r/math

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!

Paper by prominent mathematicians (each share their thoughts in separate sections; an interesting read): https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf

Here's the blog post by OpenAI: https://openai.com/index/model-disproves-discrete-geometry-conjecture/

The problem: Given n points in the plane, what is the maximum possible number of pairs of points at distance exactly 1?

Erdos famously conjectured that the answer should be n^{1 + o(1)} (essentially linear in n). OpenAI's model disproves this by constructing a counterexample that polynomially improves Erdos' bound to n^{1 + 𝛿} for a universal constant 𝛿 > 0.

Lately there have been some big announcements about AIs cracking serious theorems, and along with them, a lot of anxiety from mathematicians and researchers about what their future in the field looks like.

Am I the only one... feeling optimistic about this?

For as long as I've been around math, I've heard it described as a vast landscape- cathedrals and mountain ranges, hidden valleys, strange country stretching out in every direction. For centuries we've been exploring it on foot, in the dark, with nothing but a candle to light the next few steps.

What happens when we get a floodlight?

I think about all the structure that's been sitting just past the edge of what one human mind, or even a generation of them, could reach. Connections we never noticed. Theorems no one had the lifetime to chase down. Whole regions of the landscape we walked right past because the candle didn't carry far enough.

For anyone who loves knowledge for its own sake, who got into this because they wanted to see more of the thing. I think we're standing at the edge of something spectacular. Not the end of the adventure.

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.

Most people know this simple thing:

1 + 3 + 5 + 7 + ... gives square numbers...

Like:

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

So basically the odd numbers are like the layers which grow a square.

But there is another pattern inside the same odd numbers which I dont see talked about much. Instead of adding odd numbers one by one, cut them into groups like this:

1

3 + 5

7 + 9 + 11

13 + 15 + 17 + 19

21 + 23 + 25 + 27 + 29

So the group sizes are:

1, 2, 3, 4, 5, ...

Now add each group:

1 = 1

3 + 5 = 8

7 + 9 + 11 = 27

13 + 15 + 17 + 19 = 64

21 + 23 + 25 + 27 + 29 = 125

So suddenly the same odd numbers become:

1, 8, 27, 64, 125, ......... so on;.

which are cube numbers:

1 cubed, 2 cubed, 3 cubed, 4 cubed, 5 cubed.

That means:

1 | 3 + 5 | 7 + 9 + 11 | 13 + 15 + 17 + 19 | ...

turns into:

1, 8, 27, 64, ...

So the odd numbers are making squares if you read them normally, but they make cubes if you cut them at triangular places.

The reason is simple but kind of nice.

Take the third block:

7, 9, 11

The middle number is 9, which is 3 squared. There are 3 numbers in the block.

So the total is 3 times 9 = 27. That is 3 cubed.

Take the fourth block:

13, 15, 17, 19

The average is 16, which is 4 squared. There are 4 numbers.

So the total is 4 times 16 = 64. That is 4 cubed.

Same thing keeps going...

The nth block has n odd numbers, and the average of that block is n squared.

So the total becomes n times n squared, which is n cubed.

This also explains the famous formula:

1 cubed + 2 cubed + 3 cubed + ... + n cubed

is the same as

(1 + 2 + 3 + ... + n) squared.

Because after using the first n blocks, we have used:

1 + 2 + 3 + ... + n

odd numbers total.

And the sum of the first so many odd numbers is always a square.So cubes are hiding inside the square pattern of odd numbers.

I like this because it is not just a formula trick. It feels more like one sequence has two different geometries inside it:

read the odd numbers one by one, and you get squares.

cut them into growing blocks, and you get cubes.

what do you think guys?

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

There's a discount currently running at Springer. A lot of books that are usually under 100 are now for sale (ebook or softcover only) for 18.99 a piece (in whatever your currency is).

I am looking for recommendations in the field of operator algebras especially von Neumann algebras and their use in quantum information and quantum field theory. It can be either pure mathematics or mathematical physics.

2 books I am definitely getting are

- Quantum Entropy and Its Use by Petz D, Ohya M

- Quantum f-divergences by Hiai F

Feel free to share recommendations in other areas as well, maybe other people will find that helpful!

Hi everyone.

So I recently finished this proof today in Numerical Linear Algebra and it was incredibly dense. Every sentence had a new idea which assumed some sort of inter dimensional knowledge which I lacked. Nevertheless, I got through it and I understand it.

But it got me thinking about how i survived undergrad and ended up doing well. I studied mostly with the use of AI, and it’s got me thinking about whether I’m a fraud or not.

Throughout uni, I found myself getting stuck a lot in maths, either with proofs or examples. Whenever I get stuck I’ll ask AI to explain a proof or an idea, then we will go back and forth until I understand it then move on. I’m constantly getting stuck, I cannot read a textbook without using AI, that’s the truth, especially the grad textbooks I’m reading now. If I did my degree 10 years ago, I believe I’d fail and do very poorly, but I did incredibly well in my UG. To clarify, all my exams were in person and invigilated so I didn’t cheat, but I’m starting to think that I’m a fraud.

I literally cannot understand anything without any handholding. I got into a strong masters program, got to the top of my class, but through what? Constant handholding. I’m starting to think that i generally don’t have the mathematical aptitude for research unless I’m brute forcing ideas in my head for 7-8 hours per day with AI use.

Any advice I’d really appreciate it

In the 70s, Rota and Roman formalized the umbral calculus and, in the process, proved very deep results for the study of formal power series and essentially every polynomial sequences you can think of.

But since around the 2010s, there has been a flood of papers following the same template:

• take a known polynomial sequence,
• add one or two parameters,
• define a "new" family through a generating function,
• re-derive the same identities with the new parameters,
• publish.

Many of these papers cite Rota and Roman, even though none of the actual ideas of the classical umbral calculus are really being used.

The parameter accumulation has become so absurd that we now get outrageous names like:

• "r-Dowling-Lah polynomials"
• "lambda-Apostol-Euler polynomials"
• "Bell-Bernoulli polynomials of the first kind"
• "Chan-Chyan-Srivastava polynomials"
• "q-modified-Laguerre-Appell polynomials"
• "Degenerate Multi-Euler-Genocchi Polynomials"
• "r-truncated degenerate Stirling numbers of the second kind"
• "Gould-Hopper-Frobenius-Euler polynomials"

I'm curious how people actually view this literature.

How good is it for a layman to rediscover the core idea of a math field?

For some background, I’m in high school and I really love thinking about maths from different angles. Recently I had an idea where I imagined placing numbers on different 2D and 3D shapes, then analyzing how operations between the numbers could change depending on the shape itself. Like the geometry/topology of the shape affecting the relationships and operations.

Later I searched online and found out that something somewhat related already exists, called Cellular Homology. I don’t know if my idea is actually close to the real field or if I just stumbled onto a vague resemblance, but honestly I was really happy that I independently arrived at something even remotely connected to an actual area of mathematics.

So my question to math graduates or people deeply into mathematics is: how significant is it when someone independently rediscovers the core intuition behind an existing math field? Does this kind of thinking actually help in becoming better at mathematics, or is it more common than I think?

I’m not claiming I rediscovered the whole theory or anything huge just that it felt exciting to naturally arrive at a similar kind of idea.

Hello fellow mathematicians.

A friend and I are looking to go through Newman's proof of the Prime Number Theorem. Both of us have done complex analysis and analytic number theory at least at the level of Apostols book. BUT it's been a long time~8-10 years since we did complex analysis or analytic number theory.

So I'm looking for suggestions of books that give details of Newman's proof - ideally we'd use the same book to revise the prerequisites for understanding Newman's proof as well. I wouldn't mind a complex analysis or analytic number theory book. Preferably something thats not super terse.

This idea of going through the proof came about after we went through a proof that sum_{p<=y} 1/p > loglogy - 1 in Niven's book on Number theory - this proof uses simple and elementary arguments and is probably one of my all time favorite proofs now. It's thm 1.19 in the edition of the book I have.

I would be grateful for your suggestions.

I already asked this on r/learnprogramming but I didn't get any response:

In the intro to the book, they say there is auxiliary material related to automata and computability theory. The link provided is https://www.cs.princeton.edu/theory/complexity/ but there's no material there that I see. Hopefully it just moved, but I'd really like to find it.

Hi all, Im a junior in high school and I’ve been interested in math research and higher level math for a little while. I reached out to a math professor at a local university and he’s agreed to meet with me later in the week to talk about what I might be able to help him with this summer.

I know he has some papers on combinatorics and graph theory and specifically Ramsey numbers and that stuff.

Basically, if you were this guy and you agreed to meet with a random high schooler, what would make a good impression on you?

Neukirch uses a smaller or maybe lower-case(?) version of calligraphic O as a general notation for Dedekind domains while using the upper-case version for the integral closure of the former.

Is that just his notational idiosyncrasy, or is this a convention that others also follow? I was only aware that \mathcal{O}_{K} is usually used to denote the ring of integers of a number field K.

It seems hard to show the difference between curly O and curly o on the board, and I don't know how you would even produce the symbol on TeX, since \mathcal is upper case only.

Kind of an idle question, but I figure Neukirch's algebraic number theory book is influential enough that maybe others also use this weird letter?

I've set out on a mission to learn about chromatic homotopy theory, with the immediate goal of writing a thesis on the Chromatic Convergence Theorem, and a long term goal of getting to know the field and how it connects to other parts of stable homotopy and eventually derived algebraic geometry.

However, as I've read further and further, I've started to realize just how much stuff goes into this, and would therefore like to ask whether anyone knows a good bit about it, or would join me for a learning journey?

Note: This is clearly a pretty advanced subject, and so I will be spending over a year getting to fully know everything I need for that thesis and writing it up, then another year looking further, with the goal of seeing it in relation to AG/DAG.

If anyone could help me or wants to join, I'd very much appreciate it!

hello! from my understanding inner products generalize and standardize our intuitions of orthogonality, angels, and distances. I am a bit confused how I should interpret a space geometrically when its defined on non standard inner products. the standard inner product follows our intuition for orthogonality with a 90 degree angle, and I have a hard time imagining how this will change under different inner product definitions for the inner product space. I would imagine that it would cause us to treat a "90" degree angle as say 30 or 180.

note, from my understanding, the importance of the notion of 90 degree angle is that when something is 90 degrees in our normal everyday lives it means that we can increase one axis without effecting the other. similar to graphs, we can raise y without raising x. I can see the same underlying idea being applied for non 90 degree angels with shifted definitions

I would love help clearing these misunderstandings!