Paata Ivanisvili

Grok 4.5 just constructed an explicit counterexample to hypercontractivity for the Poisson semigroup (the square root of the Laplace–Beltrami operator) on the 4-sphere. Back in 2021, with Rupert Frank arxiv.org/abs/2101.06209 we proved that hypercontractivity holds in dimensions ≤3 and fails in sufficiently large dimensions (for example, in dimension 13). Grok's example shows that it already fails in dimension 4, making our earlier result sharp. I also tested this problem on several other frontier AI models. One of them also managed to find a counterexample, but I particularly like Grok 4.5's solution: it is explicit, simple, and elegant. The attached files were generated entirely by Grok 4.5 build (with zero intervention on my side).

@aryehazan yes it does. If you ask any reasonable AI it should be able to extend for all Lambda. And as I mentioned this argument also recovers full borell-brascamp-lieb inequality

@PI010101 does this generalize to general lambda in [0,1]? this proves it for lambda=1/2

Here is a Bellman function (dynamic programming) proof of Prékopa--Leindler, and hence of Brunn--Minkowski and the isoperimetric inequality. I have not seen such a proof via optimal control theory before, though I did search with AI and could not find one. Some comments: The argument was inspired by the recent paper I mentioned yesterday here: x.com/PI010101/statu… In that paper, they use a discrete Bellman function, defined as a supremum over directed paths. Instead of finding the Bellman function exactly, they rewrite it using mixed volumes and then use the Alexandrov--Fenchel inequality to obtain log-concavity-type properties on a family of Bellman functions. In my notes, I consider a continuous analogue of it and simply propose one candidate supersolution for the Bellman function, which luckily works nicely. One can also get the Borell--Brascamp--Lieb inequality in this way (in full generality). The idea of dynamic programming is literally everywhere: aircraft guidance, robotics, communication networks, energy systems, traffic control, financial engineering, operations research, and reinforcement learning.

Today Felipe Goncalves and Danylo Radchenko posted a preprint on arXiv resolving Bourgain-Dilworth-Ford-Konyagin-Kutzarova conjecture. One of their main results says that for any sets A_1, ..., A_n in {0,1,..,m}^d we have |A_1+...+A_n| ≥ (|A_1|...|A_n|)^p, where p=n log(m+1)/log(mn+1) and the exponent is sharp. arxiv.org/abs/2607.01458 It was known that, via a compression argument, the problem reduces to proving a highly nontrivial arithmetic inequality in many variables. Some partial cases m=1 and 2, and small n's were solved before. Congratulations to them.

One good thing about AI: academia has always concentrated opportunity at a few elite universities. Many brilliant people end up elsewhere - not because they are less talented, but because positions are scarce, timing matters, and careers often depend on being near the “right” people and problems. One real advantage of elite places is access to extraordinary students and postdocs. For many researchers, that is the most rewarding part of academic life. Some even move countries or accept major life compromises just to be surrounded by such young minds. AI may change this. Soon every researcher may have something like a brilliant, tireless student: someone to brainstorm with, test ideas, compute, object, and help push projects forward. This could make research less dependent on being at the “right” institution, and more dependent on the quality of one’s ideas, questions, and persistence.

Grok Build now renders math, formulas, and LaTeX right in your terminal

The apparent doubling was a one-day anomaly. Looking at the full final week of May, arXiv math submissions were up about 32% YoY, not 100%. My prediction (and it is already happening): we are entering the two-day paper era. Day 1: solve the problem. Day 2: verify the proof, write the paper, and submit.

arXiv math daily submissions have more than doubled since 2025

New paper with Rupert Frank on arxiv.org/abs/2605.29035 We settle the problem of finding the sharp constant in the log Sobolev inequality on the n-cycle for all n ≥ 4, by showing that it is equal to half of the spectral gap. The question goes back to Diaconis–Saloff-Coste (1995). Even n=2k were settled by Chen–Sheu (2003); n = 5 by Chen–Liu–Saloff-Coste (2008); odd n ≤ 21 by computer-assisted proofs of Faust–Fawzi (2021). The n = 3 cycle is the exceptional case.

Academia is one of the few places where, despite all its imperfections, a person could still rise mostly through talent, curiosity, and hard work (example: Ramanujan) — not only through luck, wealth, family connections etc. But what happens if access to extremely powerful AI assistants becomes the new dividing line? The news about “superhuman AI” is scientifically exciting. But socially, it raises difficult questions — especially for gifted children from poor backgrounds who may not have access to frontier models.

@ElliotGlazer Someone should really give him access to frontier models. In many cases, with a bit of experimentation using LLMs, one can improve these constants quite substantially. For example, with a little play, one can already get something like n^1.0318499…

Sawin optimized the argument a bit and extracted an explicit lower bound of n^1.014 for the maximal number of unit pairs achievable with n points in the plane, for arbitrarily large n: arxiv.org/abs/2605.20579

A remarkable moment in mathematics publishing: almost the entire editorial board of the Journal of Approximation Theory has resigned simultaneously, declaring that “the journal, as we knew it, has ceased to exist.”

Grateful to share that my NSF DMS grant has been awarded. The project studies functions of many yes/no variables and their continuous analogues, with connections to high-dimensional probability, learning theory, data science, and quantum computation. The goal is to understand approximation, randomness, boundary structure, and sharp inequalities. Many thanks to NSF and DMS for their support. @NSF_MPS #DMSFunded

Déjà vu effect in LLMs I was working on a problem, call it Problem A. At some point I managed to reduce it to another problem, call it Problem B, and then solve B. Problem B is interesting in its own right and can naturally be asked in a more general setting, depending on a parameter (t). For my application to Problem A, I only needed the case (t=2), so I never seriously thought about the other values. My feeling was that, with enough time, perhaps one or two weeks, I could probably solve problem B for other (t)'s as well. Now comes the interesting part. I gave Problem B to an LLM and asked: for which values of (t) can you solve it? It produced a solution only for (t=2). The solution looked “different” from the published one: different language, different framing, no citation to my work. But after looking more carefully, I think it is essentially the same solution, just translated into another language. This is not completely trivial to recognize unless one knows the problem well. When I asked the LLM to solve Problem B for other values of (t), it could not. I have seen this phenomenon in other examples as well. Sometimes, when an LLM appears to produce a new solution to a problem, I worry that what is happening is more subtle: the problem may already have been solved somewhere, perhaps in a hidden or disguised form, and the model is showing us the same solution from a different angle. This “déjà vu effect” makes it quite hard to judge novelty in some AI-assisted mathematics.

@roydanroy By the way I am curious to know how is co-matheamtician doing with optimization in BMO problem (public tier-4)?

Friday: AI co-mathematician 48% on FrontierMath Tier 4 Monday:

I agree that bringing in Lean would be very valuable, especially for disputed benchmark problems. My only worry is making formalization an upfront requirement for contributors. When I contributed my BMO optimization problem — public Tier 4 — if I had also been required to formalize the solution in Lean, I probably would not have contributed it. The extra time cost would have been too high.

For what it's worth, whenever our agents make a mistake in a benchmark, there's a good chance the benchmark is wrong, and it's always worth a close look. Will be curious to see how this shakes things up. Of course, an "AI assisted review" of an AI benchmark sounds a little bit like deciding whose answer you like best. Would be nice to bring in some Lean guns.

@kfountou Totally agree, strange times indeed 😄

@PI010101 Assuming you haven't uploaded your result on the internet of course! Otherwise, they could just reproduce it and use it against you!😀 Strange times!

New referee standard: ‘We reject the paper because the result is now routine with the help of LLMs’ 😄

@kfountou Yes, this is happening now. I’d have no objection to such a referee comment--provided they include a chat transcript (one prompt or many) that independently reproduces the same result with a complete, rigorous proof… not just copy-pasting my publicly available 10-page proof.

@PI010101 Is this real? I expect it to happen...

What will mathematics look like 7 years from now? I’m really curious to hear your brief opinion.