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After completing a Ph.D. in pure mathematics, 10 years ago I left academia for working in industry. There, a typical question is "What can we do to accelerate $x$?" when a project is slowed down, and the typical answer is "Let the people concerned with the issue focus on it and/or bring in some experts", which usually solves the issue.

I wonder if mathematical research can work the same way. Say, if you had $100 million to spare and really wanted to see the Riemann hypothesis resolved, what would you do?

  • Would it help to finance a special decade at some institution, where 25 leading researchers are free from everyday concerns (in particular, administrative and teaching duties) and can spend their entire time working on this problem together?

  • Would it be better to use these funds to let the 25 experts each supervise 10 graduate students over a course of 20 years? Or to support some sort of crowdsourcing?

  • Or is it just not possible to focus exclusively on one (incredibly difficult) problem and one should rather pursue whatever is doable at the moment? Is it similar to (paraphrasing Don Knuth) "Computer science is like the Great Wall of China where each workman contributes a brick"?

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closed as primarily opinion-based by Yemon Choi, Alexandre Eremenko, R W, Boris Bukh, David Handelman Sep 7 at 15:25

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ If one looks at famous open problems that have survived for a long time that were solved in the last century, most of them have the following feature: their solution took decades of mathematical breakthroughs to achieve, culminating in one final brilliant breakthrough, granting the person(s) who put the 'capstone' on the problem extraordinary credit, and everyone else who contributed much less so. For RH, there is no sign that the program that will ultimately solve it has even been initiated. $\endgroup$ – Stanley Yao Xiao Sep 6 at 19:50
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    $\begingroup$ You might (re-)read The Mythical Man-Month by Fred Brooks Jr. (either the essay or the book). If you are interested in discussing this seriously, I can come up with ideas, and you can contact me using the email address on my user page. This forum is not well suited for your question: you might see if it fares better at Academia.StackExchange. Gerhard "More People Makes It Slower" Paseman, 2019.09.06. $\endgroup$ – Gerhard Paseman Sep 6 at 19:51
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    $\begingroup$ The problem is that it asks for opinion, not fact. While it is a good question (and one I have been considering), this forum is not well suited for discussion, which is where I see this leading. If you have a good answer that is not just opinion, I encourage you to post it. Gerhard "Will Read Well Opined Answers" Paseman, 2019.09.06. $\endgroup$ – Gerhard Paseman Sep 6 at 21:08
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    $\begingroup$ It does not work with mathematics as it does in industry. Someone said: "If you bring 9 women, you still cannot make a child in one month". $\endgroup$ – Alexandre Eremenko Sep 7 at 0:31
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    $\begingroup$ This question is highly opinion-based. Not only that, any serious answer to it would inevitably touch political and ideological issues which have no place in MathOverflow. $\endgroup$ – Algernon Sep 7 at 11:22
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Wikipedia is a great project, and it is without doubt a big impactful resource. With this as inspiration, I started to collect definitions, theorems, formulas and references together with some examples, for topics regarding symmetric functions. This is skewed towards more personal interests and a bit too technical to be on wikipedia.

This has accelerated my personal research projects, as I can refer new collaborators to this page, instead of asking them to find the correct page in a book or paywalled article. I also try to keep up with the latest research, so that the information is fresh, and quickly available.

Having quick access to definitions and references, which are easily found by search engines, and viewable on a regular web page with a smartphone, should help facilitate quicker progress.

The conclusion is, funding an online resource with the purpose to quickly get a new PhD student or researcher new to the field up to speed, is probably a good investment.

Aggregating and streamlining learning existing results and knowledge is a good step to produce new knowledge, in my opinion.

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    $\begingroup$ Yes, putting everything on-line as a first (or zero-th) approximation, and then having serious people refine and streamline, ... repeatedly, if appropriate... with their personal reputations giving some kind of guarantee of veracity... gives a more efficient starting-point for further progress. $\endgroup$ – paul garrett Sep 6 at 21:44
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    $\begingroup$ One could argue similarly that MathOverflow is another such resource. While I think you have a good point, I believe Stanley's observation above is more pertinent to the question asked. I am unsure how your approach would improve the current literature assembled (and being assembled) on the Riemann Hypothesis. Gerhard "Wikipedia Does Deserve More Funding" Paseman, 2019.09.06. $\endgroup$ – Gerhard Paseman Sep 6 at 22:36
  • $\begingroup$ @GerhardPaseman Assembling a list of all lemmas (tagged with appropriate metadata) which have been used in all attempts of the RH so far, would be a good starting point I think. As a fact, I got an email yesterday regarding a research question in algebraic combinatorics, and I could basically solve the question by just referring to 3 or 4 rather obscure references (which I have encountered while compiling my web-resource), and chaining them together. $\endgroup$ – Per Alexandersson Sep 7 at 9:27
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    $\begingroup$ Some rough numbers: mathematics is roughly 1% of Wikipedia, and the budget of the Wikimedia foundation is roughly $80 million per year. So maybe this proposal costs $1 million per year? en.m.wikipedia.org/wiki/Wikipedia:1,000_core_topics upload.wikimedia.org/wikipedia/foundation/6/60/… $\endgroup$ – Matt F. Sep 7 at 10:27
  • $\begingroup$ @MattF.Maybe as a rough estimate, but a lot of people on Wikipedia edit many different types of articles, so having something like Wikipedia takes advantages of network effects. And for this sort of thing to work, one would presumably want much more detailed and technical articles than just those on Wikipedia. My guess is that the cost would be much higher. $\endgroup$ – JoshuaZ Sep 7 at 23:32
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Here's one budget for spending the money over 10 years. Obviously all the numbers are only indicative.

$\$$75 million for child care for mathematically trained people who want to work on these issues, an average of $\$$7,500 per year for 10 years for 1000 people each year. Household cleaning and food preparation could also be included. This would free up the time of current researchers, and open up the research to mathematically trained people, especially women, who are spending their time on work in the household instead of research.

$\$$10 million as prize money for unconditional proofs of known consequences of either the Riemann or Generalized Riemann Hypothesis. Perhaps this would be 20 prizes, each worth $\$$500,000, based on the lists of consequences here, here or here. The ideas in those proofs would be good sources ideas for proving the Riemann hypothesis.

$\$$10 million in travel grants to encourage global collaboration on these topics. Perhaps this would be 10 years of 400 grants per year of $\$$2,500 each, covering airfare and a week of expenses in each case.

$\$$4 million to help people write up their research or their expository works in the area. Perhaps this means that for each of 10 years there are 10 people being paid an average of $\$$40,000 per year to help write up this work.

$\$$1 million to make existing numerical research in the area more accessible, e.g. better access to tables of the zeroes, translations of relevant algorithms into nice packages in several languages.

Note all of this money may go further in countries with high levels of mathematics but cheaper costs of living. Conditions for work on the Riemann hypothesis are already relatively good for math professors at American or European universities; to make a big step forward, it may help to involve people who are mathematically talented but not in those roles, for whatever reason.

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    $\begingroup$ I'm not sure the grants for unconditional results currently known to depend on RH would be move in the direction of RH. Many of those results use methods which wouldn't directly move in that direction. For example, there are statements which have been proven via "Assume RH. Then X" and then some proves "Assume ~RH. Then X." $\endgroup$ – JoshuaZ Sep 7 at 1:41
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    $\begingroup$ JoshuaZ, if you can use that technique to prove “Assume ~RH. Then RH”, it would be good enough! Anyway, what other sorts of results would you suggest that are easier than RH but on the way there? $\endgroup$ – Matt F. Sep 7 at 2:30
  • $\begingroup$ Valid point. But often those results are things where one would be surprised if they would by themselves lead to proving RH (e.g. statements that some inequality is violated infinitely often). $\endgroup$ – JoshuaZ Sep 7 at 7:35
  • $\begingroup$ Possible options would be things like improving the ,known fraction of zeros on the line (although it seems consensus is that those techniques by themselves are unlikely to prove RH), or proving Lindelof Hypothesis or similar claims. Strictly speaking those do fall into the category of things which follow from RH, but since they are about zeta itself, we don't often think of them the same way we would the sort of results you linked to, which are purely number theoretic corollaries with no strong analytic content by themselves. (This may be an artificial distinction.) $\endgroup$ – JoshuaZ Sep 7 at 7:39

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