Aim: An invitation to explore mathematics together

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

(Disclosure: ChatGPT helped me write and refine this post.)

Inspired in part by the recent discussions on this blog, we have started a new project, AIM, an initiative to create a community-led opportunity for mathematical exploration. Since AI can make some conjectures easier to resolve, we would like to explore how these developments can be used to benefit the mathematical community, enhancing our learning and understanding. Modern developments in AI bring both excitement and uncertainty in our field. We hope to create opportunities for mathematicians, especially students and early career researchers, to shape what comes next by asking good questions, working on problems, and developing the ideas and applications that come with them.

We have collected a list of open problems in applied mathematics and related fields along with their precise statements, motivation, and references:

https://github.com/MColbrook/AIM

This repository of problems is intended to benefit the mathematical community and become a place for collaborative exploration as well as a place where mathematicians can share proposed proofs and solutions for others to see. Our primary purpose is mathematical. Rather than using a set of problems to benchmark AI systems, we want to let the community shape the questions that we pursue, whether or not AI is used, and how we learn from this work. We warmly invite readers to contribute problems to the list, directions, and solutions to those problems already in the repository. Partial results, proofs, counterexamples, references, and corrections are most welcome. Contributions with or without AI are perfectly acceptable, with careful checking and attribution.

The companion site, AIM, explained (https://mathematics-explained.com/ ), welcomes papers and videos exploring and explaining the mathematical arguments behind the solutions of these problems. A solution to a problem should not be the end of our engagement with a question, for it can open up further questions about why an argument works, whether a different formulation is more insightful, and how the ideas from a solution connect to other mathematics or applications. We want to acknowledge this work.

To encourage participation, we are also running a competition for explanatory papers and videos of up to ten minutes. The competition is split into two strands: for those who have not yet completed a first degree, and for graduate students and postdoctoral researchers at academic institutions. The emphasis is on accuracy, clarity and understanding, rather than on being first to solve or explain a problem. Entries close on 1 January 2027 at 23:59 UTC, with prizes for winning entries and a curated publication of selected contributions. The wider explanation library is open to all, independently of the competition.

Our repository and website are a starting point, and we would be most glad to hear your suggestions. Please share problems that you would like others to explore (across all of pure and applied mathematics), progress on those already listed, and other ideas that might make AIM useful to you or your students. Problems and corrections can be submitted through the repository; broader suggestions are welcome here in the comments or directly to the organisers.

Sent on behalf of the organisers, Steve Brunton, Matthew Colbrook, Maarten V. de Hoop, George Stepaniants, Alex Townsend, and Rachel Ward.