other areas like number theory… 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 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!]
constantcrying 3y ago
It is probably mostly a reporting bias. Almost all new mathematical discoveries don't have simple pictures to draw and are about highly abstract concepts which are difficult to write about for a general audience. 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.

travisjungroth 3y ago

All of those new discoveries involve running code. More people are learning to code, more libraries are being written, compute costs are going down. It’s hard to break ground on a “sit and think” problem, especially one that is approachable enough that it’s also understandable news when you solve it.

syndicatedjelly 3y ago
Here's your answer as to why these kinds of discoveries make it to the front page of HN.
- commonlisp94 3y ago
  The math literacy here is pretty poor when people imagine mathematicians to be solving college "calculus and algebra" problems.

dboreham 3y ago

Aren't all those things the same ultimately?

deleted 3y ago
[deleted]

vkazanov 3y ago

Well, algebra and calculus have been around for a while. 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.

constantcrying 3y ago
Nothing is more important for numerics than a veey good grasp of analysis.
- vkazanov 3y ago
  Yup. 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.
  - commonlisp94 3y ago
    > Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations. And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
    - vkazanov 3y ago
      Understanding that something is fundamentally an approximation is important. Users of these packages are well aware of the fact. What point are you trying to make?
      - commonlisp94 3y ago
        > fundamentally an approximation is important. 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.
  - 3y ago

gowld 3y ago

Langlands Program, one if many examples, illustrates (ha!) that nearly all math is geometrical. Math doesn't care about silly human distinctions like "Algebra", "Analysis", and "Geometry"

klyrs 3y ago

Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra... but the open questions there tend to be quite esoteric. That said, we did recently see a novel approach to solving quadratic equations ( https://www.sciencealert.com/math-genius-finally-discovers-easy-way-to-solve-quadratic-equations-after-4-000-years https://www.sciencealert.com/math-genius-finally-discovers-e... ). 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.

constantcrying 3y ago
>Algebra and Calculus are more or less "solved." Unless by algebra, you mean abstract algebra Or unless by calculus he means analysis, which is really active. Especially things like PDEs.

QuesnayJr 3y ago

I think other fields it's gotten much harder to explain the results. For example, it was a big deal in number theory a few years ago when 10 authors proved a weak form of the Shimura-Taniyama conjecture for number fields with complex multiplication. (The simplest case of Shimura-Taniyama implies Fermat's last theorem.) It would take an entire course to explain just the statement of the theorem.

svat 3y ago

> other areas like number theory… 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 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!]

constantcrying 3y ago

It is probably mostly a reporting bias. Almost all new mathematical discoveries don't have simple pictures to draw and are about highly abstract concepts which are difficult to write about for a general audience. 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.