#+title: A math PhD in the AI era (2026 edition) #+date: 2026-11-03 #+status: finished #+ailvl: 2 #+begin_abstract I was asked to give a 15-minute presentation to incoming mathematics PhD students at [[https://math.univ-cotedazur.fr][my department]] on what it means to pursue a PhD in the age of AI. These notes summarize my personal perspective. #+end_abstract The goal of this document is to give guidance, as of the start of the academic year 2026-2027[fn:outdated], to incoming PhD students. The task is, I believe, difficult, and your PhD advisor may have a very different view! Other views are listed in [[*Additional resources]]. /Disclaimer/: I am a mathematician working on AI, with a [[https://arxiv.org/abs/2602.02712][growing]] [[https://arxiv.org/abs/2605.05116][interest]] in AI safety and related areas. Nevertheless, I am not an unconditional AI enthusiast, and believe that steering how AI is used is important. At the same time, I have a hard time fully embracing arguments from colleagues who believe that any use of AI should be rejected. ** Freedom :PROPERTIES: :CUSTOM_ID: freedom :END: Academic standards are known to be relatively slow to evolve[fn:1]. This is a strength of academia as it protects long-term research programs and scientific rigor. Nevertheless, in the face of the deep changes that AI is bringing to our discipline, and more importantly to the rest of the world, the mathematical community has had to take several stances. The [[https://www.lms.ac.uk/publications/policies/AI][London Mathematical Society (Oct. 2026)]] requires an AI declaration covering ideas, calculations, proofs, and writing. The [[https://leidendeclaration.ai/][Leiden Declaration]] recommends transparency and human responsibility. Note that neither text bans AI: they ask for disclosure. I believe this is the right line, and that it would be toxic to impose or ban the use of AI. One of the core principles of research is freedom. Many of us chose academia /because/ of the freedom it offers. In my opinion, those who seek to impose their views on the entire community, whether for or against AI, threaten one of its core values. Nevertheless, if you use AI, you should understand the responsibilities that come with it. You *need* to disclose your usage to your advisor, your peers, and your audience. People have the right to know what your sources are: papers of course, but also private discussions with humans, and with chatbots[fn:: Note also that you cannot expect to be allowed to put private or unpublished work into a commercial model without authorization.]. It is bad ethics to prompt a model, submit the resulting output to your colleagues, and expect human interpretation, for at least two reasons. First, you need to attribute ideas: as you would say “X suggested this to me”, say when an idea comes from a given model. Second, you cannot offload your slop onto others to interpret. This is your responsibility. ** Understanding :PROPERTIES: :CUSTOM_ID: understanding :END: The last few months can give a bit of vertigo: the [[https://openai.com/index/model-disproves-discrete-geometry-conjecture/][Erdős unit-distance conjecture]] was disproved in May, the [[https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/][Jacobian conjecture]] in July, a [[https://openai.com/index/navier-stokes-solution/][finite-time singularity of Navier-Stokes]] was announced in September, the brutal and gigantic drop of [[https://openai.com/index/sharing-ai-progress-in-mathematics/][400 open problem solutions]] was claimed in October, and the full Kakeya conjecture fell last week. Are we, mathematicians, done? If we consider that our sole purpose is the pure production of proofs, then I believe that, indeed, our future will look increasingly uncertain[fn:2]. It is /likely/ that, as with raw computational skills or descriptive geometry in the past, technical virtuosity will be valued less in the future. I find this exciting rather than sad: I am happy to delegate computation to my powerful calculator and to look at [[https://www.flickr.com/photos/internetarchivebookimages/14759795056][descriptive-geometry drawings]] purely from an aesthetic perspective. I expect the same shift regarding highly technical proofs to leave more room for understanding. I firmly believe that producing a proof and understanding why it works are two different things. My advice is to learn how to use AI to discover new things and to check your own reasoning. Yes, you can get a direct answer to your question (which may or may not be true, see [[*Mistakes][next section]]), but probably one of the most productive uses is instant access to difficult and niche knowledge. I would bet that if you start your PhD by furiously prompting your favorite model to solve your advisor's question, you will drift away from the path of expanding your own knowledge. This is not to say that you should never use AI to generate ideas; just do so once you know the problem well enough to judge them. ** Mistakes :PROPERTIES: :CUSTOM_ID: mistakes :END: AI and humans make mistakes. Deal with that. Despite the implications in terms of PR, the OpenAI drop contained several mistakes ([[https://github.com/openai/math/blob/main/history.md#october-7-2026][3 retractions]]), and it is not clear whether the Lean formalization of Navier-Stokes follows the same path as the text version. As for Lean, I think you should familiarize yourself with the idea of formalization[fn:3]. Maybe it will not be needed in the future, but given current AI abilities, few people believe that it is sufficient to produce a pure text version of AI-generated proofs without verifiable pieces. There are two equally problematic attitudes: believing the output of your chatbot because “AI said so”, or rejecting it because “it comes from AI”, and so must be riddled with errors. A critical goal of a PhD is to develop your own critical sense[fn:4], not just to master some technical skills. The question you should ask yourself, whether an argument comes from your colleagues or from an AI, is: why should I trust this argument? To develop such confidence, you need to examine the argument, check it against your knowledge, and make it your own. ** Independence :PROPERTIES: :CUSTOM_ID: independence :END: The main purpose of a PhD is not to produce papers that solve the research program of your advisor, but to ultimately become an /independent/ researcher. That does not mean working alone, or without tools, but acquiring scientific maturity: being able to develop your own questions, intuitions, and “taste”. A PhD is a learning process where struggles and dead ends are not necessarily failures. These failures are typically one way to develop what we call intuition or taste. AI typically lowers technical obstacles (drastically), and sometimes removes whole parts of your research difficulties. This can be very helpful. At the same time, if your work consists mainly of asking an AI what to do next, you risk outsourcing not only the technical aspects but also your ability to build up scientific taste. Concretely: you risk meeting the expectation to publish without actually growing as a researcher. On the other hand, if you use it while staying intellectually engaged, it can indeed be a way to amplify your strengths and your maturation, while allowing you to explore new directions. ** Evaluation :PROPERTIES: :CUSTOM_ID: evaluation :END: How will your PhD thesis be evaluated in three years? How will a hiring committee work? I don't know, and I don't think that the mathematical community knows either. This is a central question, but standards have yet to emerge. Let's be honest: academic evaluation has never been perfect. We have all heard of “publish or perish”, and I think that AI may force us to reconsider this aspect of our work, for the greater good. You are right to fear that if you disclose AI usage, as stated in [[*Freedom][Freedom]], a committee member hostile to AI may hold it against you during a job interview. Or, on the contrary, an enthusiast may consider mathematics done without AI obsolete. People will try to use bad-faith arguments against you throughout your career. This is no reason to hide your usage: transparency is at the core of academic values, and we should (try to) preserve that form of integrity. Despite all this uncertainty, I believe that scientific integrity, mathematical understanding, critical thinking, and intellectual independence, as discussed above, will continue to matter. My one concrete recommendation: [[https://proofsandprompts.com/2026/11/02/please-stop-writing-papers-there-is-another-way/][do not try to become a paper farm]]. ** Future :PROPERTIES: :CUSTOM_ID: future :END: Your generation has an opportunity to redefine what it means to be a mathematician. I believe that we have an opportunity to work on three unfavorable properties of mathematics in academia: elitism, hyper-specialization, and publication-driven careers. I am a public servant, and I believe that I have a duty as a /professional/ mathematician. This duty is of course to produce new scientific knowledge, but also to communicate it (publications, talks, general public dissemination). I think that, as a community, we are often guilty of more gatekeeping than other disciplines. We have an opportunity to work on that right now[fn:5]. Linked to the previous issue, there is no doubt that most of us are now “hyper-specialized”. I sincerely believe that there is no hierarchy in mathematical activity[fn:6]. I think AI allows us to step back, and lowers the cost of looking at what our peers are doing. The goal of the PhD is still of course [[https://matt.might.net/articles/phd-school-in-pictures/][to push the boundary of knowledge]] on a specific problem, but I think broadening your mathematical culture is important. Even though everyone says they don't count papers, citations, or whatever line you could put on a CV, they, I mean, we do. But this is not sustainable anymore. Let's be honest: it never really was. We need to redraw the lines of what is considered a “good” scientist, or hopefully a good collaboration. My stance is the following: - Enjoy your freedom, but respect the responsibility that comes with it. - Develop the taste to understand mathematics, whatever its sources. - Watch out for mistakes along the way, and reassess your beliefs as you go. - Aim for independence at the end of your PhD. - Put your energy into doing good work. One last piece of advice: above all, do what you enjoy. Perhaps this sounds naive, given the current uncertainty, but I believe most of us chose mathematics because we genuinely enjoy it. I sincerely believe that you are in the best place, at the best time, to do mathematics. ** Additional resources :PROPERTIES: :CUSTOM_ID: additional-resources :END: Here is a selection of resources, covering a *wide* range of opinions: - William Thurston's [[https://www.math.toronto.edu/mccann/199/thurston.pdf][On proof and progress in mathematics]] (Apr. 1994, not /directly/ about AI) - [[https://leidendeclaration.ai][Leiden Declaration on Artificial Intelligence and Mathematics]] (Jun. 2026) - Daniel Litt's [[https://www.daniellitt.com/blog/2026/9/13/a-beginning-for-mathematics/][A beginning for mathematics]] (Sep. 2026) - Harvard's [[https://cmsa.fas.harvard.edu/media/2026/09/Summit-on-PhD-Math-Education-in-the-Age-of-AI.pdf][Report of Summit on PhD Math Education in the Age of AI]] (Sep. 2026) - [[https://www.acadmathsci.org.uk/wp-content/uploads/2026/10/Mathematical_Sciences_AI_AcadMathsSci_Oct2026_preprint_final.docx.pdf][The mathematical sciences in an AI-enabled world]] by the UK Academy for the Mathematical Sciences (Oct. 2026) - Association for Human Mathematics' recommendations [[https://static1.squarespace.com/static/6a9f4e1b2cbb616c586c2c24/t/6ae104291040df5c347933a6/1793131561187/AHM_Doctoral_Advising_Recommendations_27Oct2026.pdf][Preserving Our Humanity, Supporting Our Students: Doctoral Advising in Mathematics]] (Oct. 2026) - Jeremy Avigad's [[https://terrytao.wordpress.com/2026/10/05/the-future-of-mathematics/][The future of mathematics]] (Oct. 2026) - Pavel Etingof & Slava Gerovitch's [[https://math.mit.edu/~etingof/futuremath.pdf][Mathematics of the Future]] (Oct. 2026) - Clément Canonne's [[https://minorfree.github.io/Clement/][TCS, AI, and the Adults in the Room]] (Oct. 2026) - [[https://arxiv.org/abs/2610.40584][Early career researcher responses to AI in math]] by Anna Brandenberger, Hanna Mularczyk, and Dora Woodruff (Oct. 2026) - Terence Tao on [[https://teorth.github.io/tao-web/ai-views.html][AI in mathematics (and beyond)]] (last updated Nov. 2026) - The many opinions on the [[https://proofsandprompts.com/all-posts/][Proofs and Prompts]] blog, in particular [[https://proofsandprompts.com/tag/application-season/][this section]] #+begin_acknowledgement I am grateful to [[https://sites.google.com/view/damien-garreau/home][Damien Garreau]], [[https://nkeriven.github.io][Nicolas Keriven]], and [[https://simonarotanodari.com][Simona Rota Nodari]] for discussions and comments on this note. #+end_acknowledgement #+begin_ailvl GPT-6 Sol Medium was used to find references, group raw arguments, judge whether some arguments were facts or opinions, proofread the document, and highlight inconsistencies. #+end_ailvl [fn:outdated] Some or all of this may be out of touch within a year, or within a month: my apologies to the future reader. [fn:1] For instance, the San Francisco Declaration on Research Assessment (DORA 2012) is supposed to be broadly accepted, and despite this, research assessments in reviews [[https://academic.oup.com/spp/article/51/3/337/7425535][did not change that much ten years later]]. [fn:2] How you position yourself in the *creation* (constructivism) or *discovery* ([[https://plato.stanford.edu/entries/platonism-mathematics/][Platonism]]) debate on the origin of mathematics may greatly influence your perception of the current breakthroughs. [fn:3] A first step (without AI!) is the [[https://adam.math.hhu.de/#/g/leanprover-community/nng4][Natural Number Game]]. For using Lean with AI, you could for instance look at Scott Armstrong's [[https://www.scottnarmstrong.com/2026/10/autoformalization-is-now-very-easy/][blog post]]. [fn:4] In France, critical sense is [[https://www.legifrance.gouv.fr/codes/article_lc/LEGIARTI000006524457][codified]] as a major objective of universities. [fn:5] In particular, you should participate in events like /Fête de la Science/ for outreach, communicate often with your peers through the PhD seminar, etc. [fn:6] Someone in bad faith could say that, being at the bottom of the pyramid, I would naturally say that!