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(I am posting this in my capacity as chair of the ICM programme committee.)

ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's report of the ICM structure committee. The idea is that these are lectures that differ from the traditional ICM format (author of a recent breakthrough result talking about their work). Some possibilities are

  • a Bourbaki-style lecture where a recent breakthrough result (or series of results) is put into a broader context,
  • a "double act" where related results are presented by two speakers,
  • a survey lecture on a subfield relevant for some recent development,
  • a lecture that doesn't fit into any of the existing sections,
  • a lecture creating new connections between different areas of mathematics,

but these are not meant to be exhaustive in any way. So what special lecture(s) would you like to see at the next ICM?

(Unless it is self-evident, please state what makes the lecture you would like to see "special". If you would like to nominate someone for an "ordinary" plenary lecture instead, please do so by sending me an email.)

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    $\begingroup$ How about a presentation of twenty three or so open problems to guide the next century of mathematics research. :) $\endgroup$ – Sam Hopkins Aug 6 at 12:56
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    $\begingroup$ Although not directly a "research question", I think this is a great use for mathoverflow! $\endgroup$ – Theo Johnson-Freyd Aug 6 at 12:56
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    $\begingroup$ @SamHopkins We're 20 years late for this century ;-) More seriously, assuming we do have an "open problems" lecture, would you have someone in mind for delivering it? $\endgroup$ – Martin Hairer Aug 6 at 14:13
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    $\begingroup$ @Sam my understanding is that Hilbert only presented ten problems at the meeting. The other thirteen were included in his write-up. $\endgroup$ – Gerry Myerson Aug 6 at 22:45
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    $\begingroup$ I can’t imagine it would be helpful to have a talk about purported or potential crises in physics at a math conference, particularly so for the ICM. $\endgroup$ – Aaron Bergman Aug 8 at 2:41

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How about a lecture on proof assistants/formal proofs?

Most mathematicians are still skeptical of the value of proof assistants, and it's certainly true that proof assistants are still very difficult for the average mathematician to use. However, I think that much of the skepticism stems from a lack of understanding of what proof assistants have to offer. A popular misconception is that proof assistants just give you a laborious way of increasing your certainty of the correctness of a proof from 99% to 99.9999%. But that's not where their primary value lies, IMO.

For example, having a large body of formalized mathematics available could help machine learning algorithms figure out what constitutes "interesting" mathematics and help them autonomously discover interesting new definitions and concepts—something that seems beyond what computers can do now. For another example, there are increasingly many cases where editors can't find a referee for a complicated and potentially important paper because the referees are skeptical and don't want to waste time studying something that might be wrong. If proof assistants become sufficiently easy to use that authors are routinely required to formally verify their proofs before submission, then referees can focus on the more rewarding work of assessing whether a result is interesting and important instead of spending the bulk of their time checking correctness.

A good lecture on this topic could give the subject a valuable boost. Incidentally, if you want to poll people to assess interest, I would recommend polling younger people. This is one topic where I would value the opinion of younger mathematicians and students more than the opinion of senior mathematicians.

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    $\begingroup$ Personally I would love to see an interactive session with a proof assistant. I think few mathematicians have ever used one (I haven't). This might help in explaining why and how proof assistants can be useful to the mathematical community. $\endgroup$ – François Brunault Aug 6 at 18:16
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    $\begingroup$ @FrançoisBrunault If you want to watch such an interactive session now, you can choose a lecture here: youtube.com/playlist?list=PLVZep5wTamMmvdvczjrLctDM9T4nBse1M (Of course such a session designed for the ICM would be a bit different). $\endgroup$ – Will Sawin Aug 6 at 19:03
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    $\begingroup$ "If proof assistants become sufficiently easy to use that authors are routinely required to formally verify their proofs before submission"? And that such a requirement would not be considered inhumanely cruel by those authors? This would be great but are there many mathematicians expecting this to happen in, say, half a century? Personally, I am not among them. $\endgroup$ – Alex Gavrilov Aug 7 at 9:42
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    $\begingroup$ @RobinGoodfellow : Your concerns are legitimate. But regarding #1, I think proof assistants will actually help, for 2 reasons. First, they'll force the author to understand the proof well enough to explain it to a computer. You'd be surprised how often authors don't completely grasp their own proofs, yet get away with it. I expect a decrease in excessively sketchy and vague papers. Second, if the computer is checking correctness, I expect increased social pressure on the humans to explain their proofs. What use is the human otherwise? $\endgroup$ – Timothy Chow Aug 9 at 16:57
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    $\begingroup$ (continued) As for #2, this is already a problem with more conventional computer-assisted proofs, but I don't think that this is the fault of technology. Rather, it's just a fact of life that some theorems are probably only provable by means of a long and not particularly enlightening computation. Surely we'd rather have a computational proof than no proof at all? I also think that it's human nature to seek conceptual proofs, and I don't see human nature changing just because our ability to find computational proofs increases. $\endgroup$ – Timothy Chow Aug 9 at 17:00
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I suggest lectures on big and transformative ideas. For example, it would be great to have a lecture by Tim Gowers about the future of mathematics publishing, and getting away from the issues with our current model. He has spoken and written on topics like this before, e.g., in this blog post. Another option in the same vein might be an update on the Polymath project.

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    $\begingroup$ I'd too love to hear from Gowers about publishing, but my impression is that a panel would be even better, since the experience is currently distributed across various people (e.g., the editorial board of Algebraic Combinatorics will have something to say about zombie journals; whoever is behind fixing the K-Theory fiasco will probably know a lot about good and bad contracts; I'd also like to hear some failures, like OA journals folding, as told by their participants). $\endgroup$ – darij grinberg Aug 7 at 14:41
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    $\begingroup$ One of the tangible outcomes of Gowers’ foray into the crisis in publication was the creation of two journals (Sigma, Pi) where the author (or their grant/university) pays a publisher in order to have their article appear. I’m not convinced that this is a desirable outcome. $\endgroup$ – Gordon Royle Aug 8 at 2:34
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    $\begingroup$ @Gordon that was merely the first stage. Gowers' more recent, arXiv-overlay, efforts, Algebraic Combinatorics and Advances in Combinatorics, would be more to your liking perhaps. Check this blog post for more recent comments about FofM gowers.wordpress.com/2018/06/04/a-new-journal-in-combinatorics $\endgroup$ – David Roberts Aug 8 at 5:40
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    $\begingroup$ @DavidRoberts Thanks for that link, pretty much exactly mirrors how I feel about it. Although small fry in the journal world, I was an editor of the Australasian Journal of Combinatorics when we decided to go electronic with a free-to-write, free-to-read model. With APC there is always an obvious potential conflict of interest. $\endgroup$ – Gordon Royle Aug 8 at 8:01
  • $\begingroup$ @GordonRoyle I'd go further and say I am convinced that this is not a desirable outcome (Speaking as someone working in the UK, which produced the Finch report, and who has his name on a paper that appeared in Sigma for reasons not of his choosing) $\endgroup$ – Yemon Choi Aug 10 at 19:29
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A topic worthy of a special lecture, and with no obvious other place to go, is ways we as mathematicians can make our field more diverse, equitable, and inclusive. As we know, women and minorities are underrepresented in math. This has less to do with differences in talent and more to do with structural inequality in society, different access to mathematics as students, and perceptions from individuals in underrepresented groups that the mathematical community is not welcoming to them. A special lecture at ICM, drawing attention to these issues and including concrete suggestions for improving the situation, might go a long way towards making math more diverse in the future.

In addition to being the ethically correct thing to do (as being a mathematician is generally among the top jobs in terms of life satisfaction, and hence should be open to all), making math more diverse would also lead to better mathematics, as a diversity of thought and background will lead to new approaches to problems we care about. For example, lack of diversity has contributed to bad and biased algorithms, e.g., in mathematics related to criminal justice. There is already a large literature about concrete strategies to make math more diverse, including work of Uri Treisman, the book Whistling Vivaldi, the book Successful STEM Mentoring Initiatives for Underrepresented Students, and the Harvard implicit bias research. Sadly, many mathematicians are unaware of this body of research, and it doesn't neatly "fit" within our existing silos.

A great speaker for such a special lecture would be Francis Su, who has served in the leadership of both the AMS and MAA, who has worked on these issues for years, and who recently published Mathematics for Human Flourishing, a book which describes itself as "An inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourish." Another great speaker would be Dave Kung.

In the same vein, one could imagine a special lecture on how to use mathematics for social good. Several texts and resources have recently appeared on this topic, including this book, this compendium, and these curricular guides. Mathematicians might appreciate a survey of work in this direction, including pointers on how to pivot their research and/or teaching in a direction of social justice.

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The Weapons of Math Destruction would make an interesting and timely topic for such a lecture.

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    $\begingroup$ Is the book that good? $\endgroup$ – Rodrigo de Azevedo Aug 9 at 7:54
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    $\begingroup$ What is "good"? It is a relevant topic for mathematicians and society, perhaps much more so than most math lectures. $\endgroup$ – Hailong Dao Aug 9 at 17:11
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    $\begingroup$ More generally, mathematicians have very little ethics training compared to any other type of scientist (probably in part because theoretical mathematics does not function like a science). I think this is something that we need to be talking about more, and Cathy's book is one example of that. $\endgroup$ – R. van Dobben de Bruyn Aug 9 at 17:17
  • $\begingroup$ The topic is certainly relevant. Yet, that was not my question. $\endgroup$ – Rodrigo de Azevedo Aug 12 at 19:07
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    $\begingroup$ I found the book interesting and learned many things from it. $\endgroup$ – Hailong Dao Aug 12 at 20:56
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I think one lecture topic should be devoted to (some aspects of) the communication and dissemination of mathematics. Even though it is like fitting a mini conference into one hour, aspects of bringing the subject to more people is important and current practitioners and presenters should be made aware of good practices in communication.

It might be useful to invite Matt Parker or Kelsey Houston-Edwards to speak about some of their process for emphasizing and explaining a topic. We as a group might shift our perspective on what goals are important to present (by lecture, Youtube video, blog post, or preprint) a subject. Even if we cannot all become great communicators, we can try to make our areas of study accessible to those who are.

Gerhard "Is My Point Coming Across?" Paseman, 2020.08.06.

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    $\begingroup$ Great idea. It strikes me that the communication of current research in mathematics to the general public is light years behind that which is taking place in theoretical physics and cosmology for example. $\endgroup$ – D.S. Lipham Aug 8 at 22:47
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I'd suggest a lecture discussing when and how a computer can be useful to prove or disprove conjectures. As a first example, think about Euler's sum of powers conjecture. In 1769, Euler proposed a generalisation of Fermat's last theorem: for all integers $n$, $k$ greater than $1$, the equation $$ a_1^k + a_2^k + ... + a_n ^k = b^k $$ implies that $n \geq k$. The conjecture is true for $k=3$ (this follows from Fermat's last Theorem). However, it has been first disproven for $k=5$ in 1966 via a direct computer search by L. J. Lander and T. R. Parkin. The couterexample they found was: $$ 27^5 + 84^5 + 100^5 + 133^5 = 144^5 $$ Moreover, combining some results on elliptic curves, N. Elkies restricted the variables in the case $k=4$ and was able to find a counterexample using a computer: $$ 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4 $$ Here, it is intersting to notice that a computer search had not been able to find it (this is due to the fact that many parameters were involved): it was also necessary some work to restrict the situation to a more suitable case.

As a second example, consider the search for some kinds of primes: it has been conjectured that there exist infinitely many Wall-Sun-Sun primes; however, thanks to some computer searches, we now know that, if any such a prime exists, it must be bigger than $9.7 \cdot 10^{14}$.

As a third example, I will cite the search for lower bounds of de Bruijn–Newman constant: before the proof by Brad Rodgers and Terence Tao that $\Lambda \geq 0$, computer searches had established some bounds on this constant. Note also the relation with the searches for counterexamples to Riemann Hypothesis.

EDIT: Some examples of important results whose proofs required, at some steps, the help of a computer can be found, for instance, here. In some cases (e.g. Erdos discrepancy problem), a first (partial) proof involved the use of a computer, but later the conjecture has been completely proven without it. I think it may be also interesting to discuss the fact that many mathematicians, at least when the first cases of computer-assisted proofs appeared, did not accept the solutions as they were 'infeasible for a human to check by hand'.

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    $\begingroup$ I like the idea, however I don't particularly like your examples. Sure there was some interesting work involved in reducing the search space, but this still comes down to just checking if solutions in this space work. I feel similarly about the Wall-Sun-Sun primes - has the computer work brought us any results of theoretical interest, or just told us some finite set doesn't work? The last example seems more interesting - it is by no means obvious any of this can be achieved with a finite computation, and yet here we are. $\endgroup$ – Wojowu Aug 6 at 17:06
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    $\begingroup$ In my opinion some examples like the Erdos discrepancy problem and the de Bruijn-Newman constant essentially form arguments against the importance of computers, because the eventual proofs did not use any ideas generated by the computer experiments but instead totally different ones. Of course there are other areas where computers have been helpful (either in the sense of helping proofs, or in providing information about areas where proofs are not likely to be forthcoming) but one has to be careful in choosing examples. $\endgroup$ – Will Sawin Aug 7 at 1:17
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    $\begingroup$ A recent case where computer calculations were helpful to an eventual non-computerized proof is the recent proof of the sphere packings in dimension 24 arxiv.org/abs/1603.06518 where IIRC computer calculations that had been previously been done by Cohn and Elkies, Cohn and Kumar, Cohn and Miller, ... suggested properties of the key function used in the proof, which were used to rapidly find the right function once the general principles for how to construct such a function were found by Viazovska. (These works were discussed at the 2018 ICM.) $\endgroup$ – Will Sawin Aug 7 at 1:27
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    $\begingroup$ Similar is the progress on expressing numbers as sums of three cubes. There's a lot of work to trim down the search space and be clever about the algorithm. $\endgroup$ – David Roberts Aug 7 at 3:50
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    $\begingroup$ A great recent example of a computer-assisted proof is the Kaluba--Kielak--Nowak proof that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for $n\geq 5$. This seemingly completely abstract statement relies on an intensive computer calculation, but this came as a complete surprise to the community. $\endgroup$ – HJRW Aug 7 at 15:06
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Especially since we lost Michael Atiyah in 2019, I would like to see a talk dedicated to the unity of mathematics. The idea of addressing the "tower of Babel" tendency of increased specialization is always needed, I think. This can be accomplished in several ways already suggested. Perhaps by giving an overview, or a list of visionary questions, or imagining new ways to accomplish a sense of unity in the diversity of the subject. Maybe a lecture entitled "the unity and diversity of mathematics". Such a title may even bring in topics mentioned such as inclusiveness, etc.

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    $\begingroup$ Just as aside comment (I hope don't disturb) I add the following reference. The article (in Spanish) La joven promesa española de las Matemáticas by Laura Moreno from the web page of the newspaper El Mundo (date 2 OCT. 2017) refers in its third paragraph a quote of the words that Sir Michael Atiyah provided as advice to Ismael Sierra. $\endgroup$ – user142929 Aug 9 at 6:35
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During the lockdown I've seen an online talk by Pierre Pansu about persistent homology. Roughly (I'm not the right person to explain it) this is a robust and recent computational way to compute homology, at several scales, with the aim to ignore "noise". It's for instance used in shape recognition. Pansu's talk (which was in a geometric group theory seminar) was explicitly to advertise its used in pure math, and precisely in geometric topology / group theory, where it ought to bring new computational methods, more powerful than naives ones (e.g., if one wishes to under the shape, e.g., computing homological invariants, of small pieces of Cayley graphs). The talk was great and motivating (more than my poor summary!)

PS MathSciNet search for "persistent homology" (anywhere) yields papers: 0 in years $\le 2004$, 25 in 2005-2010, 100 in 2010-2015, and 200 in 2015-2020.

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    $\begingroup$ Absolutely agree that we need to focus on the pure-math applications, particularly after seeing the applied-math applications being hyped to death. $\endgroup$ – darij grinberg Aug 7 at 11:42
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    $\begingroup$ I didn't claim that we need to focus on the pure math applications, and even less said anything negative about the applied math applications. Pansu advertised the applied math applications, and suggested possible new uses in pure math. He was then talking to an audience in pure math and more specifically in geometric group theory. I'm not suggesting one should do exactly the same. Actually these two aspects address respectively "a survey lecture on a subfield relevant for some recent development and a lecture creating new connections between different areas of mathematics. $\endgroup$ – YCor Aug 7 at 11:54
  • $\begingroup$ Was it this talk? youtube.com/watch?v=XMrsMlF73jc $\endgroup$ – JCK Aug 7 at 22:52
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    $\begingroup$ @JCK no this was in April 2020, but the second half of the talk was close to your link. The first half of the talk was really a quick introduction to persistent homology. His slides (in French) are here. $\endgroup$ – YCor Aug 7 at 23:11
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    $\begingroup$ @PiyushGrover you're playing with the word "applied", and I actually rather said "use". One might use/apply/adapt methods coming from rather applied math (traditionally applied, say, image recognition), to, say, compute Betti numbers of some manifolds defined by arithmetic means. $\endgroup$ – YCor Aug 8 at 8:15
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Particularly in memory of John Conway, whose creations were mathematically interesting and nontrivial, while of potential appeal to a wide audience: a lecture on developments in accessible mathematics. The idea would be to present progress in solving old problems and new challenges in areas that could be reported by the nonspecialist media, to give the public a taste of what mathematicians do.

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Maybe a panel lecture on tools for online collaboration.

A lot of people now know about and attend online seminars (as listed on researchseminars.org), and there has been some panel discussions already (e.g. this one). But as time goes by, probably more maturity is developing.

One could also be insterested by other aspects:

  • Machine-Learning-inspired live subtitles, which could help Alice and Bob collaborate when neither speak well enough the languages that the other speaks ;
  • prospects for automatic speech-to-$\LaTeX$ for taking live notes, or writing a draft
  • ordering equipement for a whole bunch of universities together, to get a better deal from providers

Indeed, these tools make positions at smaller universities perhaps more attractive than they used to, since daily collaboration/interactions is not restricted to departmental colleagues. They even make collaboration between academics and people from other places more possible (e.g. people working in public agencies, or the private sector).

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How about a survey lecture on the impact of algebraic geometry in mathematical physics? Second proposal: A survey about the impact of mathematical algorithms for computational simulation in science and engineering.

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In their recent ICM paper, Numbers, germs and transseries, Proceedings of the International Congress of Mathematicians, Rio de Janeiro, 2018, Volume 2, edited by B. Sirakov, P. N. de Souza and M. Viana, World Scientific Publishing Company, Singapore, pp. 19-42, Aschenbrenner, van den Dries and van der Hoeven discussed the ambitious program they are engaged in for extending asymptotic differential algebra to all of the surreals. During the last decade, there have been a wide array of advances in the theory of surreal numbers. I'd like to see a talk discussing those advances as well as the future prospects of Conway's theory.

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Empirical processes are key to certain subfields such as high dimensional statistics, compressed sensing,... Even though the field of empirical processes is far from being new, I believe that presenting recent results by Naor, Latawa, van Handel or others, while having a view on recent applications could be beneficial to many.

Further, challenges arise both in applications and in theory and a talk (with two speakers?) could have its place at the ICM. It could either be a survey lecture or a lecture presenting connections, or even a survey of the connections. It could help more 'applied people' dig into some theoretical aspects or the other way round.

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Being less facetious than Sam Hopkins, I would content myself with a Hilbertian selection of major problems within the confines of fuzzy mathematics.

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It may we worthwhile for the community to debate the following.

Do either (or both) of the following tend to diminish the importance of aesthetics in mathematics?

  • The "unreasonable" usefulness of mathematics in "real" life.
  • The pursuit of mathematical research as a career.

In particular, as a consequence, has the overall aesthetic quality of mathematics diminished over the last century or so? One could make the case that the aesthetic quality of mathematical work has become inaccessible, not only to the common man, but also to the lay man and (to a surprising degree) to the working mathematician from a different area of mathematics.

The former influences mathematics by defeating Weyl's famous claim that given a choice between what is useful and what is beautiful, he would choose the second.

The latter leads mathematicians to relentlessly tunnel down rabbit holes and not come out to meet and to exchange notes with other rabbits!

Of course, there are counterpoints! (Else this would not be worth debating.)

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    $\begingroup$ Weyl's quote is about truth and beauty. Not utility and beauty. $\endgroup$ – Lucia Aug 7 at 18:24
  • $\begingroup$ @Lucia Thanks for the correction. I will leave it as it is begging poetic licence since looking for truth is halfway to looking for utility! $\endgroup$ – Kapil Aug 8 at 4:56
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    $\begingroup$ No; I don't think you capture Weyl's meaning at all with that misquote. Weyl certainly did not have applications in mind. More relevant would be to quote Oscar Wilde from "The picture of Dorian Gray": "We can forgive a man for making a useful thing as long as he does not admire it. The only excuse for making a useless thing is that one admires it intensely." $\endgroup$ – Lucia Aug 8 at 5:00
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    $\begingroup$ @Kapil I just wanted to say, I like your suggestion. I agree that the need to get a job, and to be seen as doing "useful" work (grant organizations might be to blame here) has probably reduced the amount of mathematics done for aesthetic reasons. But, there are also WAY more mathematicians now than 50 years ago, so we probably haven't seen a drop in aesthetic quality, at least for the papers that "rise to the top." Still, I like that you draw attention to this issue. You might enjoy the Journal of Humanistic Mathematics. $\endgroup$ – David White Aug 9 at 12:23
  • $\begingroup$ @DavidWhite Thanks for the suggestion of reading JHM. $\endgroup$ – Kapil Aug 10 at 1:33
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An expository debate between Peter Scholze and Shinichi Mochizuki on the veracity of the latter's claimed proof of the abc conjecture.

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    $\begingroup$ Why debate? A fistfight would probably be more fun for the spectators? $\endgroup$ – Jens Reinhold Aug 7 at 9:37
  • $\begingroup$ A public debate is probably not the best way to go about this. However, I would like to hear a talk outlining Mochizuki's proof and its criticisms, specifically on whether the doubts expressed by Schloze have been explained away in the version of the paper which will be published soon. $\endgroup$ – D.S. Lipham Aug 13 at 0:53
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    $\begingroup$ @D.S.Lipham to my knowledge, and I have been following rather closely, Mochizuki's papers have not changed in any substantial way. He has been updating them continuously on his website and providing close commentary on the changes (from saying when italics have been removed from a single word, to when specific subsubsections have been "clarified"). Nothing has been done about the complaints of the community as focused through Scholze and Stix, apart from saying on his blog they don't get basic logic. $\endgroup$ – David Roberts Aug 14 at 2:13
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I would suggest a short lecture on the usefulness of MathOverflow in Mathematical Research. The very question is indicative of the importance that ICM has given to MO. It would be better if some great problems and answers of MO and their impact in the larger body of Mathematics be lectured upon.

In addition, I suggest a lecture on Influence of Combinatorics in Mathematics. This is on the basis of my observation that recently the number of papers on arXiv is maximum in Combinatorics (the second is, I think Number Theory). Along with this, there are several papers in other topics wich crosslist to Combinatorics as a secondary topic. This clearly shows the wide influence of Combinatorics on all of Mathematics.

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    $\begingroup$ why the downvotes? $\endgroup$ – vidyarthi Aug 6 at 16:59
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    $\begingroup$ How does "most arXiv papers are in combinatorics" show the "wide influence of combinatorics on all of mathematics"? For all I know from this it could equally be the case that combinatorics is a very large but very isolated field of maths. (Of course I know this is not the case, but again, I don't see how cited arXiv data support that) $\endgroup$ – Wojowu Aug 6 at 17:10
  • $\begingroup$ @Wojowu I mean several papers in other fields are also crosslisted in combinatorics along with their primary subject ( like many number theory papers are also cross listed in math.CO tag along), and number theory is as usual widely regarded as the'queen of mathematics' $\endgroup$ – vidyarthi Aug 6 at 17:29
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    $\begingroup$ Different fields have different publishing habits. $\endgroup$ – François Brunault Aug 6 at 18:00
  • $\begingroup$ @FrançoisBrunault yes, but why not combinatorics wield an influence on Mathematics (and Mathematicians)? $\endgroup$ – vidyarthi Aug 6 at 18:02

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