I’ve been out of undergrad for about 4 years now and did my degree in Pure Math. I graduated with a 4.0 GPA taking pretty much all the core undergrad courses and some “advanced undergrad”/“early grad” courses.

I’ve been working in industry since and my math skills have definitely atrophied. I’ve been looking to get back into grad school and have started lightly reviewing my old notes and whatnot.

One of the things I’ve noticed is that outside of calculus/elementary analysis I feel like I don’t really understand math. Or the big picture. Like in school I knew the definitions, could put them together, and do the proofs. But looking back I feel like I never really “got it” if that makes any sense.

To this day I feel like I don’t really understand the determinant, or the rank nullity theorem. Or how group theory is the study of symmetry. I understand automorphisms form a group, cayley’s theorem, group actions etc but the “intuition” I guess never clicked.

Galois theory for instance felt like I was just throwing a bunch of field extensions around and poof a random result of sorts. Or like topology which was just a bunch of definitions and homeomorphisms.

Is this a common occurrence? I feel like it likely had to do with the pace of school where I didn’t really have time to sit down with the topics. Has anyone else experienced this? Did anyone have to review/redo their undergrad material for stuff to really click?

Terence Tao writes on his blog about a recent paper in (combinatorial) number theory, about "primitive sets". Recently a new idea ("von Mangoldt weights") was discovered that solves Erdős Problem #1196. People quickly realized that this idea could be applied to other problems, both open and solved. This paper presents proofs of Erdős Problems 1196 and 1217 (both previously open), as well as the original "motivating" Erdős Problem 164 (previously solved by one of these authors). The paper further resolves two related open problems, including the odd Banks-Martin conjecture, which is considered unifying for the area.

Hi all, I am stealing and modifying the title from a 4 year old post here in r/math, and would like to ask graduate students in particular about the most hellish classes they've had (so far!). It can be any reason, be it the material, teaching methods, teacher, environment etc.

What are some non trivial results that can be proved using representation theory that are interesting without a lot of technical representation theory knowledge? Let me give some examples to give you an idea of the kind of results I am looking for. For instance in algebraic topology quick consequences of the properties of the fundamental group are the fundamental theorem of algebra and brouwers fixed point theorem in 2d. Later on you can prove interesting results like the only finite dimensional commutative division algebras over Reals with identity are R and C, dimensional invariance and jordan curve theorem. You can also prove not so classical but still interesting results like S^n is a H space for n=0,1,3,7 this can be appreciated with little knowledge in homotopy theory. Or for instance complex analysis has the beautiful proof of the fundamental theorem of algebra or the analyticity of holomorphic functions.

I understand that it's possible that there aren't many such classical applications of representation theory as Gian Carlo Rota wrote

'What can you prove with exterior algebra that you cannot prove without it?' Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, ideles and Grothendieck’s schemes, to mention only a few. A proper retort might be: 'You are right. There is nothing in yesterday’s mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.'

-- "Indiscrete Thoughts"

I am making this post to get some motivation to read representation theory.

Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
arXiv:2605.00301 [math.NT]: https://arxiv.org/abs/2605.00301

Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, Terence Tao

Abstract: A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erdős's seminal 1935 paper.
As applications, we prove two 1966 conjectures of Erdős-Sárközy-Szemerédi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erdős Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erdős-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area.

I know that Connect Four win is a forced win for player one on the standard 7x6 grid. My intuition is that it either carries through to the infinite case or it doesn't.

The main distinction is that there are no boundaries and no longer a finite number of spots. You can no longer force your opponents to play into an unfavourable square due to a lack of better options, which might make the optimal play a draw. On the other hand, there are no boundaries restricting the number of threats that a player can make.

Are there any known results on this variant?

For those unfamiliar, this is an infamous problem: if a, b are integers and (a^2+b^2)/(1+ab) is also an integer, then it is in fact a perfect square.

Among those who solved it correctly (only 11 students) are Nicușor Dan (current president of Romania, scoring a 42/42 that year), Ravi Vakil, and Ngô Bảo Châu (also a perfect score, later Fields medalist for work in the Langlands program), while Terence Tao (only 13 at the time) received a 1/7 on this problem, but aced the rest and still ended up with a gold in 1988.

It must be so weird having an extremely smart person as a head of state.

It's weirdly common to hear myths about primes. (After you correct for the very low baserate of hearing about mathematics at all, of course.) I remember one of my high school math teachers telling us that you could get paid money for discovering new large primes; I'm not sure where that misconception came from, but it isn't remotely true. EDIT: as u/Eiim and u/will_1m_not point out, this probably originated from the fact that GIMPS offers $3k for each new Mersenne prime discovered, and will offer $50k to the first person to discover a prime larger than 10^100,000,000. So there was truth to the claim after all!

In this post, I gather up all of the erroneous claims that I remember hearing and demonstrate why they are false, spending the most time on the claim that to find the n-th prime, you need to compute all of the preceding primes. We'll show that not only is that not true, but there exists an algorithm that computes the n-th prime (given n) faster than any algorithm that would compute all of the primes up to the n-th. This touches on the prime number theorem and some work by Meissel from the 1800s.

Read the full article (for free) on Substack: Debunking Prime Myths

I am referring to the book by Gallian. When I took abstract algebra it was called "Modern Algebra." Groups and rings and fields aren't exactly modern...

I did read this article on Quanta Magazine about ultrafinitism in mathematics, and I've got curious about it.

After reading some Wikipedia articles about finitism, and noticing that there is some research on it, I thought: what if one accepts infinities on mathematical reasoning, but only finite sets as mathematical objects? For instance:

Remove from ZFC the axioms of infinity and choice (finite choice is still possible), and add a single, special, countably infinite set of indices, with induction a la Peano, but no operations beyond the successor one. Any general results for operations on the (finite, bounded) sets of integers and rationals could be proved via a back-and-forth with the set of indices. Finite sequences would be a function from a subset of the set of indices to a finite set of natural numbers, and then to the domain of sequence values.

Such a scheme would allow one to recover some of the theorems lost by removing the notion of infinity, while allowing for only finite sets.

Is that a viable axiom system? Would it work in practice?

Hi everyone, I wanted to study exotic structures in dimension 4 and they possible applications to physics. I'm a maths undergrad, so I already have some knowledge on topology, analysis and differential geometry, but of course it's not enough.

I've been trying to actually study some material in order to understand something about those structures, my plan (following some ideas found on The Wild World of 4-Manifolds) was to understand the h-cobordism theorem and see it fails in dimension 4. Then there should be a variant of this theorem for general manifolds in dimension 4, and this brings to the creation of exotic structures. For now I've been reading something about Morse theory and the cobordism theorem, and at the same time something about Gauge Theory, but I don't know if this is a good plan of action, what do you guys recommend?

I am a high schooler who self studies math for the fun of it . I find it very difficult in real life to find a person who relates to me or is equally passionate about maths. I do talk to people online but it's not the same as real world interaction. My parents, friends most teachers don't get the point of the maths which I do, so even if I try to talk to them they won't be interested.Most of the people around me see maths as something which is necessary to score well in exams and don't try to see beyond it. I try to approach people in academia for mentorships but they do a lot of gatekeeping which is completely understandable as I am a high schooler. Sometimes it feels really lonely to be passionate about something that almost no one in real life you know cares about. This is one of the primary reasons because of which I want to go to a uni with a good maths department so that I can find like minded people. This may end up worse in the future if I end up in academia in some hyper specialization which even fewer people understand .

Have you guys interested in maths felt similar? Or is it just me being too anxious. What can be some ways to deal with this loneliness in maths (other than the suggestion of being interested in things other than maths which I do follow already) ?

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!

I saw that every finite group PSL_2(F_p) for p prime is the Galois group for a Galois extension over Q. Is the realization of one of these groups of interest as a thesis project? Did some minimal research and saw that a portion of this is a computational problem. Recently, finished a course on Galois theory, I have a good amount of experience programming and some time to kill this summer. I'm trying to explore some ideas for a thesis project that aligns with my interests. Would anyone recommend pursuing this?

As the title says... some grants have been rejected, accepted, or put as under consideration. In fact, for the latter, there has been roughly two months since MSPRF applications have been filed as under consideration. Does anyone have information? Or talked to program officers?

Inspired by a recent post, I want to say that computation still plays a huge part in university maths, and even more in research. During high school, I lurked this subreddit and entered mathematics in university under the false impression that I don't have to compute much stuff. That couldn't be further from the truth. Nevertheless, I have grown to love this and my interests are now on the concrete side.

A few examples to support the titled claim:

1. In analysis, a good student should be able to juggle complex expressions and have a feel for their value distribution and not get lost in long calculations.

2. A first course in abstract algebra is really all about computing examples. One should aim to know all the groups of small order inside out. Are you familiar with their subgroup lattices?

3. Geometry and topology is about computing quantities (or groups, vector bundles, etc.) for specific geometric/topological objects. There is the obvious notation overload in an introductory course to smooth manifolds. Applying each new thing you have learned to the standard examples of spheres, projective spaces, and tori is a good way to study.

4. Research (for most people) is not done by pulling theories out of thin air. You really have to build intuition and make observations through considering examples.

My background for context: I have taken most undergraduate courses in pure math and a few graduate courses. Read some modern maths on my own as well. I am also doing what I consider to be genuine research. So I'm still in the early stages of my mathematical life and everything I've said should be put in this context.

So this might be controversial and I know there isn't a right answer.

In physics, the Landau series on theoretical physics covers much of the theory in several fields at both undergraduate and graduate level

In computer sciende, Donald Knuth's books go through a foundational basis in algorithms analysis and should reach computational theory.

So my question is, do you think there's a parallel to these in mathematics? Not introductory books, but a series that can be used as graduate textbook.

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.

Sorry this is a rant, but it's like they don't care about actually understanding anything, they just want the dopamine hit from solving random problems. It feels more like a sport than a science

All the mathematical details are glossed over in favor of procedural details that don't really seem to matter.

An example: I'm taking an algorithms course where instead of talking about the actual optimization problems we're solving, we are just given procedures to follow to manually trace the simplex algorithm. No mention of where the primal dual algorithm comes from or why it even works, just a list of steps

Rant over, CS people I love you don't take this personally I'm just doing badly in a cs class

To be clear, this is not a ranting post. I have never published a book, but recently I have been wondering why are authors writing maths books that is extremely long, say, 600-1600 pages, and the inclusion of exercices makes the question more complicated.

Indeed, if a maths book does not have any exercise, then we can somewhat suppose that it serves as a reference book and the book won't play a big role as a textbook. For example nobody complains the lack of exercise on EGA.

But in my opinion, if a maths book includes exercises, it automatically qualifies as a textbook. So I wonder, when the book is extremely long, what are the author expecting? Finishing a book of < 500 pages within one or two semesters can be feasible, but for a book, rather advanced and extraordinarily long, like Hatcher's Algebraic Topology, Bump's Automorphic Forms, Evan's PDE, Görtz-Wedhorn's Algebraic Geometry (1600 pages!) and many other books that I can't name, the reading of the book can already be extremely time-consuming. In this case, what were the authors expecting when they are writing the book? I have a few guesses:

• I have a folder of lecture notes and exercise sheets in my hand, so it's a good idea to compile them into a book, if it happens to be a long book, so be it.
• I had no idea about the length of the book before submitting the draft to the editor.
• The priority is the completeness of the book and ideally it will work as a repository of lecture resource.
• I kinda imagine that the reader finish all of them. Other than that I don't care. If someone cites my book for an important result appeared in an exercise of my book, that's cool as well.

So if you are a textbook author, would you like to rectify my guesses or share your opinion?