What should we tell our students?

8 October, 2026 in guest blog, math.GM, opinion | Tags: , | by | 58 comments

[This is a guest post by Álvaro Lozano-Robledo. This blog post was initially written in a different file format and converted using AI. — T.]

TL;DR: Keep calm and carry on studying math.

I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog. After giving much thought to what topic I should write about to maximize impact, I decided to take this opportunity to reach out to the students: particularly to those undergraduate and graduate students who just a few months ago were dreaming of an academic career in mathematics, but their dreams may now seem distant and, for some, apparently impossible to ever become a reality. This post was inspired by a message (quoted below in its entirety, with permission) that I received from a student desperately looking for advice and guidance. This is not the only such message I have received (and I suspect that many of us are receiving many similar requests), but it is perhaps the most heartfelt, and the one that has moved me the most. Please also note the urgency of the message. Students are making decisions now.

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AHM Statement on OpenAI’s October 6 Release of Mathematical Documents

7 October, 2026 in guest blog, opinion | Tags: , | by | 120 comments

[This is a guest post by the Association for Human Mathematics, reposted from their statements page. This blog post was initially written in a different file format and converted using AI. — T.]

Yesterday, on October 6th, 2026, OpenAI — which is currently defending lawsuits against accusations of illegal plagiarism, copyright infringement, and trademark dilution — released a repository of manuscripts purporting to contain solutions to a number of high-profile problems in mathematics.

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The barriers of perception

7 October, 2026 in guest blog, math.GM, opinion | by | 3 comments

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

“This problem has been tried by several famous mathematicians.” “There is a heuristic argument for why these methods cannot work.” “Getting this algorithm would give new circuit lower bounds.” Observations like these can take on a life of their own, almost like a game of telephone. A limitation of a particular approach, or an implication whose difficulty we do not fully understand, becomes a reason to believe that a problem is beyond reach, and eventually a reason not to think about it at all.

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Hexagon

6 October, 2026 in advertising, guest blog, media | Tags: , | by | 16 comments

[This is a guest post by Ben Antieau, cross-posted from his blog. — T.]

There was no personal problem, no world problem, whose eloquent solution did not exist— somewhere in some hexagon.

—Jorge Luis Borges, “The Library of Babel” [1]

New repository

I am writing to announce Hexagon, a new repository for research works in the mathematical sciences and theoretical computer science. This has been a joint effort of many people over the last couple of months, starting in early August 2026. We felt that the proliferation of results proved with LLM assistance required a new model for collecting the associated papers. Besides wanting these works to be citable and reliably stored for long periods of time, we wanted to allow lower barriers for authorship and for community interest than the arXiv. We also wanted for these works to have a home besides random GitHub repositories or X/Bluesky/Mathstodon/LinkedIn posts.

In order to support the development of this project, we founded the Hexagon Mathematics Foundation in the state of Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3). The Board of Directors consists of Mohammed Abouzaid, François Charles, Bryna Kra, David Savitt, and Lauren Williams. I serve as the first Executive Director. And, we have a wonderful Advisory Board, where Kevin Buzzard, Akhil Mathew, Johannes Schmitt, Steinn Sigurðsson, Nikhil Srivastava, Ravi Vakil, and Rachel Ward serve. For more details, see our who we are page.

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Changes to the Erdős problems web site

6 October, 2026 in guest blog, math.GM, media, update | Tags: | by | 10 comments

[This is a guest post by Thomas Bloom, crossposted from the Erdős problems forum. This blog post was initially written in a different file format and converted using AI. — T.]

AI is changing everything, for better and/or worse, and the rate of change is dizzying; in recent months this has been particularly evident in mathematics, where AI has gone from being essentially useless to helping solve some of the hardest problems in mathematics in less than a year.

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Postdoc position on formal verification and algorithm discovery for numerical analysis

5 October, 2026 in advertising, guest blog, math.AP, math.NA | by | 1 comment

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

About us and the position

The Chair of Numerical Modelling and Simulation at EPFL is dedicated to the design and analysis of numerical algorithms for partial differential equations. Our research is oriented towards the development of novel and innovative numerical techniques aiming at improving the integration between numerical simulations and geometric modelling and processing.

Proof assistants such as Lean, together with recent AI tools, are changing how mathematics is done. We are opening a postdoctoral position to explore what formal verification and algorithm discovery can bring to numerical analysis. There is no fixed project. The goal is to find out, through concrete experiments, where these approaches help in the design and analysis of numerical schemes for PDEs, and where they still fall short.

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The Future of Mathematics

5 October, 2026 in guest blog, math.HO, opinion | by | 27 comments

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

“Mathematics underwent, in the nineteenth century, a transformation so profound that it is not too much to call it a second birth of the subject—its first birth having occurred among the ancient Greeks…”

Howard Stein, in “Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth-Century Transformation of Mathematics”

“The report of my death was an exaggeration.”

Mark Twain

“May you live in interesting times.”

(traditional)

I recently attended a meeting of innovative science and technology startups supported by Convergent Research, the organization that oversees the Lean FRO, a nonprofit that develops the Lean theorem prover. The meeting was designed to stimulate discussion, and when I introduced myself as a mathematician, many participants were eager to talk about the impact of recent events in AI on mathematics and reactions in the mathematics community. They were surprised to hear that I find the tone of the community responses on blogs like this one and Proofs and Prompts generally positive and encouraging, even though we all recognize that fundamental aspects of our day-to-day professional lives are bound to change. These discussions have helped me shape some of the thoughts I would like to share here.

There is a narrow view of what mathematicians do, encapsulated in our daily workflows: we try to solve problems, and when the hard problems are too hard to solve, we make up easier approximations, solve them, and then vary the parameters. That practice has been disrupted by the events of the last few months, in the sense that the kinds of results that would have, a year ago, made for perfectly respectable publications can now easily be generated with the help of AI. This has left us worrying about what it will mean to do mathematics going forward, as well as how to train and support the next generation of mathematicians to do whatever that is.

The history of mathematics offers us a broader view. What has remained stable, despite centuries of changes, is that mathematics is a culture of rigorous reasoning and communication, providing us with language and abstractions that let us think and communicate more reliably and efficiently. Surely such reasoning is still important, even in the age of AI. The fact that many of us find mathematics aesthetically pleasing doesn’t diminish its practical utility, but rather is explained by it: I expect that the reason that doing mathematics feels so good is that it is the exercise of capacities that are so fundamental to our survival as a species that they are wired into our DNA. If that’s right, mathematical thought isn’t going away any time soon.

The challenge is that solving the kinds of problems we have been solving for decades becomes decoupled from the goal of enhancing our mathematical understanding when we let AI do the work. The question, therefore, isn’t whether we still need mathematics, but rather how to pursue mathematical understanding in the age of AI. I will provide three general answers.

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On classical solutions and singularity formation in incompressible fluids

4 October, 2026 in expository, math.AP | Tags: , | by | 5 comments

[This is a guest post by Diego Córdoba and Luis Martínez-Zoroa. This blog post was initially written in a different file format and converted using AI. — T.]

Euler formulated the equations for an inviscid incompressible fluid more than 250 years ago. Published in his 1757 memoir, they belong to the earliest systems of partial differential equations in mathematical physics, following d’Alembert’s work on the wave equation. Understanding the evolution they describe has required the work of many generations of mathematicians. One of the central questions is whether an initially regular fluid flow can develop a singularity in finite time. The recent work on Euler and Navier–Stokes obtained with the aid of large language models has brought renewed attention to this question. These exciting developments build on decades of mathematical research, including works from recent years, when the field has continued to be specially active. In this post we describe some contributions to that body of knowledge, including our own. We will go over some of that recent mathematical literature, and discuss the ideas of constructing blow-ups through a cascade of vortex layers.

Partial differential equations provide a common language for problems ranging from wave propagation and fluid motion to geometry and general relativity. For an evolution equation, the Cauchy problem asks us to determine the future from prescribed initial data. A local existence theorem gives a solution for a short time; the next question is whether it continues for all time or develops a singularity at a finite time. This distinction between global existence and finite-time blow-up is already visible in the elementary Riccati equation

\displaystyle y'(t) = y(t)^2, \quad y(0) = y_0 > 0, \quad y(t) = \frac{y_0}{1 - y_0 t}.

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What does the advent of powerful AI models mean for mathematicians like me?

3 October, 2026 in guest blog, math.GM, opinion | by | 18 comments

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

Recently I have read a stream of articles questioning the future of mathematics: what constitutes a proof, what we value, how we retain the human enterprise of creating knowledge. Many of these articles are written by mathematicians making significant progress on the conjectures that shape their fields, whose work is now imperiled by the capabilities of frontier AI models. I am not one of those mathematicians. There are many mathematical problems that I have revisited over two decades, but none will grab headlines. Nevertheless, I care deeply about their resolution, as do others in my field. What does the advent of powerful AI models mean for mathematicians like me? I have yet to succeed in prompting a solution to one of my long-term problems, though AI models have helped fix an incorrect lemma and suggested a helpful reorganization of a paper. They have pointed me in directions in which I had not thought to look, and spurred me to learn new mathematics. Combining AI with the skills I have honed over the years is making me a stronger mathematician.

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AIM: an invitation to explore mathematics together

2 October, 2026 in advertising, guest blog | by | 20 comments

[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.

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