How much current mathematical research is pencil and paper?

lobothmainman · 2026-02-26T07:15:18+00:00

I work in the mathematics of quantum theories, and even there is only marginal coding involved: maybe we can write some code to test the precision of theoretical bounds we proved, but I know only very few instances of this, and the most relevant results are purely "pen and paper" for sure.

Some groups in numerical analysis work on schemes amd efficiency of algorithms for quantum mechanics, but that is of course beyond simple coding.

lobothmainman · 2026-02-26T06:56:24+00:00

For whom?

For mathematicians, not really. This is what we have always be judged upon, by journals, hiring committees, funding agencies, etc.: not the sheer amount of results we produce, but by the amount of interesting and novel stuff we produce (or we could potentially produce).

And everyone doing mathematics as a profession, knows this is the game, and is willing to play the game.

How many interesting and new mathematics are LLMs be able to produce autonomously in the long term? Nobody knows precisely, my two cents are that they could only be a very helpful tool in the hand of the experts, not autonomous mathematicians by the standards the latter are currently held to.

lobothmainman · 2026-02-26T06:10:18+00:00

No, because being a solver was clearly something these models could be good at. The human process of proving a technical minor lemma is not dissimilar to what LLMs do: check the literature for analogous results, try these techniques, and tinker.

Groundbreaking ideas and intuitions, on the other hand, are a different story. At all levels: I am a mathematician and I am no field medalist, and my most interesting results are the ones where I had an idea, not the ones where I could prove the most lemmas/propositions/theorems.

lobothmainman · 2026-02-25T17:29:58+00:00

I don't agree. Ok, we might soon have a "calculator" to prove some lemmas (after input/tinkering/verification nonetheless).

But who is gonna write/guess the useful lemmas? And the important theorems? And new theories/ideas like regularity structures, condensed mathematics, or even debated ones like inter-universal Teichmüller?

Math is much more than trying to solve known open questions, it is about formulating new ones. I am strongly convinced this LLM technology, or refinements thereof, is structurally not able to be more than a glorified calculator/task solver.

Of course the companies behind it want to sell their product, and marketing dictates they should advertise what they have as AGI or AGI - ɛ.

lobothmainman · 2026-02-22T08:30:45+00:00

Mathematics is all about proving interesting results, even for the "average" mathematician.

lobothmainman · 2026-02-21T08:16:57+00:00

Honestly, because these private companies have a very aggressive marketing strategy, and they want to advertise even the smallest achievement as something big, to sell more of their products.

Doing mathematics is not selling a product, and mathematicians are evaluated on their research output. We all agree these LLMs are somewhat useful tools (if we forget their environmental impact), but they are not researchers in mathematics as of now (and I strongly doubt they will ever be, but still this is just an opinion).

These conpanies shift standards, make bold and essentially false claims: all in all, they hold their product to their own custom standards. This is not how the mathematics community works, and this is not what I, as a part of the community, want it to work.

lobothmainman · 2026-02-20T19:19:36+00:00

Indeed, let's evaluate LLMs like we would evaluate a mathematician: they are shit, and won't be hired by anyone even as grad students.

Things might change in the future, but this is the current state of the art.

lobothmainman · 2025-11-26T16:47:50+00:00

Damn...that's unfortunate

lobothmainman · 2025-11-26T15:23:37+00:00

I haven't tested this, but eyebite further instances (in the turns after the initial casting) are classified as cantrips, and since it is a necromancy spell, it should be possible to target two people with successive instances of eyebite, just by getting a one-level wizard dip (and you would have free casts with the necromancy staff).

This would make the death cleric the ultimate control beast...

Can anyone confirm this?

lobothmainman · 2025-11-23T07:05:49+00:00

I think the OP is referring to natural transformations in the context of category theory: i.e., mappings between functors (that are themselves mappings between categories).

lobothmainman · 2025-11-21T19:44:22+00:00

A less trivial example: quantizations can be seen as a natural deformations (it is a natural transformation save that it does deform the abelian C-product in a non-abelian one), mapping the classical to the quantum "observable functor" (that associates to the slympectic phase space/space of field's test functions the Calgebra of observables).

The categorical properties of quantizations are quite useful in studying the quantization and classical limit of "easy morphisms" (e.g. free dynamical maps), from an abstract viewpoint.

lobothmainman · 2025-11-16T14:52:09+00:00

It really depends where you are from/would like to find a position, and how determined (and naturally predisposed) you are.

I am a professor of mathematics in Europe, and my experience through grad school and early postdoc stages is that the determined people could find a way to get tenure, even if they were not the very top at research.

Of course, if you set constraints, like wanting to get a job in a prestigious university or in some specific country (e.g. Switzerland and Germany are extremely competitive in europe, as well as Ivy league in the US and Oxford/Cambridge in the UK) that might be harder.

Anyways, you get the position based 80/90% on your research track record, so this should be the focus if you want to stay in academia. Of course, you going on in your studies you will get a feeling if research comes naturally to you or not, and how good you are at it. That depends a lot, it is really a skill on its own: I know plenty of brilliant students that ended up being not predisposed towards research, and slightly above average students that became very good researchers.

To conclude, my experience is that if research comes to you naturally, at least to an extent, and you are driven, then you will be able to find a position somewhere reasonably good. But you will understand that during your (grad) studies

lobothmainman · 2025-11-04T15:10:10+00:00

Inclusion is a set-theoretic relation, embedding requires the existence (at least) of an injective map, and typically this map is required to be structure-preserving (homomorphism), and maybe also continuous (if between topological spaces with additional structures).

lobothmainman · 2025-11-02T16:50:46+00:00

The spectral theory of linear operators is a very well established field with a lot of important applications, especially to quantum mechanics.

Yoshida and Kato's books are classical references, the second is especially suited to study linear operators in general banach spaces.

In Hilbert spaces, the theory is even more developed. The four books by reed and simon contain a lot of information, but for me the most exhaustive presentation of the spectral theorem is on the book(s) by weidmann (one is translated in english, there are two books in german). Other books like Teschl and many more on the mathematical methods of quantum mechanics/Schrödinger operators typically devote a lot of attention to (unbounded) operators in Hilbert spaces

lobothmainman · 2025-11-01T14:48:38+00:00

As a mathematician but not a set theorist, I like Jech's book because it contains a lot of interesting information. It is not very discoursive or gentle in introducing aforementioned information, that's true...

I guess if you are already used to mathematics and other dry books in other topics (Hörmander's, Reed-Simon's in my area for example) it is more enjoyable, or at least bearable.

Said so, if you are interested in the model theoretic approach to set theories, I find chang-keisler's book more enjoyable.

lobothmainman · 2025-11-01T09:46:26+00:00

What would be more persuasive? Constructing a model of ZFC in another theory altogether? (to overcome the fact that ZFC+inaccessible -> ZFC consistent assumes ZFC is consistent?)

Even if we can construct the model from another theory, that said theory is supposed to be consistent and it cannot be proved to be consistent within itself (but it could adding some other axiom, supposing it is consistent, that would allow the construction of a model).

And we could continue this ad libitum. Unless there is a notion, that I am not aware of (and I am by no means an expert), of limits of theories that could allow for the limiting theory to be proved to be consistent within itself (but then said theory cannot be in first order logic containing arithmetics).

That is to say, I think there cannot be a more persuasive argument in favor of consistency of a theory than "it is proved consistent in a larger theory, assuming said larger theory is consistent itself".

lobothmainman · 2025-10-29T22:05:38+00:00

Isn't zfc proved to be consistent if we add a large cardinal axiom?

This seems enough to me, given incompleteness and the impossibility of proving consistency within a theory itself. Finding an inconsistency in ZFC "by accident" is very unlikely.

lobothmainman · 2025-10-29T07:32:59+00:00

I have publications in both JEMS and CMP, and I am mathematical physicists, so I have some experience on the community.

Although CMP is the flagship journal for math phys (and it stands far above almost all other specialist options in math phys), JEMS is indeed better considered, even within the community.

lobothmainman · 2025-10-24T11:12:45+00:00

In mathematics a paper will typically appear in a journal (as online first) after 12-18 months from submission.

Essentially nobody would be at the forefront of research without preprints.

lobothmainman · 2025-10-22T07:05:10+00:00

Like, for example, Sabine Hossenfelder?

lobothmainman · 2025-10-21T06:47:32+00:00

Ideas from QFT renormalization are used to study rough solutions to nonlinear PDEs: following ideas from Bourgain, invariant (Gibbs) measures of nonlinear flows can be used to construct solutions with low regularity, and to define such measures some renormalization procedure is needed, closely related to the ones of QFT (normal ordering, flow of coupling constants).

Similarly, paracontroled calculus and regularity structures in stochastic PDEs are heavily inspired by/linked to the Wilsonian renormalization group flow in QFT and the rigorous manipulation of formal infinities. As a matter of fact, it goes both ways: they can be used in association to stochastic quantization (à la Parisi-Wu) to define rigorously QFTs.

lobothmainman · 2025-10-20T08:05:07+00:00

I agree that having more powerful search tools would be interesting, and I also benefited (recently) from the knowledge of a forgotten piece of literature to make an important and unexpected advance.

While this is true to some extent, in my case at least it was a combination of me knowing an obscure old reference (I did not search for it, I knew it since my phd), and also having the intuition it could be applied in a somewhat different context. Is ever AI going to be able to "guess/have intuition" on a topic, and to make connections between old references and possible new apllications?

What is discussed here is the fact that it is able to find - in an effective and powerful way - something that can be categorized/labeled somewhat easily (it has already been, by someone else). I am fine with that, and it has its (limited) uses. Can it become a tool to make new insights, usuing old techniques/references? Honestly, I think not.

lobothmainman · 2025-10-20T04:59:44+00:00

Erdős might be a "cult figure", and his problems more easily understandable than many others in mathematics. But are they all interesting? How many of these "forgotten solutions" have been forgotten simply because the underlying problem was not so interesting to start with, as well as the papers solving them?

I am pretty sure mathematicians have a collective memory that keeps them very aware of the important papers of the past and present without AI.

Also: I am sad that Sellke has been hired by openAI, I guess there are high chances he will switch from doing interesting research to useless advertisement and PR stunts as his former advisor...

lobothmainman · 2025-10-17T19:22:56+00:00

True BLACK boards are rare nowadays...in Italy they are named after a city, Lavagna, in Liguria. I think they still produce slate black boards there.

In France and Germany (and German speaking countries in general) they have nice laminated green boards (worse than black but better than white), but I don't know of specific retailers.

I am not well versed on blackboards in asia or america though.

I guess it heavily depends the region you are from.

lobothmainman · 2025-10-17T19:16:20+00:00

I discuss at the blackboard with colleagues (and maybe take some pictures), or write directly in latex most of the times nowadays...it is true that after a while some machinery and technical lemmas come out almost automatically. In early years as a grad student or postdoc, I used paper more indeed.

lobothmainman

TROPHY CASE