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This is a great overview, but there is a big gap between the narrative style overview here and the technical details of the proof in the paper. The conclusion i

by ximeng 5y ago

This is a great overview, but there is a big gap between the narrative style overview here and the technical details of the proof in the paper. The conclusion is in very vague terms ("far fewer integers"): Helfgott and Radziwiłł’s solution to the logarithmic Chowla conjecture marked a significant quantitative improvement on Tao’s result. They could sample over far fewer integers to arrive at the same outcome: The parity of the number of prime factors of an integer is not correlated with that of its neighbors. It would be really interesting to have a commentary to go along with the proof to help bridge this gap. This could explain why choices are made in the proof and help to visualise the implications. This article comes close in some ways, for example with the examples regarding multiplicativity of the Liouville function. OK, it is a 100 page paper, so understanding the details of the proof is going to be tough. It would still be fun if there was a way as an interested reader to step up from reading this type of article to at least understanding the abstract of the paper.

CaptainNegative 5y ago
The article goes into significantly more depth than is quoted above, building motivation first by describing the approach initially taken by Tao (as I understand it, analyzing the combinatorial expansion of a particular graph on the integers) followed by giving an overview of the new approach (analyzing the spectral expansion of the difference between Tao's graph and a simplification thereof) and why the simplification was reasonable. Obviously the overwhelming majority of the gory details were spared, but given the content I thought the author did a good job giving a high level picture accessible to a relatively general audience.
- ximeng 5y ago
  Just to be clear - I love the article and want more.
  - ianai 5y ago
    I’d actually refer the interested reader at that point to just try reading the actual paper past that point. It’s probably not going to be easy or even all that legible at first, but it’s where to start. Backfill with outside sources from there.
    - konschubert 5y ago
      But, to support GPs suggestion: Wouldn't it be nice if there was something that would help hold the hand of an educated-but-not-expert reader when reading the actual paper. "just read the source" isn't the answer to a lack of documentation :) (usually)
      - ianai 5y ago
        It’d be nice but I don’t know what you’re asking for would look like exactly. Academic papers usually have an introduction just past the abstract and may be the closest to what’s requested here. This is supposed to get the reader prepared for the meat of the paper. In a lot of math papers, it acts as a 101 prepper/refresher for the core concepts used in the paper. There’s also the wolfram site for math. It’s been a while, but it functions something like an encyclopedia of all math topics. I’ve used it to quickly backfill a reference many, many times. Reading an article or proof is not exactly analogous to “reading the source” as both articles and proofs are meant to be written to be read by a human with an agreed upon background and lead them logically to the conclusion(s) of the article. Articles and even proofs can be intimidating, I get that, but they are surmountable with a suitable time investment.
        ximeng 5y ago
        I think maybe Wikipedia is better here than Wolfram's site, it has entries for Ramanujan graphs and expander graphs for example. The paper does actually seem readable, especially at the beginning, but it's dense and long. Somewhere between the article and the paper in terms of complexity and technicality would definitely be interesting to me, although I realise it's a lot of work to produce something like that. Interestingly the paper acknowledges collaborators who have helped via MathOverflow. I wonder if a collaborative site for explaining maths papers in more detail could be viable. After all, even xkcd.com has explainxkcd.com. Like you, I am not sure what this would like, but it would be fun to find out.
        deleted 5y ago
        [deleted]
- ChefboyOG 5y ago
  Quanta is one of the most consistently high quality publications operating today, for my money. Their Joy of X podcast with Steven Strogatz is highly recommended for anyone interested in math.
WoahNoun 5y ago
>an interested reader to step up from reading this type of article to at least understanding the abstract of the paper. I don't think this is a reasonable expectation for most modern results in mathematics. There is just too much background to cover. Quanta's narrative style is the absolute best at it, but often leaves people with technical backgrounds outside of math wanting more. But I don't really think that's possible.
- ximeng 5y ago
  I agree in general, and Quanta is great, including this article. I do wonder if more could be done in this case, where the abstract (https://arxiv.org/abs/2103.06853 https://arxiv.org/abs/2103.06853) has relatively simple terminology. I guess I'm just greedy for more of the same.
hardlianotion 5y ago
There is a rather interesting exposition paper put out yesterday by Helfgott: "EXPANSION, DIVISIBILITY AND PARITY: AN EXPLANATION" https://arxiv.org/pdf/2201.00799.pdf https://arxiv.org/pdf/2201.00799.pdf Have only scanned it, but it seems well motivated and readable.
- ximeng 5y ago
  This is exactly the kind of thing I'm thinking about, thank you. It still looks challenging but certainly more accessible than the original.
- gilleain 5y ago
  This is great. Not only the acknowledgements section to MathOverflow (as mentioned in another thread) but also the section with a drawing of an elephant > Incidentally, in earlier drafts of the paper, we did not have a “writer” and a “reader”, but a mahout and an elephant <<picture of man and elephant>> They were unfortunately censored by my coauthor. As this is my exposition, here they are :)

by ximeng 5y ago

CaptainNegative 5y ago

The article goes into significantly more depth than is quoted above, building motivation first by describing the approach initially taken by Tao (as I understand it, analyzing the combinatorial expansion of a particular graph on the integers) followed by giving an overview of the new approach (analyzing the spectral expansion of the difference between Tao's graph and a simplification thereof) and why the simplification was reasonable. Obviously the overwhelming majority of the gory details were spared, but given the content I thought the author did a good job giving a high level picture accessible to a relatively general audience.

ximeng 5y ago
Just to be clear - I love the article and want more.
- ianai 5y ago
  I’d actually refer the interested reader at that point to just try reading the actual paper past that point. It’s probably not going to be easy or even all that legible at first, but it’s where to start. Backfill with outside sources from there.
  - konschubert 5y ago
    But, to support GPs suggestion: Wouldn't it be nice if there was something that would help hold the hand of an educated-but-not-expert reader when reading the actual paper. "just read the source" isn't the answer to a lack of documentation :) (usually)
    - ianai 5y ago
      It’d be nice but I don’t know what you’re asking for would look like exactly. Academic papers usually have an introduction just past the abstract and may be the closest to what’s requested here. This is supposed to get the reader prepared for the meat of the paper. In a lot of math papers, it acts as a 101 prepper/refresher for the core concepts used in the paper. There’s also the wolfram site for math. It’s been a while, but it functions something like an encyclopedia of all math topics. I’ve used it to quickly backfill a reference many, many times. Reading an article or proof is not exactly analogous to “reading the source” as both articles and proofs are meant to be written to be read by a human with an agreed upon background and lead them logically to the conclusion(s) of the article. Articles and even proofs can be intimidating, I get that, but they are surmountable with a suitable time investment.
      - ximeng 5y ago
        I think maybe Wikipedia is better here than Wolfram's site, it has entries for Ramanujan graphs and expander graphs for example. The paper does actually seem readable, especially at the beginning, but it's dense and long. Somewhere between the article and the paper in terms of complexity and technicality would definitely be interesting to me, although I realise it's a lot of work to produce something like that. Interestingly the paper acknowledges collaborators who have helped via MathOverflow. I wonder if a collaborative site for explaining maths papers in more detail could be viable. After all, even xkcd.com has explainxkcd.com. Like you, I am not sure what this would like, but it would be fun to find out.
        deleted 5y ago
        [deleted]
ChefboyOG 5y ago
Quanta is one of the most consistently high quality publications operating today, for my money. Their Joy of X podcast with Steven Strogatz is highly recommended for anyone interested in math.

WoahNoun 5y ago

>an interested reader to step up from reading this type of article to at least understanding the abstract of the paper. I don't think this is a reasonable expectation for most modern results in mathematics. There is just too much background to cover. Quanta's narrative style is the absolute best at it, but often leaves people with technical backgrounds outside of math wanting more. But I don't really think that's possible.

ximeng 5y ago
I agree in general, and Quanta is great, including this article. I do wonder if more could be done in this case, where the abstract (https://arxiv.org/abs/2103.06853 https://arxiv.org/abs/2103.06853) has relatively simple terminology. I guess I'm just greedy for more of the same.

hardlianotion 5y ago

There is a rather interesting exposition paper put out yesterday by Helfgott: "EXPANSION, DIVISIBILITY AND PARITY: AN EXPLANATION" https://arxiv.org/pdf/2201.00799.pdf https://arxiv.org/pdf/2201.00799.pdf Have only scanned it, but it seems well motivated and readable.

ximeng 5y ago
This is exactly the kind of thing I'm thinking about, thank you. It still looks challenging but certainly more accessible than the original.
gilleain 5y ago
This is great. Not only the acknowledgements section to MathOverflow (as mentioned in another thread) but also the section with a drawing of an elephant > Incidentally, in earlier drafts of the paper, we did not have a “writer” and a “reader”, but a mahout and an elephant <<picture of man and elephant>> They were unfortunately censored by my coauthor. As this is my exposition, here they are :)