Suggestions for special lectures at next ICM

Question

(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.)

Answer 1

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.

---

EDIT: Kevin Buzzard ended up giving precisely such a talk at ICM 2022: The rise of formalism in mathematics.

Answer 2

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.

Answer 3

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.

Answer 4

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

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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 $$ a_1^k + a_2^k + \cdots + a_n ^k = b^k $$ 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'.

Answer 9

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.

Answer 10

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).

Answer 11

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.

Answer 12

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.

Answer 13

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.

Answer 14

A lecture by Ken Ribet similar to his talk as the outgoing AMS president, updating the work on the Taniyama-Shimura-Weil and Serre conjectures, and modern proofs of Fermat’s Last Theorem.

https://youtu.be/mq9BS6S2E2k

Answer 15

Other idea could be a lecture about the site MathOverflow itself. This is my proposal for a special lecture, the slides could be about

• presentation and what's MathOverflow;

• brief history with comments/anecdotes/curiosities from senior users;

• papers that were borned/inspired in MathOverflow;

• MathOverflow's community;

• statistics about the site;

• MathOverflow in social networks;

• extracts (if it's possible) about answers and/or unsolved problems;

• importance of Meta and critics;

• a call to joint/recruit more new contributors;

• perspectives or future projects...

• underrepresented subcategories of mathematics;

• policy of moderation (adding a greeting/thanking for the moderator team from the community of users of MathOverflow;

• inherent problems of the site, for example downvotes without comments/feedback;

I don't know what of these could be suitable and legitimate for the owners of the site, moderators and community of the site.

I would like to dedicate with all respect this proposal to the Ukrainian casualties of the war started previous week.

Answer 16

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.)

Answer 17

An expository debate between Peter Scholze and Shinichi Mochizuki on the veracity of the latter's claimed proof of the abc conjecture.

Answer 18

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.