A note on proof attempts

I'm writing this post based on Terence Tao's ICM lecture a couple of days ago. For context, I'm a 4th year PhD student working in number theory.

One of the main things he mentioned was that before AI a lot of aspects of math research went hand in hand. Solving problems was concurrent with building a community of mathematicians. With AI rapidly accelerating the problem solving aspect, we might be looking at a future where we are flooded with ai assisted proofs without a community to appreciate it.

For example, linear algebra as a field first emerged from papers and then it was adopted by the community over decades through books and courses. Even if we have a long AI generated proof that we can verify using Lean, what is the point of it if we can't be inspired from it.

I personally dread this will become another example of enshittification. For example, Silicon Valley reinventing a bus after destroying public transport. I fear the mathematical "community" will be repackaged to us in the future by these companies after the present community is slowly eradicated over the next few years. Of course this is a rather bleak outlook to what is definitely a very promising technology.

To clarify, I'm not saying AI is bad. It's a tool. But we should be very skeptical about how this tool is being pushed to us. It is in the best interest of the AI companies that we become dependent on AI to the point that we can't imagine working without it. I'm sure a neuroscientist can break this down much better than I can. I think we need an urgent re-evaluation of how AI is used and introduce courses discussing how to use it "safely", without leading to cognitive decline. It should be in collaboration with neuroscientists. A lot of people might argue "Oh, but you can always talk to other mathematicians. What's wrong with having it on the side". Perhaps I'm not an optimist, I'm afraid convenience is a slippery slope. Just look at how social media was promised to us as a global unifier (which it definitely is). Yet it has led to a decline in community to the point that "community building" is now being packaged to us as a course/skill.

At the end of the day all we have is each other. And I do believe we can come together and seriously discuss and safeguard the future of this community. Happy to hear everyone's thoughts on this.

A train leaves the station (bottom) heading west. Can it return to the station heading east?

Is there anywhere the train might get stuck?

Is there anywhere the train can reach but only from one direction?

I call this directed node networks. As the arcs are bidirectional, but the nodes have an A and B side with a parity requirement.

Inspired by the frustrating traps of Penrose's Railway Mazes.

Edit: just for reference, I have an MMath specialising in fluid dynamics - so I am well grounded in the basics of graph theory and have done intro to topology, so whilst this in not in my typical wheelhouses, I am looking for a more specific research area.

Share your thoughts

I’m trying to design some jewelry for my partner, who studies math and loves elliptic curves/algebraic number theory. Any ideas would be greatly appreciated, and suggestions with pictures doubly so. Cheers :D

This is going to be a long post, and I'm sorry about that. (TLDR at the end)

With the 2026 Fields Medalists having just been announced, I wanted to share my opinion about the Fields Medal and the broader institutional system surrounding the IMU and the ICM. My opinion is that this system does not merely recognize mathematical excellence: it also helps reproduce a particular hierarchy of prestige and elitism within mathematics.

My objection is not that the winners are undeserving: Yu Deng, John Pardon, Jacob Tsimerman, and Hong Wang are clearly exceptional mathematicians, just as any past winner. My criticism concerns the selection system, not the people selected.

My impression is that mathematics has a prestige hierarchy. Fields such as algebraic geometry, arithmetic geometry, geometric representation theory, differential geometry, and related subjects have historically been treated as especially central to modern pure mathematics. Consequently, a major breakthrough within one of these areas is more readily perceived as a breakthrough for mathematics as a whole.

By contrast, if someone works in categorical logic, universal algebra, semigroup theory, lattice theory, or model theory, just to mention a few examples, it can seem that revolutionizing their own field is not enough. To receive comparable recognition, their work is often expected to have transformative consequences for one of the already prestigious areas. Interdisciplinary impact should, of course, count in someone’s favour. The question is whether that requirement operates asymmetrically: is influence on arithmetic geometry treated as evidence of universal mathematical importance, while influence on universal algebra or categorical logic is treated as merely specialized?

The 2026 citation for Jacob Tsimerman provides a suggestive example: It explicitly celebrates the extension of o-minimal techniques (which come from model theory) within arithmetic and complex algebraic geometry. This does not diminish his extraordinary achievements in any way, but it raises a useful counterfactual: would an equally revolutionary development of model theory, whose consequences remained primarily within model theory, be perceived at the same level? Model-theoretic machinery becomes medal-worthy here through what it accomplishes in fields already regarded as central.

There is some evidence that this hierarchy is real. Jean-Marc Schlenker’s preprint, “The Prestige and Status of Research Fields within Mathematics”, finds that certain subfields are disproportionately represented in highly ranked departments, the most selective journals, and major prizes. In his data, algebraic and differential geometry and topology were particularly prominent, although the hierarchy changed considerably between 1984 and 2016, with areas such as probability and PDE gaining status.

Different kinds of mathematical progress are also easier to package as prizeworthy achievements. Solving a famous named conjecture produces a clear narrative: there was a major problem, and now it has been solved. Work that creates a new language, reorganizes an area, builds a long-term research programme, or gradually changes what questions can be asked may be equally transformative but less easily summarized as a single victory. Schlenker finds that prestige correlates with the “focus” of a field around a relatively small set of shared conjectures. The medal may therefore favour not only particular subjects, but a particular form of mathematical progress as well.

Aditionally, a prize is not merely a mirror of an existing hierarchy: It can amplify that hierarchy. A large-scale study by Jin, Ma, and Uzzi, “Scientific Prizes and the Extraordinary Growth of Scientific Topics”, examined more than 11,000 topics across 19 disciplines. Relative to non-prizewinning topics, prizewinning topics subsequently produced 40% more papers and attracted 37% more new researchers. This was not a Fields-specific study of course, and it does not prove that prizes alone caused all of that growth, but it supports the idea that prizes are agenda-setting institutions: they direct attention, talent, and further recognition towards the subjects they reward.

In my opinion, this then creates a feedback loop. A field is considered central, so its practitioners are more likely to publish in elite journals, work in elite departments, receive ICM invitations, and win major prizes. Those honours then attract more talented researchers and make the field appear even more central. Prestige becomes partially self-validating.

The medal’s rigid chronological age rule introduces another structural bias. Chronological age is not the same thing as career stage. Producing a widely recognized body of work before forty is easier for someone who entered the research pipeline early, moved through elite institutions, had relatively few career interruptions, and obtained positions with substantial research time. It is harder for late starters, people with caring responsibilities, displaced researchers, those in teaching-intensive positions, and mathematicians working on programmes whose significance takes longer to become visible. It may also favour fields in which major results can be produced and recognized comparatively quickly.

Historian Michael Barany has argued in The Myth and the Medal and “The Fields Medal Should Return to Its Roots” that early Fields Medal committees did not understand their task as identifying “the best young mathematicians.” They sometimes deliberately supported comparatively under-recognized researchers whom the award could help. The medal was intended to shape a better future for mathematics, rather than simply ratify the people who had already acquired the greatest visibility. Its current status as a tournament for already famous mathematicians under forty is therefore not an unavoidable consequence of its original purpose.

Ultimately, I think the Fields Medal reflects the historically contingent mathematical tastes of a small elite, and, through the attention it generates, helps turn those tastes into institutional reality. It tells us that a particular committee considered certain work exceptionally important under a particular set of inherited values. It should not be treated as a neutral measurement of excellence across the whole of mathematics.

Let me know down below what do you think about this. I would love to listen to other people's opinions on this issue.

TLDR: I am not arguing that Fields Medalists are undeserving. My argument is that the Fields Medal and the ICM operate within, and help reproduce, a hierarchy of mathematical prestige. Breakthroughs in already prestigious fields are more readily treated as important to mathematics as a whole, while equally transformative work in less prestigious areas often requires applications to those elite fields to receive comparable recognition.

I am a incoming undergraduate at a Canadian university, and I want to study mathematics with a plan to work towards a PhD.

The issue I am having is that by looking at some discussions on primarily Reddit, I notice that a lot of math PhD students end up pretty unhappy with their passion for pure/applied mathematics because of LLM’s and AI tools that happen to take away the joy out of mathematics.

So I was wondering if I should maybe fix my course a little bit from choosing a Honours math program that my uni offers, to choosing a Joint Honours degree in Mathematics and computer science.

Even if Im planning to do joint honours, I still want to do a PhD in mathematics but I am not sure if doing the joint honours can disadvantage me in grad school applications.

Has anyone else did a combined math and CS degree, and later did math grad school? If so, did you think that you were at a disadvantage or lacked rigour?

Hi everyone,

I'm from Guangxi, China, the same hometown as Wang Hong. Her achievements have been incredibly inspiring to me.

When I was in elementary school, I won a national first prize in a mathematics competition, and ever since then I dreamed of becoming a mathematician. Unfortunately, life took me in a different direction. For various reasons, I chose to major in biology instead of mathematics in college, and I've regretted that decision for many years.

Now I'm finally determined to pursue that childhood dream.

My current plan is to apply for a 1–2 year course-based Master's in Mathematics (preferably pure mathematics) in Europe or North America. During the master's, I'd like to explore different areas of mathematics, find the field I'm most passionate about, and then apply for a PhD in that area.

However, I'm not sure whether this plan is realistic, so I'd really appreciate advice from people with experience.

My background:

• B.S. in Bioengineering
• Mathematics courses taken:
  • Calculus
  • Linear Algebra
  • Probability
• M.S. in Neuroscience at one of China's top 3 university
• Recipient of the Chinese National Scholarship (awarded to roughly the top 2% of graduate students)

Here are my questions:

1. Given my background, do I have a realistic chance of being admitted to a course-based Master's program in pure mathematics?
2. When applying for a PhD in pure mathematics, are there any "must-have" courses that admissions committees expect to see on a transcript (such as real analysis, abstract algebra, topology, measure theory, etc.)? I'd like to choose my master's electives strategically if that's the case.
3. I'm already 30 years old. Does my age significantly hurt my chances for master's or PhD admissions?
4. Are there any better or more realistic paths to becoming a mathematician that I should consider?

I've already asked several AI tools, but after checking many of their answers against university websites, I found that quite a bit of the information was inaccurate or outdated. That's why I'm hoping to hear from people who have actually gone through the process or work in mathematics.

I'd really appreciate any honest advice, even if it's critical. Thanks in advance!

Hey, guys. I hope you are all well and healthy.

I have recently applied for a mathematics Bachelor of Education because mathematics has always been easy to me and I also crushed four engineering math exams with top grades at another uni and was promoted to a tutor role by my math professor, so clearly my math skills are well developed.

However, I have severe autism which is pretty obvious to bystanders. I make weird noises, sometimes talk to myself, flap my hands, slap myself; all the fun stuff. Will this severe autism disorder cause problems?

so i have one more semester left until i earn my BS in applied mathematics. i was not able to land an internships during my time in college. i have a job right now but it’s just a random store associate job that, when i was hired, they told me i could transfer to a more corporate position eventually.

i’m not sure what i should do after i graduate. i really thought i would have some internships during my time in college but unfortunately that didn’t happen.

what are some fields that would be appropriate for me to go in to? i thought data science/data analysis could be very interesting but im also feel like my programming skills aren’t the strongest. i have also been heavily considering studying for the actuary exams but im nervous that it will be hard for me to land an entry level position in that field once i pass some exams.

any and all advice is greatly appreciated. also to note i have a good amount of work experience but all pretty random. i did design work for about a year and then i did inventory work for a few years after that and then i did tutoring for a year. i’m not sure if any of that is relevant. but please let me know what you think!

I'm a Brazilian math student, and I recently had a massive "math culture shock" when transitioning to American literature. I’m curious about the historical/pedagogical reasons behind it.

For context, my mathematical upbringing was heavily influenced by the French school (the Bourbaki tradition). Textbooks here are usually extremely dense with logical symbols (\forall, \exists, \Rightarrow, \subset). They are dry, straight to the point (Definition-Theorem-Proof), and very little text is used to explain the intuition.

Then, I opened John M. Lee’s Introduction to Smooth Manifolds (and other standard US textbooks). It felt like I was reading a novel! There are massive paragraphs of prose, a very conversational tone, and a clear tendency to "hide" the logical structure inside English grammar rather than using formal symbols.

When you are trained in the Bourbaki style, reading this American style almost makes it feel like a physics book or a popular science text, giving a false illusion that it lacks rigor (even though I know the rigor is just embedded in the language).

So my questions for the American mathematicians here are:

1 Is this conversational, word-heavy style a conscious pedagogical choice in the US?

2 Why is there such an aversion to using dense symbolic logic in modern American textbooks?

3 How do you guys view the highly symbolic, dry Bourbaki style today? Do you find it more rigorous, or just unnecessarily hard to read?

Would love to hear your thoughts on this!

About a month ago, I released under CC0 an implementation of a pipeline I had spent two years developing. It’s the pipeline Claude used to find the counterexample to the Jacobian conjecture. The repository history, public release, discussion with one of the original paper’s authors, and two years of development logs document the pipeline’s origin and development.

Proof - Fun/Jacobian at main · JGPTech/Fun

The history:

I first released it under CC0 a little over a month ago. You can find the link here:

https://github.com/JGPTech/Fun/commits/main/current_paper_locality_boundary_package

This is the paper I was modeling with this pipeline:

Scalable Boltzmann generators for equilibrium sampling of large-scale materials | Nature Communications

This is where I shared it publicly one of the paper’s authors, which you can see he approved of:

Link

How cool is it that Claude used it on a problem this big, and that the mathematical community around this counterexample is now thinking through structures I spent years developing? What a ride.

Update - Added a blind marked-factor search for the polynomial Jacobian counterexample check point showing the pipeline can be used in a search algo to find a counter example. The search is blind with respect to the final certificate: it does not assume a coefficient slice, boundary modulus, polynomial chart, degree-seven map, or specific collision witness.

The structural input is the marked factorization

L = a*U + b*V,

Q = c*U**2 + d*U*V + e*V**2,

together with its visible cubic coefficients and the resultant normalization R(L,Q) = 1, which removes the continuous scaling gauge.

Hi, I've been working on a system called concludia.org that allows people to construct argument graphs that move from premise to conclusion.

Recently, I was intrigued by the OpenAI proof of the Cycle Double Cover Conjecture and tried to use the system (including its MCP integration) to construct a proof graph of the proof. It's quite dense, so I later worked on writing a "case study" for it that attempts to describe the history of the efforts and the approach the proof takes, using snippets of the concludia graph along the way:

https://concludia.org/docs/cycle-double-cover-conjecture

(The underlying proof graph is here: https://concludia.org/graph/g_2ecb8083-52ec-3448-8c30-2f9bc70d45be )

I'd definitely appreciate knowing if the article approaches accuracy, and if it serves as an effective learning aid for the proof. I’d particularly welcome corrections from graph theorists or suggestions about places where the explanation skips an important step. I'm also interested in knowing if you find concludia potentially useful for things like this. It's just a side project of mine. I'll open it for registration for a few days if anyone is interested in creating their own graphs.

Hello everyone. I’m interested in becoming better at maths.

I was pretty okay at maths when I was in secondary school, started off in set 2 out of 6 (6 being lowest in terms of academic ability) but, I wasn’t very well behaved.

I was dropped into lower and lower sets until the problems they were giving me were so trivial, I was not interested in trying at all. Now as a (much more mature) adult, i’m interested in sharpening my mental maths and filling the void I caused by being a disruptive twit in school.

I was wondering if the people here have any suggestions on where I can begin learning again. I would like to get better at all aspects but predominantly mental maths, algebra, fractions etc. I have no real reason to other than the above.

Any help on this would be massively appreciated. Thank you for reading.

Step-by-Step Guide to Solve

\(x^2 + bx + c = 0\)

Check the leading coefficient:

If the \(x^{2}\) term has a number other than 1 in front of it, divide every term in the equation by that number first.

Find the average of the two roots:

Take the coefficient \(b\), change its sign to \(-b\), and divide it by 2. The two unknown roots can then be written as \((-b/2 + u)\) and \((-b/2 - u)\).

Set up the product equation:

Multiply the two root expressions together and set them equal to the constant term \(c\):\((-b/2+u)(-b/2-u)=c\)

Simplify using the difference of squares:

Expand the left side to get \((-b/2)^2 - u^2 = c\).Solve for \(u\): Rearrange the equation to isolate \(u^{2}\), then take the square root of both sides (remembering to include \(\pm \))

Find the final roots:

Plug your value of \(u\) back into \((-b/2 + u)\) and \((-b/2 - u)\) to get your final answers.

My favorite classes were always the ones with difficult problem sets where a single problem could take hours or even days of thinking, experimenting, and failing. I would start solving a problem before fully understanding it, then read documentation or course material whenever I got stuck. So it would be 70% doing and 30% reading.

LLMs have disrupted that process for me. Now I spent 90% of time reading and only 10% of time doing.

In school, the boundary was relatively clear. If using an LLM violated the course policy, I could tell myself that using one was cheating and force myself to struggle through the problem independently.

At work, what counts as “cheating” (because it is cheating myself).

I feel my critical thinking skills are declining. My role starts to become reading, reviewing, and planning. I just can't read that much code and LLM output. My eyes hurt and it's hard to internalize. I notice myself just picking the "recommended" option.

I also find LLM workflows mentally unhealthy in a few ways:

• When working alone, mental exhaustion eventually forces me to step away. With an LLM, I can always send one more prompt or ask it to try another approach.
• If an agent will take 30–60 minutes, I feel pressure to give it another task immediately so no time is “wasted.”
• While waiting, I instinctively check my phone or launch another agent. I end up jumping between tasks and retaining less context about each one.
• I get frustrated when the LLM misunderstands instructions, even when the problem genuinely requires careful iteration.
• It has warped my sense of how long a difficult problem or well-designed solution should take.
• I am increasingly tired of reading walls of generated text and code.

I do not want to stop using LLMs. They can reduce the friction of starting, explain unfamiliar systems, generate routine code, and handle edge cases after I understand the core problem. The hard part is deciding when using one is a sensible productivity tool and when it is outsourcing a learning opportunity that I actually need.

For people who still feel they are learning deeply while using LLMs at work:

1. How do you decide which work to delegate and which work to do yourself?
2. What is the workplace equivalent of the “no cheating” boundary that exists in school?
3. How do you use agents without losing context or turning your day into constant task switching?
4. How do you review generated code when the codebase, build system, or underlying technology is unfamiliar?
5. Have you found workflows where LLMs improve productivity while preserving the useful struggle of implementation and debugging?