Open problems, open mathematics

[This is a guest post by Antonio Auffinger. This blog post was initially written in a different file format and converted using AI. — T.]

Despite writing papers in pure mathematics, much of my time in the past decade was spent talking (mostly listening, to be accurate) to biologists, computer scientists, and physicists. Theoretical physics and computer science are grounded in mathematics and thus share much of our language and, to some extent, our culture. The barriers between biology and mathematics are an order of magnitude higher. I am convinced that biologists are part of a different species. The language, incentives, culture, training, and everything else you might think of appear to be completely disconnected from the way we do mathematics. Yet, the similarities are abundant: the pursuit of understanding, the excitement of discovery when new patterns or phenomena emerge, and the hours needed to make minor advances, sometimes leaving us with just frustration. The intuition of setting up a new experiment feels like witchcraft to me, much like our way of finding connections between different abstract objects feels like magic to them.

What does talking to biologists have to do with the future of mathematics?

It could be beneficial to look at what is happening and what has happened with our neighbors, even if none of us can predict where we will be two years (or even three months) from now. A first lesson that I learned is that a large part of biology is technology-driven. Fields completely change, emerge, and die in a matter of years. The advent of CRISPR, RNA-seq, and cryo-EM, for instance, suddenly allowed humans to observe and manipulate phenomena that were previously inaccessible. These tools generate precious data, and the incentives often favor a culture of seclusion, where discoveries are frequently not shared until the final product is complete.

This has made me appreciate the culture of mathematics. Mathematics has never been free of competition or secrecy, but we abundantly share. We share ideas, we share problems, and we share entire skeletons of approaches with our colleagues, with visitors we just met, in talks, in public forums, and on YouTube. I suspect the enormous effort often needed to produce a proof, even with a full outline, has helped sustain this openness. Mathematicians and mathematics have deeply benefited from this open attitude. Many new connections and major discoveries have started with honest conversations at coffee breaks, in hallways, or on hiking trails. Sharing ideas or arguments before they are fully formed allows others to see pathways we missed, point out obstacles, or take the problem in directions we had not imagined. Also, it is simply more fun to do math together.

My worry is that this unselfish openness will become a thing of the past. If proof generation becomes a fast, accessible commodity while our ways of giving credit remain unchanged, mathematicians (especially those still building their careers) may feel encouraged to optimize locally in ways that weaken our culture of sharing. In the past few weeks, I have had colleagues reach out for advice and tell me they will no longer post on arXiv. I have witnessed trainees posting rushed papers online for fear that others could quickly carry out strategies already outlined in previous work. I am also part of the problem, as I have started advising my students to be extra careful when sharing work in progress.

Biology also offers examples of communities deliberately changing these incentives. During the Human Genome Project, the Bermuda Principles called for the rapid public release of sequence data. Later, the Fort Lauderdale Agreement tried to balance this openness with proper recognition of those generating the data, placing responsibilities not only on researchers producing and using the data but also on funding agencies. Although it is focused on data, it is an example where the status quo was changed by community intervention.

I do not know what the right analogue is for mathematics, but I believe we need to start thinking about it. I encourage the community to reshape our incentives and the way we give credit, to ensure broad accessibility to these new tools, and to make openness a reasonable choice, especially for those still building their careers. Senior mathematicians must engage in serious conversations about ethical use with their trainees. In turn, trainees should be encouraged to truly explore, because many of the solutions to the issues we currently face as a community will come from them.

None of this is an argument against the use of AI in mathematics. I believe these tools will raise the ceiling of the things we can discover, leading mathematicians to ask new questions and understand new phenomena.

This brings me to the second lesson I learned from my biology colleagues: many of the questions I hear from biologists need mathematics. New mathematics. There is room here for topologists, number theorists, dynamicists, algebraic geometers, analysts, etc. This is not just because of our capacity as proof builders, but also because of our capacity for abstraction and for understanding phenomena. Applied sciences are generating complex, time-dependent data that require new methods and theories. This is also a two-way street. Decades ago, topology and knot theory unexpectedly provided a framework for understanding how enzymes untangle and rearrange DNA. In the other direction, attempts to understand population genetics and how gene frequencies drift over time helped motivate new classes of infinite-dimensional stochastic processes, including measure-valued diffusions. Today, geometry and probability underlie dimension reduction methods that biologists use daily to make sense of massive single-cell datasets, such as t-SNE and UMAP.

I am not suggesting that mathematicians need to pivot to biology or any applied science. The pursuit of mathematics for its own sake is the absolute bedrock of our field and that must remain intact. However, as AI changes the landscape, looking outward presents an incredible opportunity to discover new questions, new phenomena and new mathematics, and perhaps AI might even lower the barriers to taking that leap.

Mathematics has much to offer the other sciences, and much to learn from them. I am looking forward to what comes next. Proofs may increasingly be generated by machines, but the ultimate purpose of mathematics remains exactly what it has always been: to ask questions and to understand.