“... the KLS conjecture (named after its creators, Ravi Kannan, László Lovász and Miklós Simonovits) asks a simple question: Suppose you want to cut a convex shape — maybe an apple without dimples — into two equal-size portions, and you’re planning to put one aside for later. The exposed surface is going to turn brown and unappetizing, so you want to make it as small as possible. Among all the possible cuts, which will minimize the exposed surface?

“It’s not too hard to answer this question, at least approximately, if you are limited to straight cuts. But if you’re allowed to make curved cuts, all bets are off. In dimension two, mathematicians know that the best cut will always be a straight line or an arc of a circle. But in dimension three, the best cut is understood only for a few simple shapes, and for higher-dimensional shapes, mathematicians usually don’t even have a hope of finding the optimal cut.” (Erica Klarreich, 2021, Quanta Magazine (link).)

The original paper, by Kannan, Lovász, and Simonovits, is here: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput Geom 13, 541–559 (1995). https://doi.org/10.1007/BF02574061

As discussed in https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/, earlier this month, three separate groups of researchers each released proofs, and a fourth researcher released a Lean 4 formalization. AFAIK, all four were developed in collaboration with various AI models.

• Song, Z., & Zhang, X. (2026). An O(1) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447
• Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a proof of the Kannan-Lovasz-Simonovits conjecture. https://arxiv.org/abs/2610.05474
• Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728
• Lean 4 formalization: https://github.com/HongruZhao/KLS500

From the Microsoft blog post:

On the same day that the authors [Song & Zhang] posted their complete proof, a separate group of leading experts [Bizeul, Klartag, & Lehec] posted a full proof that they developed in parallel. Their work built upon the findings in the first version of the Microsoft paper [Song & Zhang], calling it “the crucial step in the proof.” A third proof [Balasubramanian & Kasiviswanathan] which appeared separately on the same day relies on a closely related idea.

The CV was last updated this year as it contains a 2026 honor.

In 1901, Hilbert proved there is no smooth isometric immersion of the hyperbolic plane H^2 into Euclidean 3-space. (https://en.wikipedia.org/wiki/Hilbert%27s_theorem_(differential_geometry)))

John Douglas Moore, in his 1969 PhD thesis under Shoshichi Kobayashi, Some Geometric Applications of Exteriorly Orthogonal Quadratic Forms, conjectured the natural generalization: that there is no isometric immersion of H^n into E^(2n-1). He also worked toward the conjecture in his 1972 paper Isometric Immersions of Space Forms in Space Forms. (https://msp.org/pjm/1972/40-1/pjm-v40-n1-p17-s.pdf)

Recently I noticed Moore posted a preprint paper on arXiv last November, Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem. (https://arxiv.org/abs/2512.00973) It would appear he has proved his conjecture after 56 years! [edit: I should have been more measured with that statement, as doubts were quickly raised after I made this post; see comments below.]

He writes on his research interests page that the proof is "based on the generalized Gauss-Bonnet formula developed by Allendoerfer, Weil and Chern." (https://web.math.ucsb.edu/~moore/researchinterests.html)

I find this incredibly cool but don't have much sense for how big of a deal it is; if anyone out there does, let us know! On a personal note, as an undergrad I took ODEs and vector calc under Moore, and later did a summer research project with him on minimal surfaces. He was so warmly encouraging and a huge positive influence on my math trajectory, easily one of my favorite professors ever.

If this result holds up, it's a beautiful capstone on a long and accomplished career.

I tried making a triangle of numbers and then marked the primes. I saw that the primes in the left were more frequent than primes in the right. I can also extend it by writing some code but does it ever happen that the primes in the right exceed the primes in the left or do the primes in the left increase their gap against the primes in the right

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I was thinking about my real analysis class when I took it. I think it could have been improved in many different ways. More practice problems, more explicit instruction instead of trying to have students figure it all out.

If the aim of constructing number systems is to explain the geometric line as a set of points, wouldn’t ZFC + (V=L) be a natural choice, perhaps even the best one? In this framework, analytic geometry, differential geometry and topology all work just as they normally do, without the need to grapple with metamathematical questions such as the continuum hypothesis. (Moreover, this rules out measurable cardinals, large-cardinal phenomena and similar feverish pies in the sky, but for the purposes of describing the continuum, that seems a reasonable price.)

I have always found it a little surprising that popular mathematics books rarely discuss Gödel’s set-theoretic universe, which seems to offer such an intuitive and straightforward foundation for mathematical analysis.

I've been studying Shafarevich's Basic Algebraic Geometry for some time now. He's a really famous mathematician (I first really learning about his results after hearing about his famous theorem on the inverse Galois problem), but I've been studying on my own so I could only guess how his name is pronounced, and I was wrong. Today, watching a youtube video, it became apparent to me that his name is read ShafaREVich, which the Cyrillic with accent mark (Шафаре́вич) seems to confirm.

Not really that uncommon for me -- I was pronouncing Arrhenius wrong for years after teaching myself chemistry in elementary school, and I wasn't corrected until I took AP Chem.

Also, even among mathematicians, Dirichlet seems to be particularly contentious, as I heard widely divergent pronunciations from my math professors.

Anyway, any other mathematical names that are commonly pronounced wrong? Euler's probably the most common among nontechnical people to encounter and pronounce wrong.

Chern's an interesting case, since his name is spelled in the old Gwoyu Romatyzh system, which uses letters like r to indicate Chinese tone -- modern pinyin spells in Chén, but even he himself seemed to have accepted the normal English pronunciation of his name with an r.

One of my favorite essays on mathematics is Arnold's "On teaching mathematics." In it, he describes what he views as defects of the traditional treatments of real analysis and calculus in universities; and in particular, how the emphasis on rigor removed discussions of the beautiful geometry of Riemann surfaces and how they relate to calculus. As a specific example, he describes how one can tell which integrals are easy or hard to solve by looking at the genus of the associated Riemann surface.

This week on Hidden Phenomena, you can learn what Arnold meant by this comment: what is the "genus of the Riemann surface associated to an integral," and how does it help you understand if the integral is easy or hard?

I had my first grad-level exam this week and I bombed it pretty awfully. It was a 50min exam with ~12 problems, a mixture of proofs and counterexamples. I would say I solved 2 or 3 of them and made starts on 2 or 3 more; we haven't gotten grades back yet but I'm expecting something like a 20/100. The thing that's annoying me is that none of the problems were all that hard. I can tell that I could solve them all, only it would take more like 3-4 hours instead of 50min.

Part of me wants to think that the time frame was totally unreasonable, but another part of me thinks that it was a skill issue. I'm guessing a lot of members of this sub have dealt with similar situations, so I'm wondering if there is any advice for getting quicker at answering these early-grad-level types of exam questions?

The title today is a reference to Douglas Adams' description of writing a book as "putting a hundred thousand words in a cunning order." It's technically a correct description, but it doesn't really describe how anyone actually does it in practice. Just so, while it is technically true that one builds a proof by putting many inferences in a clever order, it doesn't describe how to actually do this.

A proof, like a book, is a collection of narrative threads---understand how to build narrative, and you will start to understand how to write a proof.

Read the full post (for free) on Substack: Putting a Hundred Inferences in a Cunning Order

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Hello! I need some problem books to accompany the book Differential Geometry of Curves and Surfaces by M. Di Carmo. Thank you!

Same as above

Is it elementary or notoriously massive and is there any approaches(feel free to discuss about any heavy machinery approaches too) around which you think it(the diophantine equation) can be showed to have finitely many solutions given the specific constraint?

I am looking for a text/lecture series on the theory of locales (pointless topology) . I don't want to go into much depth I just want to get a broad picture . My goal with this is to be able to understand topos theory better , as Andre Joyal in a lecture at IHES suggests that before learning topos theory it's better to have an idea of the theory of locales as there are a lot of parallels .

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EDIT : Discord link attached

Holy fuck this book is hard as shit. I am a 2nd y math PhD studying this along with Hida's Geometric Modular Forms and Elliptic Curves rn. Looking for people to form a study group devoted to this. Let me know if you're interested and maybe we can form a gc/discord :)

(This book is "Arithmetic Moduli of Elliptic Curves to be clear")

Discord link : https://discord.gg/RXbRvTyR8

A brief explanation of proof assistants

I have seen Kneser's method of extending tetration to real numbers, but that doesn't seem to be right as I can't see how 2^^0.5 is not 1.55961...

As a starting point, we can see that 1+1=2 and 2x0.5=1. Also 1.4142...x1.4142...=2 and 2^0.5=1.4142. Similarly 1.55961...^1.55961...=2 and 2^^0.5 should be 1.55961...

To see integer values for all of them, the smallest number after 4 will be 256 where 256x0.5=128, 256^0.5=16 and as 4^4=256, so 256^^0.5 should be 4

We can get values at 0.5, 1.5, 2.5, ... for all numbers by this method, but there should be some way to make the curve continuous as well. Also a^^b should also satisfy a^(a^^b-1), so 2^^1.5 should be 2^1.55961 which is ≈2.9477... and 256^^1.5 should be 256^4 which is 4294967296

Is there any other way to extend tetration to real numbers

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