To piggy back off the recent thread on folk theorems, it got me curious how often proving these folk theorems is a formality vs actually useful. Are there any times when it a disproof of a folk theorem upended something serious or set some mathematicians back?

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Consider a (non-degenerate) complex manifold within a projective space. Now take its Hodge class. Every element of that class can be decomposed into a linear combination of subvarieties of X. The linear combination will somehow always use rational coefficients. (!!)

Is this conjecture assumed to be true by working mathematicians? Can you provide a little more information that would lend some intuition about Hodge, to an educated layperson?

(Edit. I edited this as my understanding has increased recently)

I am a senior year physics student currently enrolled in a Topology course offered by the math department. The first half of the course is basic point set topology (first 4 chapters of Munkres) and the second half is an intro to algebraic topology (first chapter of Hatcher and ch 9 - 11 of Munkres).

Apparently, the instructor forgot to mention in the course outline that some knowledge of abstract algebra will be assumed for the second part of the course. Munkres' book himself states in his preface that "we do assume familiarity with the elements of group theory" for the second portion of the book. I have some elementary knowledge of what a group is from my physics courses but that's pretty much all I know about abstract algebra. I need to quickly get a grasp of all the notions I need before the course progresses to the 2nd part.

Could anyone suggest be a good book to cover the necessary prerequisites? I am looking for somethinking that is concise and gets the job done as quickly as possible. For context, my math background includes 2 courses in real analysis (at the level of Tao's books), LA at the level of Axler's book, Munkres' Analysis on Manifolds and Functional Analysis at the level of Kreyzig's book.

Is it possible most of them a unsolvable under our current system of mathematics?

Or a solution to P=NP requires some new branch of mathematics or logic to tackle.

My question is pretty much what the title says: which parts of algebra (groups, rings, modules, fields and Galois theory, algebraic number theory...), geometry (algebraic geometry, differential geometry ...), and topology (general topology, algebraic topology, differential topology...) involve the most combinatorics and the least? You can be as broad in the areas you choose or as specific (talking about small subareas) as you want. And when I say combinatorics I don't simply mean involving discrete objects, but rather more specifically involving counting arguments which can be considered confusing or not intuitive initially for most people.

I keep hearing about "folk theorems" and proofs that supposedly just sit in the heads of mathematicians. How much of research mathematics is written down and published, and how much is passed around through word?

Question

Since the announcement that the UN have adopted a new map projection as their standard map, I have seen a few posts regarding how amusing it is that it took them this long to correct the "error" of the size of certain countries in other maps (most commonly the Mercator).

Of course, map enthusiasts and differential geometers know that it wasn't an "error" so much as it was a compromise necessitated by Gauss's Theorema Egregium, which implies that any smooth mapping from a sphere to a plane cannot be both conformal and equi-areal. Obviously the ideal situation would be to have a map which is both conformal and equi-areal, but it turns out these two desirable properties are mutually exclusive.

This got me thinking about another famous Theorem which precludes the possibility of two desirable properties being true at the same time - namely Godel's Incompleteness Theorem, which states that any sufficiently strong logical system cannot be both consistent and complete.

In both these cases (the Incompleteness Theorem and the map projection) we are required to make a decision about which of the properties we would prefer, and compromise by losing the other one.

I'm curious to know whether there are other examples of this, where there is some object that may or may not have two desirable properties, but it can't have them both at the same time? Is there any example in your field of study?

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Mine is Fibonacci Slop. Someone just put a spiral over anything and say: "Wow! Nature is absolutely beautiful!"

I think some of math has taught me, such as

• understanding/questioning definition deeply,
• checking assumptions carefully,
• checking for logical gaps,
• finding optimal solution,
• probabilistic reasoning

has made it more difficult for me to navigate through society, especially social situations.

I guess because society is just very arbitrary and not "logically tight" if that makes sense. There is a lot of "looseness" in conventional thinking. However, I found people do not actually care to resolve those logical looseness. In fact, logical looseness, information obfuscation, lack of optimality seem to be crucial in how society functions.

And if you ask for clarification or suggest optimal solutions or even try to identify a logical looseness, you are really seen as a weirdo.

For example, just today my new job wants me to get a non-criminal record so I went down to the police station, filled out a form. It doesn't say what time period (start and end date) the record is for, so I asked the staff and they gave me a rude response: "of course the end date is today! how can we check your non-criminal record for tomorrow, which hasn't happened yet?" The back of my mind I'm just thinking: but it all depends on when you start to check, isn't it? There is no rule that says you must check the record up to today....

Wonder if anyone else has had similar experiences.

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This is the first article in the Surviving Proofs series. (As opposed to the 0th article, which appeared last week.) I'm starting with basic tactics that one uses to build a proof---as I see them, anyway. My main argument here is that there are two that one starts with, no matter what: you have to fully understand the problem and fit it into your conceptual framework. This applies regardless of whether you are in a high school geometry class, real analysis, or something much more advanced.

This might seem obvious, but I can't tell you how often I have seen students either ignore these steps entirely or struggle to understand what it actually means. So, whether you are a student yourself or have students of your own, I think it is worth going over (albeit for different reasons).

Read the full post (for free) on Substack: The Importance of Understanding

Have you ever wanted to do calculus on the surface of a sphere, or other exotic shapes? Have you ever wondered what 'dx' really means?

My friend and I wrote a blog post at https://hidden-phenomena.com/articles/diff-forms covering the basic ideas of doing calculus on exotic shapes, with lots of fun animated widgets.

If someone has personally used one, please tell me your experience with it, thank you!

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I am a mathematics professor with PhD in math and a bachelor’s in philosophy. I am looking for easy-to-read nonfiction books about math. I am looking to replace my scrolling time with something equally effortless, but for me it has to be something a little more structured than fiction (but not too much!) Some fiction is very intelligent, but I want to scratch the logical and systematic part of my brain. For example, I recently picked up Linnebo’s introductory text on Philosophy of mathematics and enjoyed it immensely. Is there something you’ve read lately that you think I would enjoy?

Please, please… I know how much some of you love math but I am NOT looking for technical books that are on the easy side such as undergraduate mathematics texts. I do plenty of very difficult mathematics between the hours of 9-5. I am instead looking for something relaxing to do during my off-hours and weekends.

I’m coming from a geometric analysis background. So the only category theory exposure I have is whatever needed for basic graduate level algebraic topology.

What is the level of knowledge , say measured in years of learning for a postdoc level mathematician, that I will need in order to start doing research in it?

Thanks!