13 Lectures On Fermat 39;s Last Theorem

[Venice Sassone]

ICM2022 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

(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.)

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.

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.

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.

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.

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.

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.

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.

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.

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 $9.7 \cdot 10^14$.

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'.

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!)

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.

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).

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.

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.

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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