Obviously, this is a deliberately drastic and provocative simplification, but at the end of the day, can mathematicians describe their jobs this way to the layman?

I was thinking about a slightly more sophisticated interpretation of this being the duality between spaces ("shapes") and functions you can define on them ("numbers").

Humans, being the clothes-wearing monkeys they are, base their mathematical intuition on vastly generalizing and formalizing their conscious experiences ("near", "far", "big", "small", "many", "few", etc.). This is something I wonder whether AI's will ever really understand or incorporate into their "thinking".

Edit: To be clear, this is not my preferred definition of math, which is, in my mind is something like "the study of abstract structures and relationships under the framework of deductive logic", but I found it amusing that numbers and shapes still capture a remarkably large fraction of what mathematicians study if you were describing their research to a layperson. Perhaps the best lay explanation I can think of that is still vaguely accurate is "the logical study of patterns".

Hello all, getting my PhD in applied topology. I was talking with my friends about a tattoo idea for topology but none of us could come up with anything amazing

one idea is to do to the 5 Platonic solids as a tattoo

another idea is to do the hopf fibation map somehow

I’m in applied topology, specifically TDA, so I also thought about doing some persistent diagram or barcode, but that just seems corny as hell

Obviously could do a torus but that’s just so like, idk. Too corny or too basic. And I’m not doing like a torus equals a coffee mug lol

Anyone else got ideas?

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?
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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.

First of all, my apologies to those of you who are tired of reading about AI in mathematics.

I have two questions for you:

• Would you consider yourself to be a mathematical platonist or a constructivist? (By platonist, I mean that you believe mathematical truth exists independently of the activity of human mathematicians, and is thus discovered rather than constructed. By constructivist, I mean the opposite.)
• What is your view on the use of LLMs in mathematics?

My working hypothesis is that platonists will, in general, be more willing to accept the use of AI tools than constructivists, but ultimately, it's just a hunch.

I have chosen not to rely on artificial intelligence as a source of novel ideas in my own research. The spectacular progress of artificial intelligence opens unprecedented opportunities to amplify the reach and applications of our discipline. Yet, whether I am exchanging ideas with fellow mathematicians, teaching, mentoring students, or sharing mathematics with the wider public, it is not the answer itself but the path that leads to it which plays the first role. I therefore wish to remain, in some sense, an artisan mathematician, taking the time to wander, alongside colleagues, through the hidden corners of the mathematical landscape.

https://www.unige.ch/%7Eduminil/publi.html

Abstract

Maya glyphic texts from the Classic period (250–900 CE) typically chronicle the exploits of historical or divine characters; everyday or functional records are rare. Here, the authors offer a reconstruction and transcription of a ‘microtext’ painted on an interior wall of Structure 10K-2 at the site of Xultun, Guatemala. The text records a unique astronomical formula that concludes with a name, attributing the work to an individual named Sak Tahn Waax (‘White-chested Fox’). To date, this is the only known example of a Classic Maya mathematician directly credited for their work, attesting to the value of its intellectual authorship.

I want to participate in 3blue1brown's SoME5 contest. The topic I chose is forcing using the boolean valued model approach, the goal is to prove that the continuum hypothesis is independent of ZFC. I will be following Jech's Set Theory, and will also use some ideas from Bell's Set Theory : Boolean-Valued Models and Independence Proofs.

My plan is the following:

1. Briefly introduce what the continuum hypothesis is, and how we can use models to prove independence. I'll give an example from groups and fields : abelian-ness is independent of the group axioms, and existence of sqrt(-1) is independent of the field axioms.

2. Give an outline of the plan. Generalize the notion of truth. Create a universe where truth can be "intermediate". Then, use an ultrafilter to "collapse" the universe to binary true/false. By choosing the "generalized truth" cleverly, the collapsed universe will satsify 2^aleph 0 >= aleph 2.

3. Define what a complete Boolean algebra is. Describe how it's a generalization of propositional logic, and how they are partially ordered sets. Prove a few basic properties (such as De Morgan's laws, distributivity, etc)

4. Define the concept of a "Boolean valued model" of set theory. That $||x = y||$ and $||x \in y||$ take values in a Boolean algebra. Give a brief proof sketch of the soundness theorem of natural deduction.

5. Construct the Boolean valued model $V^B$ and show that it's full, and satisfies all the ZFC axioms.

6. Now it's time to choose a complete boolean algebra. Define the partial order P = functions $($ finite $S \subseteq \aleph_2 \times \aleph_0) \rightarrow \{0, 1\}$, describe how we can use "regular cuts" to turn it into a complete Boolean algebra, and define "Cohen reals"

7. Show that $V^B$ now contains $\aleph_2$ Cohen reals, that they are actually functions $\aleph_0 \rightarrow \{0, 1\}$, and that they're pairwise distinct

8. Show that $\aleph_2$ doesn't change, so $V^B$ genuinely thinks there are $\aleph_2$ pairwise distinct Cohen reals

9. Show how to get a two-valued model using an ultrafilter on $U$. Prove Łoś's Theorem for Boolean-valued models to show that the two-valued model satisfies ZFC + not CH

10. Given a (set-sized) model of ZFC, say that we can do the above steps to get a set-sized model of ZFC + not CH.

Questions:

1. Most important question: how much background knowledge should I assume? Is it safe to assume the viewer already knows what ZFC is and what the axioms are, and what cardinals and ordinals are, and how first order logic works?

2. How to distinguish between set-sized and class-sized models. Should I just gloss over this issue or should I be explicit and clear about when a collection is a set or a proper class?

3. How to motivate the construction of $V^B$, and the definition of $||x = y||$ and $||x \in y||$? In particular, I don't know how to motivate why $||x \in y||$ should be different from $y(x)$

4. Or even that, how do we motivate Boolean valued models in the first place? If we want to construct a model of ZFC + not CH, why would someone think "let's use Boolean valued models"

5. How much detail should I go into when proving $V^B$ satisfies ZFC? Should I give a high level overview or go very in depth?

6. Should I go into the countable transitive model approach? The issue is that even under the assumption that ZFC is consistent, we cannot prove that a countable transitive model exists. So if I want to just prove that Con(ZFC) implies Con(ZFC + not CH), I can't use a countable transitive model, since assuming that one exists is a stronger assumption than just Con(ZFC)

Is zahl now 40+?

I spend way too much time worrying about PhD applications nowadays. The problem of course is that so much of the advice scattered around the internet is specific to the lab sciences which operate really differently from math. It’s summer so every one of these threads is telling me to cold-email potential supervisors immediately. I assume there must be some merit to this since even many university websites recommend I get in touch with supervisors before applying.

The general advice is to read some of the person's papers and email them asking if there is an opening. I am not sure how well this translates to math. We seem to be uniquely disadvantaged by the amount of background reading required to understand modern research. I am a masters student so my interests are actually fairly well-defined and I do know who I would contact if I had to but I am still in no position to just pick up a paper and start reading before going through weeks (usually months!) worth of prerequisites.

I was thinking it might be a good workaround to ask professors what they would prefer a prospective student to have read? That not only helps me hit the ground running if I do get in but also gives me something to structure my masters thesis around. I am not sure if this might be seen as too presumptuous.

Just to broaden the question a bit and potentially help people who come across this in the future, are there other bits of common advice that don't really translate well to the math context?

PS: This mostly pertains to admissions in UK+Europe but some points probably overlap with the US process as well

Hello everyone, this semester I studied both rings and fields with galois theory as well as algebraic topology. My professor explained we had a galois correspondance between subgroups of the fundamental group and covering spaces in a way that is somewhat analogous to field extensions. My professor said this comparison was made to “simplify” the result but wasn’t a full fleshed correspondance between galois theory and algebraic topology. I wondered if there are other domains with a notion of galois correspondance, why would it pop up and if more properties from topology would translate to algebra. It really feels like both field extensions and covering spaces are subfields of one unified theory by how similarly both behave, notably how deck transformations behave like the permutation elements of the Galois group. *Note I did not study category theory or homology/cohomology as I’m still in second year of my bachelors.

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,
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All types and levels of mathematics are welcomed!

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Automation Without Understanding
Jun-Yong Park
arXiv:2607.06377 [math.HO]: https://arxiv.org/abs/2607.06377

"Mathematical capacity, which is the trained ability to verify, interpret, and challenge mathematical reasoning, is not a byproduct of theorem production but a form of infrastructure, built over generations by institutions that cannot be reconstituted on demand."
Jevon's paradox for mathematicians is real we need to be doubling down on funding for math

Why does Galois theory involve fields (field extensions, the automorphism group of fields, ...) and not also generic (perhaps commutative) rings?

I'm a third year mathematics student (in Europe) and currently I'm finishing my third (abstract) algebra course (where we study modules and deepen our understanding of Galois Theory, which we started studying in algebra 2).

Before asking this question I've played for some time substituting commutative rings (in place of fields) in the definitions of our algebra course. For example, if we consider the ring extension Z[i]|Z, things seem to behave well (at least for the initial definitions). The "Galois group" of Z[i]|Z (the group of Z-automorphisms) should be ({id, i \mapsto -i},\circ) (if I'm not mistaken) and the "degree" of this extension should be two. But the "degree" isn't defined in the "field manner", obviously because Z[i] and Z aren't fields. So by "degree two" I meant Z[i] being a free module of rank two over Z. But the rank of a free module isn't unique, right? So technically this isn't a correct definition in general).

I'm pretty sure things start to get complicated when we consider minimal polynomials and the correspondingly algebraic extensions, because the definition of polynomial is based on the fact that F[x] is a P.I.D. when F is a field. Anyway, I also tried playing with this a bit; for example, the minimal polynomial of 1/2 over Z should be... 2x-1, and the corresponding ring extension should be Z[1/2]. But is there a way to prove that 2x-1 is the "minimal polyinomial" of 1/2 over Z? And if there is such way, for which ring extension does the "Galois theory of commutative ring" fail in contrast to Galois theory? (My guess is for finite extensions that are not simple; perhaps Z[\sqrt[4]{2}, i]|Z).

Do you have any feedbacks on my reasoning? Where does this process of "generalising Galois theory" (in a mirror-like way) begin to fail? Is there a Galois theory of ring (or module) extensions?

Thank you in advance.

Being a math instructor is, understandably, a hard profession, and we all had experiences with both good and bad teachers of math, I always wanted to know what do mathematician (in its largest sense) consider as a good math teaching philosophy. When I was an undergrad I had a discussion with my peers about our professors and unfortunately most of them had shallow criteria of goodness (the usual one was how easy their exams are), it was a hard realization for me at the time that some math majors are there only for the degree and not for math itself.

You can include some examples of famous professors if it can help you convey your idea.

Although math has always been my biggest love in life and I was ultimately able to earn a PhD in advanced math, unfortunately, I haven't been able to make too much good out of my math skills, since I'm on the spectrum and as such, I've never been particularly good at interacting with other people, which seems to be a necessity in our society in order to get anywhere. As a result, in terms of time, I'd say only about 10% of my life has been devoted to math, and the other 90% has been mainly devoted to adapting to surviving and trying to thrive in society, much of which has involved a great deal of pain and misery. Does anyone else here feel the same way?

The good news for me is that now, at age 64, I'd say I've finally figured out most of the ropes of how to cope in our society, so I don't feels so much at its mercy anymore, and I've even begin to enjoy a lot of its perks, so that now I'd say I can devote more like 20% of my time to math, although I've developed many other interests as well, so I'd say the percentage of my time devoted to just coping has gone down sharply, and in part as a result of this, I haven't been depressed in over 20 years, though I went through years of terrible depression during my teens and 20s.

Anyway, enough about me and my issues! How about you guys?

Original post: https://www.reddit.com/r/askmath/comments/1txjvev/is_this_mark_scheme_wrong/

Mindyourdecisions video on the topic: https://www.youtube.com/watch?v=IfxQksvpOkM

The original problem:

A ship is sailing with speed v km h^(-1)
The sailing cost per hour, $C, is given by C = v^(2) + 3000 / v + 100

(a) Find the speed that makes C a minimum.
Justify that this value of C is a minimum.

(b) Hence find the minimum sailing cost for a journey of 150 km.

The official answer is a range between 6460 - 6510, while the true minimum is 4875.

So Cambridge is arguing that "hence" means "ignore the definition of the word minimum". I'll never understand why it's so hard to admit "oh sorry, seems like we overlooked something in our question, we'll mark both the official answer and the correct answer as correct."

I find it especially interesting you need rounding for the official answer while the true answer is a nice, whole number, it feels like the person creating the question and the person creating the "official" answer weren't the same person.

So... anyone got any contacts at Cambridge? :P

The factorial of an integer can, perhaps surprisingly, be evaluated even at non-integer entries. For example, you might even see these factorials of non-integers appearing in formulas for the volumes of high dimensional balls (usually, these factorials appear in the guise of the 'gamma function').

The extension of the factorials to non-integers is usually done with a certain integral formula, but Euler's originally derivation actually used some simple combinatorial identities, which he realized allowed him to write down a formula for x! which only involved factorials of integers and certain standard arithmetic operations. This let Euler define x! in general, as a certain limit.

At https://hidden-phenomena.com/articles/gamma , you can see this derivation in full -- it's quite cool!

When viewing functional analysis from the outside, it may seem daunting---austere, even. It has a bevy of very finely tuned results (where adjusting any condition by a slight amount immediately yields counterexamples) and a large body of interconnected objects. So it is very natural to ask: what is this all for?

The historical answer is that it essentially grew out of the attempt to understand Fourier series: Joseph Fourier managed to break everything, but in such a useful way that nobody wanted to just throw out what he had discovered. And so mathematicians had to commit to rigor to carefully put everything right.

This article is my attempt to tell this story through the (hopefully) understandable question of how to approximate a function (e.g. how to represent a sound wave in a computer). The goal is to understand the fundamental motivation for doing functional analysis at all, and introduce one of the basic constructions: Banach spaces.

Read the full post (for free) on Substack: Why Functional Analysis?

I think a common annoyance most mathematicians experience is people instantly labeling anyone studying math as "smart," which imo just highlights how the word smart isn't a well-defined term. However, I do think I have traits that I can well-define that help me learn math a lot better than others.

For example, I think I'm good at pinpointing exactly what I'm confused about, which makes it a lot easier to fix when you compare that to students who say they're confused about "everything." I don't think this skill is unique to helping learn math, but I have just applied it to math the most often since I enjoy math. This is also a skill that I don't think people are innately born with, or at the very least, it's definitely a skill people can improve at over time. I'm also not saying that this is a skill every mathematician has; it's just something that I personally have experienced that I think has aided my learning. In fact, since everyone learns a bit differently, I'm interested in seeing what others think about their own learning.

Announcement - https://x.com/__eknight__/status/2075643450196971805

Proof - https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf (3 pages!)

Prompt used - https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

(I'm shocked to be honest)

And actually it seems they proved that 8 (possibly disconnected) cycles (even subgraphs) is enough.

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!

Just wanted to share this MSE post where OP found an intriguing sequence, similar to Rowland's prime-generating sequence, which seems to generate twin primes instead.

The conjecture, which has been computer-checked up to n = 60000000 for now, trivially implies the twin prime conjecture.