Again, a week full of developments around AI and math. More than 700 papers have been published on more than 300 open math problems. The good news first: AI seems to make sign mistakes as well, resulting in a withdrawal of three papers . Nevertheless, the problems that have been ‘resolved’ have stirred mathematics in each corner, some people quickly checking the outcomes in relation to their (sometimes lifetime) work.. Now, I may be witnessing this from a relatively comfortable position, but this may not be the case for younger researchers, our PhD students and postdocs, who find it hard to see how the mathematical landscape is evolving, even where this is going in a few months, or even weeks from now. So here are some thoughts, in an attempt to shed some light on this.

First of all, my own experience with the students that I encounter in my department and around. They actually seem very hesitant to use genAI, for several reasons: ethical, environmental, but also because they don’t want to deprive themselves of the pleasure of discovery. On the other hand, they are also aware of the necessity to use it, driven by the competition they are in. Clearly, for the careers and promotion of our young researchers, in the past decade the focus was put heavily on brilliance and excellence, based on number of publications, proposals, etc. Again, and this I am also witnessing myself: many (many!) more publications are submitted to journals, which are generated by AI, and also research proposals overload the funding system, as it is very easy to write these with AI-assistance.

My glimpse into the future is that we, mathematicians, should adopt a different style of working. Concerning publishing papers: maybe similar to our colleagues in physics, after a quick pen-and-paper creative exploration, we may have to stir ‘the machine’ in the search for the right Lemmas and Theorems. And, in the other direction, we may have to identify the ‘data’ worth further research, among a ‘zoo’ of genAI-generated Lemmas and Theorems.

Our publication practice should then also become more hybrid in nature, where we may strive for a combination of ‘human’-written or -stirred research, showing the skills of the authors, and AI-assisted parts, speeding up the research. This may be realized by a non-linear publication style, where one ideally zooms into different levels of details of the manuscript. This may be realized by the type of publications we see in other fields, such as physics, biology or chemistry: a short letter highlighting the main theorems and conceptual insights, which is then followed by additional material including more details; the latter possibly contains AI-generated material, for which the authors at all times keep their responsibility of course.

For our next generation, the challenge is to explore the balance between their own, and the AI-generated ideas. Excellence will then get a different meaning, precisely in this two-faceted combination. For publishers of mathematics journals, one should facilitate the exposition of these two levels, allowing editors and referees to judge articles based on both of these quality criteria. And, in line with this, for national science foundations the challenge will then be to quantify excellence based on both of these quality criteria, while being extremely careful to distinguish the two facets!

Remains that also I am curious to see where all of this is going, what the landscape will be in a year from now, or even months, or weeks. Most importantly is that we keep discussing this together!

During the past few months as many of you I have of course followed the development of the use of AI in mathematics. Or abuse; when it is unclear which part of a paper, thesis or exercise sheet is written by the (human) author or student. Many expert opinions and concerns have already been shared on the internet (see https://terrytao.wordpress.com/ for a good up-to-date collection or a start) so I won’t repeat those here, or dwell on those aspects (how important they may be). Rather I would like to illustrate the use of AI in a working example, and how I found it to be extremely useful as a powerful assistant in understanding new mathematical structures. Fortunately, there are more instances of mathematicians using genAI as I do, and I think this may actually indicate a way forward in mathematical research.

Two years ago I proposed an extension of K-theory invariants from C*-algebras to so-called operator systems, a structure I encountered somewhat before that in my work with Alain Connes on spectral truncations. The mathematical details are not so relevant here, more important is that it led to the question in the field whether these abstract invariants could be computed in some examples. I made some progress in some cases, and could for instance prove (announced here) that the first K-invariants of the Toeplitz matrices are labeled by their matrix signature. However, a full understanding was missing, and the situation was similar with other known finite-dimensional examples, notably Fourier truncations of the circle and graph operator systems. As I did not find the time to start a full analysis of these cases, which all seemed to require their own techniques, I at some point decided to try genAI for the Toeplitz matrix example. I had some idea to base a proof on, and after quite some going back-and-forth to the prompt I managed to put together a proof. Via handwritten notes I convinced myself of the proof, and then phrased it in the preprint I have now put on the arXiv. This is computational work that could have taken several weeks, if not months, and which was now a matter of days. I was really amazed by the speed, and also started to appreciate the level at which the LLM would search its way through the literature, and correct itself when I was pointing out some imprecision. And vice versa of course.

But then I realized that there are also these other interesting open cases for which I also wanted to know the K-theory invariants. And that is where it is really amazing, the ease with which it produced also these other computations and proofs. The background literature was of a completely different nature, with which I was only vaguely acquainted. However, it pointed to the right sources and steps in a potential proof, which I could then check, again by pen and paper, and then turning it into another section in the papers.

The result is a rather extensive paper, containing a variety of techniques, not necessarily easy to read (even though not complicated, simply the diversity of methods which made it less uniform). This led me to consider that the way in which we write papers may have to change in the future of AI-assisted mathematics. Indeed, even if we agree not to let AI write the papers for us (something we unfortunately witness already as journal editors, or as readers of the arXiv) it is relatively easy to write papers with interesting results, something which could have taken way more time in the old days. So I also tried an alternative, more human-reader friendly option: I wrote a ‘covering’ short 6 pager, where the core of the message is put to the foreground. Namely, I took some three-dimensional cases from the classes of examples considered in the longer paper and wrote a concise, but self-contained treatment for them. The reader now has the option to just grasp the main message, and for instance continue developing the main theory from there, or take (parts of) the extensive paper to dive into the details. Maybe we can adopt such a non-linear narrative also in our paper-writing?