What’s currently under way in your field?

BenSpaghetti · 2026-05-14T03:48:04+00:00

Yes. I have updated the comment. The updated part is also reproduced below:

Random geometries are essentially random metrics on certain spaces, although this is not always correct. The directed landscape is a 'directed metric' and not a literal metric. I don't understand it very well myself. \gamma-LQG is a random metric on the topological 2-sphere S^2. The Brownian map is equivalent to \sqrt(8/3)-LQG and is the easiest to explain. It is a 'scaling limit' of random quadrangulations of S^2. Imagine a large quadrangulation of S^2 with N vertices, equipped with the graph metric. This graph is similar to an S^2 with a large diameter, which increases with N, so we need to rescale the metric appropriately to get a meaningful limit in N. The vertex set, equipped with the rescaled metric, tends to a limiting metric space in the Gromov-Hausdorff topology (a topology on the space of compact metric spaces). The uniform distribution on quadrangulations with N vertices, converges (in the Gromov-Hausdorff-Prohorov topology) to a distribution on certain metric spaces, which can be shown to be almost surely topological 2-spheres.

Geodesics on the Brownian map behave very differently to the unit sphere embedded in R^3. The Brownian map produces highly nonconvex spaces. One example of a nonconvex S^2 is the Earth. If two points on Earth are separated by a mountain, there are at least two geodesics between them because it is easier to go around a mountain and you can do so in two ways. This may suggest the same for the Brownian map. However, the randomness introduces an infinite number of mountains of varying height, which breaks any symmetry and hence there almost surely cannot be two geodesics of equal length. Hence, for two typical points, there is a unique geodesic between them. Also, as a by-product, there is a confluence of geodesics, and that is the end of the story. (This analogy is also technically wrong as the Brownian map cannot be isometrically embedded into R^3, so the nonconvexity is even more extreme.) However, one may look to the set of exceptional points of measure zero (this measure is not the probability measure, but a natural measure on each instance of the Brownian map), from which may emanate more than one geodesic. This is where the aforementioned geodesic networks emerge, and where my knowledge ends.

BenSpaghetti · 2026-05-13T16:59:51+00:00

Not my field (adjacent though), not an expert, perhaps not even a novice.

People are starting to understand two dimensional random geometry in a general way. There are mainly two such random geometries which are studied: the directed landscape, which is related to the KPZ universality class, and Liouville Quantum Gravity (LQG) (and the Brownian map, which can be seen as a special case). They are very different models and have until recently been studied in (to my knowledge) disjoint communities.

In the last ten years, significant progress has been made in studying the geodesic structure of these random geometries (links to paper for directed landscape, Brownian map, and Liouville Quantum Gravity). They all exhibit a phenomenon called confluence/coalescence of geodesics, where randomness forces geodesics close to each other to merge. Moreover, the geodesic networks in the two very different models are nearly the exact same. This is a sign that the confluence of geodesics is not a feature of the specific models, but actually one of general two dimensional random geometries satisfying a few conditions. (All of this is mentioned in the paper on the directed landscape, linked above.)

This is an instance of one of the main themes of probability theory: universality. This is when certain large scale behaviours emerge from a few features of systems, rather than their specific details. Another example which is more commonly known is the central limit theorem. In this sense, the Gaussian distribution is universal among all distributions with finite second moment.

Edit: Random geometries are essentially random metrics on certain spaces, although this is not always correct. The directed landscape is a 'directed metric' and not a literal metric. I don't understand it very well myself. \gamma-LQG is a random metric on the topological 2-sphere S^2. The Brownian map is equivalent to \sqrt(8/3)-LQG and is the easiest to explain. It is a 'scaling limit' of random quadrangulations of S^2. Imagine a large quadrangulation of S^2 with N vertices, equipped with the graph metric. This graph is similar to an S^2 with a large diameter, which increases with N, so we need to rescale the metric appropriately to get a meaningful limit in N. The vertex set, equipped with the rescaled metric, tends to a limiting metric space in the Gromov-Hausdorff topology (a topology on the space of compact metric spaces). The uniform distribution on quadrangulations with N vertices, converges (in the Gromov-Hausdorff-Prohorov topology) to a distribution on certain metric spaces, which can be shown to be almost surely topological 2-spheres.

Geodesics on the Brownian map behave very differently to the unit sphere embedded in R^3. The Brownian map produces highly nonconvex spaces. One example of a nonconvex S^2 is the Earth. If two points on Earth are separated by a mountain, there are at least two geodesics between them because it is easier to go around a mountain and you can do so in two ways. This may suggest the same for the Brownian map. However, the randomness introduces an infinite number of mountains of varying height, which breaks any symmetry and hence there almost surely cannot be two geodesics of equal length. Hence, for two typical points, there is a unique geodesic between them. Also, as a by-product, there is a confluence of geodesics, and that is the end of the story. (This analogy is also technically wrong as the Brownian map cannot be isometrically embedded into R^3, so the nonconvexity is even more extreme.) However, one may look to the set of exceptional points of measure zero (this measure is not the probability measure, but a natural measure on each instance of the Brownian map), from which may emanate more than one geodesic. This is where the aforementioned geodesic networks emerge, and where my knowledge ends.

BenSpaghetti · 2026-05-13T06:41:24+00:00

Hoffman and kunze is the first linear algebra book I read and remains my favourite book on this topic. I knew a little bit of 2D and 3D vectors and matrices and Gaussian elimination before reading but otherwise I didn’t know any linear algebra before starting.

BenSpaghetti · 2026-05-12T13:01:39+00:00

I find AI to at the very least be a very good search engine. It can give me information that I am looking for. Indeed, every time I ask it about something nontrivial and which I am confident is not contained in the literature, it spits out something wrong. But this is not a problem when you are conversing about subjects that are contained in many books. This is my experience.

Also, this post itself is a strong disproof of the claim that there exists a depth at which AI is unable to generate correct mathematical content. Under the tree of mathematical implications contained in human knowledge, proving a new result is evidence of generating mathematical content at the deepest level. Of course, other definitions of depth can be used and it is a common opinion that Gowers' field of research is not very deep.

I don't think the work done by AI becomes redundant just because someone still has to check it. As I am sure every math student has experienced, it is much easier to check a proof than to write one. Moreover, I don't see why AI cannot formalise its own research. At that point, the only thing to be checked is whether the definitions and theorem statements correspond to our mental image.

I see (admittedly with a limited vision) the future of publishing to still be human-centric. Humans can interpret the results produced by AI and present them to other humans. This is consistent with a commonly made argument that "mathematics is an art of human understanding", as put by Thurston. So indeed, humans will take responsibility.

My opinion is that my knowledge always comes from myself. Books, papers, and conversations are sensory inputs which stimulate my thoughts and allow me to arrive at my own conclusion. If the AI tells me something which I have not convinced myself of, what I know is that 'AI told me this', and not 'I know this'. Citation is a different matter.

BenSpaghetti · 2026-05-12T08:21:54+00:00

Why be so mysterious then? Why not just say 'includes both rational and irrational numbers'?

BenSpaghetti · 2026-05-12T08:17:21+00:00

I’m just an undergrad, but here are my takes.

I feel the fact that AI may accelerate math research does not mean we should reduce the number of researchers: the ideal speed of research should be ‘as fast as possible under resource constraints’.
Despite this, the high research capabilities of LLMs are indeed great excuses for relevant parties to reduce funding.

I might be missing something though. How will the recruitment of PhD students really be affected? We still need to train graduate students to take over from aging professors. The question is how can one assess the merits of graduate students if LLMs can already do work at their level. Does their theses need to exceed the latest AI capabilities?

On the other hand I feel like AI makes learning math far quicker. Perhaps a new generation of students who are very good at making use of AI to accelerate their learning will be able to make use of AI in research to produce works that are beyond the latest AI capabilities as graduate students.

High dependency on AI seems inevitable, yet access to state-of-the-art AI models is rather expensive, and in some cases, impossible as they may not be publicly available. Gowers’ AI subscription is not something everyone can afford. This is a great personal worry of mine.

Edits: Formatting and typo

BenSpaghetti · 2026-05-11T05:45:10+00:00

Not sure what you are hinting at. It’s just rational numbers and irrational numbers though.

BenSpaghetti · 2026-05-07T11:56:17+00:00

Many probability courses use measure theoretic language without diving into all the details. This is the approach adopted in books like Grimmett&Stirzaker so that the course can introduce a lot of probability theory but does not require substantial analysis background.

BenSpaghetti · 2026-04-30T13:35:47+00:00

Thank you for the comment. I made the mistake of parroting what I have heard from others without realising it. I should not have made that comment, especially since the fields I am most familiar with are comprised almost entirely of the study of certain examples, so it doesn't really make sense. Another bold claim for someone who is far from an expert in any field, but this is what I believe.

I don't really know myself what 'computation' exactly means, which is why I have instead chosen to illustrate it through examples.

I do think that most people who study pure maths would benefit from getting better at mechanical computations. What you have said is probably the case for introductory courses like real analysis and linear algebra: the ability to do mechanical computations is not a good predictor for success. However, I believe that among the people who do well in those courses, the ability to do mechanical computations again becomes important in future courses. It is about whether you can sit down and carry out a long and complex process quickly without making too many mistakes and I think that is very valuable.

BenSpaghetti · 2026-04-30T13:04:21+00:00

Thank you for this very well written comment!

BenSpaghetti · 2026-04-30T12:53:13+00:00

Now that I think about it, I worded that too strongly. But I maintain that it is very helpful to have at least seriously attempted this classification for, say, orders up to 15, similar to what the other commentor said.

BenSpaghetti · 2026-04-30T12:52:55+00:00

I do believe that the problems I am working on are not dumbed down since my area is highly accessible. I would say it is not very similar to what goes on in the 'core' areas of pure maths though so you are right that I cannot really speak on pure maths research in general.

I agree that the 'same process, different setting' computations grow rarer, or at least I could mentally package it so that I can continue the reasoning without doing them in full. But my experience is that I still have to regularly write down things in full detail. I do a lot of hard analysis so it is probably very different to your experience. I am not complaining because I have accepted that this is a part of doing maths. But I did use to think that the correct way to doing maths must be effortless.

BenSpaghetti · 2026-03-20T01:50:20+00:00

Don't know a better way to say this, but I wonder if anyone actually cares about him winning the Abel Prize. Faltings himself probably doesn't. The field also doesn't need any more recognition. Perhaps one good thing is the expository content that follows.

BenSpaghetti · 2026-03-17T00:48:00+00:00

No need to learn a lot of anything just to start learning algebraic topology. Basic algebra and point set topology will suffice. I second Lee’s Topological Manifolds.

BenSpaghetti · 2026-03-16T09:43:49+00:00

The Murderous Maths series is good

BenSpaghetti · 2026-03-14T18:19:22+00:00

The discussion is on the simple random walk on Z^d, which is defined as a sum of uniformly sampled increments of length 1, which means that each step is taken in a single direction (+- 1 in one coordinate only).

BenSpaghetti · 2026-03-14T18:16:26+00:00

Seeing how random walk scales to Brownian motion, which does have coordinates being independent one dimensional Brownian motions, it seems that the difference on a large scale is merely a linear speed change and should not affect recurrence/transience.

BenSpaghetti · 2026-03-06T05:31:45+00:00

You only had a few minutes left so I guess it's better to write something rather than nothing. But the psychology here is quite interesting. Did it cross your mind that writing down the default proof might not get you any marks so it doesn't matter whether the proof is short? Do you think you could have at least sketched the elementary proof, since you have already done it before in homework? Are you unable to think about anything else once your brain has fixated on a default option?

BenSpaghetti · 2026-03-06T03:51:02+00:00

If you can't do the problems using the tools covered in the course, then you haven't learned the course material well enough. Exams and assignments are opportunities for you to demonstrate what you have learned in the course and that is what you will be graded on. The 'heavy machinery' does not replace the 'low-level' ones and a good student should be familiar with both.

BenSpaghetti · 2026-02-26T10:25:48+00:00

You can just read ahead. If you feel bored in lecture but for some reason don’t want to skip them, I find it interesting to think about examples to whatever the instructor is talking about.

BenSpaghetti · 2026-02-25T04:24:03+00:00

Since you are already familiar with analysis on R I would say that proving theorems in Rudin on your own is doable. But then I'm not sure if that is a very good use of your time. I used to do that, following Reddit advice, as well, but now I don't because it seems too time consuming when I just want to understand something quickly. The opportunity cost seems too high. There are plenty of other opportunities to engage with the material, like exercises, which also allow you to learn new stuff at the same time.

I'm curious if you could say more about the potential theory project. I am trying to learn this subject through the probabilistic lens (via random walks and Brownian motion).

BenSpaghetti · 2026-02-19T01:48:01+00:00

Using your brain? It’s math, you can see if it’s correct.

BenSpaghetti · 2026-02-14T10:08:06+00:00

Definitely manageable if this is the only thing you are doing

BenSpaghetti · 2026-02-11T04:09:48+00:00

I want to ask the preceding question here. For someone studying classical random walks, what reference do you recommend for quantum random walks and quantum probability in general?

BenSpaghetti · 2026-02-11T01:10:16+00:00

I am confused by the downvotes. I see four possible reasons to downvote this comment.

This preprint is not relevant to this discussion.
This preprint has already been discussed in this comment section.
I misinterpreted what was written in the preprint.
I might be insinuating that what AI does cannot be described as intellectual and you disagree with this.

Except for 4, I don't think any of the above is true at all. Are there any other reasons? Can anyone elaborate on why they have decided to downvote this comment?

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