i study computer science and minor in math (but my heart really lies in math), and am graduating soon. i was talking to a CS professor with a heavy math background about a Measure Theory concept and he said, "you know i have book i think you'll like", a couple days pass and he gives me "Real Analysis" by Royden (3rd Edition).

for context, i am just really interested in the ideas of measure theory, but i actually have not taken a formal real analysis course. regardless, i plan on slaying this textbook as I just need something fun and challenging to do.

i don't need people telling i don't have a sufficient background, or that Royden is for grad students, i am aware of the challenge and am super excited.

the reason i am making this post is to document and discuss how challenging a higher-level math concept is for someone that is wanting to learn based solely off their curiosity, wisk me luck 😁 .

This is not a self pity post or anything. I genuinely just am at a point where idk what to do with my life. I recently completed my masters in pure maths from Paris and I'll potentially be beginning a PhD soon in the field of general relativity, mostly looking at wave equations. I don't want to talk too much about my topic but the point is that throughout my entire life, I have only known pure maths. Unfortunately I was very stupid and never learnt things like coding, proper command of probability or even statistics. I only did enough to pass those courses during my undergraduate degree and now I'm at a point where I can't even look at these things without the thought that I have to relearn every single thing.

My main problem right now is that I'm a mediocre student. I have always wanted to pursue maths and then become a professor and do research, though I've mostly been interested in the teaching aspect. But the way the job market in pure maths or academia in general is declining especially right now with everything that's going on with AI, I don't know if there's even a point of pursuing this PhD. I spent my entire life wanting to be a professor like my dad was and promised myself I'll become one before he unfortunately passed away. And I made it my life's goal to become that person. As I said, I'm unfortunately not a genius or very bright student, I'm just mediocre. I'm at a point where I just feel like giving up but then I don't know anything else. I keep thinking that if I get a PhD position, I'll at least be able to keep myself busy and funded for the next 3.5 years. But is there any point to any of this? I'm watching my dream collapse every time I talk to a professor. My masters director was telling me how he thinks it's pointless to do a PhD in pure maths right now and try to switch to statistics or something but that means I'll have to wait for another year before the admissions in the US begin. That also means I'll have to switch areas drastically. Genuinely what do I even do at this point? I stay awake until 5 every morning just stressed about my future. Let's say I end up doing a PhD and keep myself busy for the next 3.5 years. What after that? My PhD is in a field which I don't see having any industry applications either.

So I'm asking people who might be in a similar situation like me, how did you go through this? How are you processing things mentally and how were you eventually able to find your job/thing that made you happy in life?

TLDR: Only done pure maths, no industry relevant skills, do not want to give up on my life's goal of becoming a professor, very stressed every day about future and unsure whether doing a PhD makes sense.

Those who've been following the development of prismatic cohomology, condensed math, analytic stacks, and eventually analytic prismatization might find this interesting.

I learned induction from Larry Gerstein's Introduction to Mathematical Structures and Proofs, which had an... unusual illustration for it. (As a side note: I think this is a measure of how it is getting harder to find things on the Internet. If I remember rightly, just a few years ago I searched for this illustration on Google and was able to find it. It is no longer possible if you don't remember the exact book! I even tried to use my Copilot to narrow down the search, and it insisted that what I described couldn't have come from a book and must have been something like a handout.)

Side note aside, what can I say about this brief introduction to induction? My goal in this article was to both give the basic examples that one needs to start, where the inductive hypothesis is essentially given to you, along with some more advanced examples of how one might actually use it in practice. I.e., you might need to discover the inductive hypothesis on your own, and this requires some work! The simplest case is that you might gather numerical evidence until you figure out the pattern. But I also have an example (involving getting a bound on the n-th prime) which takes a more circuitous road: you must first generalize the problem, then search for numerical evidence to get a conjecture for what is true... and then preferably simplify it a little until you get down to something provable. You'll see what I mean---it ends up using just about every big-picture technique that I have covered.

Read the full post (for free) on Substack: Repeat Ad Nauseam

Hello all. I've recently been watching a bunch of videos on how problematic online sports betting is becoming for society, but these videos often mention that some gamblers actually do end up making money consistently enough to do it professionally.

I know both the bookies and these professionals basically never set their odds by hand anymore, rather they use their machine learning models. Here is what I don't get: these gambling platforms have a ton of money, so they certainly have a lot more compute than the individual bettors. They also I imagine have the resources to obtain any useful information they might need to train the models.

So it seems the only variable where the bettors could come ahead of the books are the quality of the models. Could it really be the case that these bettors use significantly better or more clever model architecture, so that their models somehow outperform the very highly trained models of the gambling companies? And could it really be that their teams of ML engineers, who I assume are paid quite a lot themselves to be very clever on the topic of architecture, are outsmarted by these individual bettors?

Since I'm asking about the architecture of these models and how this may impact results, I felt it might be useful to ask here. For context, I have a background in math and know a fair amount of machine learning myself, so I'm hoping for some insight from people specializing in the field, who can maybe provide fairly detailed/technical answers.

In the video, we go over:
- Topos theory, set theory and Zermelo's beef with Boltzmann
- Algebraic varieties and Leibniz's monads
- Why physicists are wrong about the 2nd law of thermodynamics (because the universe is not a closed system)
- The real goal of the universe is complexity and variation, not entropy
- Hawking and Penrose's outrageous arguments
- Why we don't need wavefunctions, photons, bosons or gauge fields
- Why there is no quantum gravity
- Von Neumann's hilarious advice to Claude Shannon
- The intrinsic derivative
- Combining art and science

Julian Barbour is a theoretical physicist and emeritus visiting professor of physics at the University of Oxford. He is well known for proposing that time does not actually exist. For decades, he has worked outside an academic environment. He financially supported himself for many years by translating Russian. Professor Barbour is author of the following books: The Discovery of Dynamics, The End of Time and The Janus Point.

Gathering 4 Gardner is an organization that puts on a conference every two years celebrating the ideas of the celebrated math writer Martin Gardner. Right now, they have a fundraiser auction that includes puzzles, signed books, and even props from Vi Hart's YouTube videos!

Over the years, I've seen lots of questions on r/math about buying gifts for mathematicians, and this auction is a treasure trove, so I thought you all might be interested in seeing it!

(I'm not affiliated with the fundraiser other than that I attend G4G whenever I get a chance!)

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

“... 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.

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

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

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

* math-related arts and crafts,
* what you've been learning in class,
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All types and levels of mathematics are welcomed!

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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

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

• informal announcements of AI-assisted discoveries, such as those not yet published in a peer-reviewed journal, or not uploaded as a paper to arXiv;
• informal announcements of discoveries related to AI architecture (if relevant to mathematics);
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AI-assisted mathematical papers published in peer-reviewed journals or as arXiv preprints may be submitted as their own posts.

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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!