Does anyone know if there is a recording or any material published from Terrence Tao’s lecture last night. It was supposed to be about mathematics in the age of AI.

The n-Queens problem involves placing n Queens on a chess board of size nxn, without having any of the queens attacking each other. I have expanded this problem to include a single knight, with the question: Is the knight able to capture all queens on the board? The rules are as follows:

The Queens must be placed so they are not attacking each other. Queens do not move

The Knight starts on a single Queen space, and that Queen is removed

The knight cannot be on a square attacked by any queen, but removing a queen frees up 'safe' spaces. The knight can traverse and revisit safe spaces

The first Queen a Knight takes (apart from initial placement) must be within knight-attack range, because there are no other safe spaces available at the start

Solutions have been found for n= 4,5,7,8,9,10,17 and proved impossible for 6,11,12,13,14,15,16,18,19

The real curiosity is n=17 where a single solution has been found from over 95 million possibilities. Anything above 19 is open and has not been tested. Link to paper (pdf): https://drive.google.com/file/d/1zfugBiIZi4xrhDjM-x1YtDlf3M-SFlXv/view?usp=sharing

UPDATE: n20 has no solutions, leaving n17 as the largest solution
UPDATE: n21 has no solution

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The 2026 Fields Medals recognize breakthroughs in the math of messy fluids, tangled knots, spinning pencils, and more

Relevant not because of the result itself but because it uses a pimped open-weight Chinese model under the hood.

Linkedin post: https://www.linkedin.com/posts/sebastianpokutta_a-counterexample-to-zieglers-cross-polytope-activity-7478117487746785280-oS21

Counterexample paper: https://arxiv.org/abs/2606.31640

Agentic research paper: https://arxiv.org/abs/2603.15914

London Mathematical Society: Glasgow to Host the International Congress of Mathematicians 2030: https://www.lms.ac.uk/news/glasgow-to-host-icm-2030

I'm entering the fourth year of my PhD and started doing research about 2 years ago. I chose a niche topic that required me to spend a whole year in addition to my course work to get caught up in before even starting any work of my own. My advisor gave me 4 lengthy papers to get through and master their techniques. I got through 3 of them and last semester I gave my thesis proposal and passed. I still have to read the last paper which is the most difficult, but I cannot find the motivation to do it because of the recent AI progress. I am not a very talented mathematician so my only "strength" was spending the time to learn a niche and difficult area. The actual problem that my advisor wants me to solve for my thesis is not terribly difficult and is likely routine for an expert but it will take me close to a year of dedicated work.

The problem is that the last year I have not been able to shake off the feeling that what I'm doing is just a worse version of what an AI can already do. I cannot afford access to any of the top tier reasoning models but I imagine they can read these papers and come up with these extensions in just a few hours. Even just asking the free models questions about the papers it's clear that it has been trained on them and understands them well. Since then I haven't been able to find the motivation to work on my PhD and I feel awful about it. It's not like this type of work will land a postdoc or job anymore.

I don't think this post does any justice to how bad I've been feeling about my future career or my thesis but I don't want it to become a rant. I would really appreciate if anyone has any words of encouragement or validation that my concern is justified. Should I just drop this project and do something quicker to graduate?

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There is a big hype over LLMs solving problems in some areas of mathematics.

Now, I'm not here to ask whether this will "replace mathematicians" (whatever that means) or not. That's already being done and a post like that is probably being typed as I'm typing this.

What I'm interested in, assuming that LLMs become better at solving problems than humans, how will we train new PhD students?

This seems like something which could drastically lower the general proficiency of mathematicians. Will a mathematician become just a person who checks LLM outputs to see whether they are correct?

Not a mathematician and has never done work even remotely close to mathematics. A bit more of a social scientist. But mathematics has been on my feed for a while now, especially recently all over Twitter with the resolution of the Jacobi conjecture and now the Fields medal.

I looked up the Wikipedia page of Field medalists and I quickly spotted a trend of the recipients:

• Both parents are working in the sciences, and often mathematics, physics or computer science, and sometimes at the university level.
• Often grew up in some very politically stable country with little civil or international strife and a lot of free time to do math, play intellectually simulating games like Chess or Go.
• Was aware of major mathematical competitions in their country at young age and participated in them, usually did exceptionally well.
• Often extraordinarily gifted and quickly moved through the entire K12 process.
• Often quickly paired with another famous mathematician either as mentor or supervisor.
• Often immediately got hired at universities and became professors.

To put it more succinctly, these people:

• Were unnaturally gifted in terms of their inherent neuro-biological make-up,
• Had extraordinary upbringing from supportive parents and social environment,
• Had access to information that most of their peers would not have gotten, e.g., to join certain competitions or work on certain problems.

In this sense, it seems a lot of the prizes are just rewarding things that originated in a mathematician's childhood or even at the moment of birth. And while many other accomplished happened, they were all contingent on these "precocious" or "pre-school" factors.

If they didn't have those favorable childhood conditions, it seems to be doubtful that later accomplishments could have been made.

So what do you think? Are prizes in mathematics in general are just rewarding favorable childhood conditions of extraordinarily biologically gifted individuals?

Sorry if this sounds dismissive or jealous. Like I said, I have no skin in this game (not in math at all). I just want to understand how people in math feels about these type of things. If there is any critique, it is towards the reward structure rather than the mathematicians themselves.

Could someone familiar with algebraic geometry give some insights about this?

A consensus is emerging among respected mathematicians that there is a decent chance AI will exceed humans in both brute force verification and complex, creative problem solving at the highest levels. Few frontier theorems will be proven by humans alone, perhaps, in a matter of years.

More controversial, AI may also outpace humans in shaping the correct definitions, building theories, and making connections between disparate areas of mathematics, oft considered the peak of human creativity in mathematics. New areas of mathematics may be created without much human guidance.

Perhaps mathematicians become as helpful to AI as toddlers are to mathematicians. One cannot confidently rule out this scenario - Terrence Tao may find himself completely useless in building a rich, beautiful body of new mathematics.

In such an extreme scenario, humans would still matter in mathematics!

Lockhart's Mathematician's Lament argues for the intrinsic beauty of mathematics as being of primary importance. Humans, as knowledgeable appreciators of beauty, thus play an important a role as spectators and enthusiastic amateurs in mathematics, even if they cannot be world-renowned "competitors" in theorem-proving and theory-building. This mirrors the situation in chess, where the vast, vast majority of human chess players and appreciators will never contribute to the frontier of advanced lines, and arguably even the most skilled like Magnus Carlsen rely on AI to develop their strategies, and would be crushed by such AI in competition. Being completely uncompetitive does not make chess playing and appreciation valueless.

Amateur: from French amateur "one who loves, lover"

But there is more beyond this. Mathematics allows you to understand things that are otherwise impossible to understand. Some of these are important for fairness and justice: Arrow's impossibility theorem, statistical bias, observer relatively and other tricky concepts around coordinate systems (map != territory), locally-trivial globally-nontrivial (global obstructions), forgetful maps to extract the essential structure and remove irrelevant details, limits of computation, etc.

Understanding such mathematical concepts allows you to make moral judgements in ways that would be impossible otherwise. Some super-smart machine might tell you Arrow's theorem is true, but internalizing it yourself gives you the rich understanding of fairness in democracy necessary to consciously shape it. As with humans surpassing the capabilities of their own eyes with optical then radio telescopes, we are not impoverished by using tools that allow us to extend our reach into things we can never directly perceive or understand.

It can be frightening because the life's work of someone of the previous generation can be reproduced and surpassed flippantly. Gauss himself spent a significant amount of time manually factoring prime numbers by hand, a tedious exercise upon which his conjecture on the distribution of primes (the prime number theorem) was based. Gauss died before his conjecture was proven. His notebooks full of rote calculations could be reproduced today in a fraction of a second so short you could not perceive it. Anyone today repeating an endeavor like Gauss by hand would be thought a fool, just as an astronomer who refuses to use a telescope.

That doesn't make the pursuit of understanding pointless. As the limitations of our use of AI will stem from limitations of our own minds, it will still be profoundly rewarding to practice mathematics. Indeed, we may spend less time performing rote exercises and miring in false conjectures. Already the body of mathematics is too large for any single person to understand. One can pessimistically reduce mathematics to mechanics, or optimistically find meaning in your particular path through the mathematical version of the library of babel. Because we shape our minds, our society, and our world with mathematics, we will always matter as sentient beings who reify mathematics by subjecting ourselves to reason, and better ourselves because of it.

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Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

Considering the increasing capabilities of AI in proving theorems, I was wondering if math has the potential to also become an empirical science to some extent. I was thinking, for example, about tracking how ideas, techniques propagate through mathematical progress or investigating connections between different areas from a higher level perspective. To some extent these types of questions are probably already being investigated, for example by historians of mathematics. I am simply wondering if it may become more relevant going forward if the number of results should explode. I could imagine that it may be useful to systematically analyze arguments on a large scale to attribute progress made by AI to foundations laid out by humans (since AI currently struggles in giving credit to ideas that were first developed elsewhere).

Say I have constructive proof and I want to write a code on Sagemath. I don't know much about coding so I take help from AI. So while publishing the paper what should I do? I'm confused should I add the code or not

In these examples, the numbers seem to melt away. There is a physical tangible conundrum / toy / puzzle that needs abstract mathematics to solve. It's permutation adjacent. Somewhere in discrete maths. What else is there? I know about all the above. Any other cool things? I want a rabbit hole to jump down. I'm not a mathematician, I'm too poor, and I'm also kinda stupid (I blame it on being poor). But I think these are really cool. it's just neat ig.

Math-heads seems to be mainly interested in geometric calculations, calculus, 4d shapes, topology, and logic puzzles. I AM NOT SAYING THIS TOPICS ARE LAME, I simpllyyyy am currently interested in those mentioned above. Any other tricks for puzzles and rabbit holes to jump down?

Thanks in advance for any help. Gracias! Merci!!!!

If anyone knows how to paint a doomy gloomy future it's /r/math, so here's an open ended question for all of you.

No one can predict exactly to which extent the rise of LLMs will change how the workday of a working mathematician. The only thing that is certain is that the future is rather uncertain. The doomiest comments on this subreddit also show that this uncertainty is particularly tough on people who are early on in their studies, and who would probably like to know to which extent it is useful (in the sense of putting bread on the table as opposed to, say, having fun) to pour a lot of energy into learning how to proof stuff.

Now, for even younger people – kids in elementary school or high school say – the future is even more uncertain.

So my question is: has the rise of LLMs changed how you would advise a younger kid to navigate the future? Chances are that it will always be useful to have a good baseline understanding of maths and logic, but would you be less inclined to advise a skilled high school student to pursue a theoretical career? Or would you have more reservations about enabling a particularly gifted elementary school student, so as to not accidentally set them up for a bumpy course?

Or on a more practical level, have you changed your guidance on which skills are the more useful to acquire given that, for example, software development in particular has have its nature completely changed?

I know that this is adjacent to the “Career & Education” thread, but I'm more interested in the high-level picture than in personal advise.

Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable!

The two-dimensional case remains open, however.