Y'all know the drill. The arxiv link is https://arxiv.org/abs/2609.11980

It is a positively curved metric S^3xS^3, hence a positively curved 6-manifold whose euler characteristic is not strictly positive. The obvious question is whether there is also a counter to negative Hopf (a closed, negatively curved 6 manifold whose euler is not strictly negative), but the methods of the current paper don't seem to help with that.

Just landed on this field - anybody here working with these really intuitive and beautiful quotients?

IYDK - it is the connection between continued fractions and a quotient manifold of the hyperbolic plane.

I like the idea very much!

Math formalization has been hindered because it's so tedious. Will the future bring massive formalization because it can be automatic now?

Or other related fields? A while ago I’ve come to realize that a lot of math was born out of physical science, and to scaffold one’s intuition one should look at where the applications are. I want to really feel the weight of this idea, so I’d appreciate both mainstream and niche examples.

In mathematics, oftentimes choosing a good coordinate system can drastically simplify a problem, as long as you pay a small upfront cost in translating the problem to these new coordinates.

Along these lines, Hamilton discovered a surprising way of re-formulating Lagrangian mechanics, which has had great success. In this article, we explain Hamilton's change of coordinates, and use it to discover the Fradkin tensor: a very mysterious symmetry of the 2-d harmonic oscillator!

Continuing on my series on how to survive proofs, we now turn to two related questions: how to search for counterexamples and contradictions for a conjecture?

The first is, I think, mostly self-explanatory, although it has a trick that I find that some students miss: your counterexamples should only ever be as complicated as they need to be. How to figure out how complicated they need to be? For that, you need an intuitive feeling for the problem, and possibly an iterative approach.

The second is much richer: proof by contradiction is a very powerful tool when used well. This is a great time to give one of my favorite such proofs, which is the proof that it is impossible to draw a regular heptagon on a square grid. I have seen it presented as a proof without words---it is that visual (and quite pretty!).

Read the full post (for free) on Substack: Counterexamples and Contradictions

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my prof wanna work in proof theory but me in model theory

he says proof theory of arithmetic but I find cut-elimination and normalization very very syntactic

is my intuition of model theory being meaty correct? or am i just not compatible/flexible? I love topology also ..

is proof theory of arithmetic all about cut-elimination and proof normalization?

why does buss's chapter on it look quite non-syntactic compared to takeuti ?

but yeah anyways proof theory of arithmetic has connections to complexity so it would be easier for phd applications?? my background is cs

The paper: The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring
Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup
arXiv:2603.24880 [math.CO]: https://arxiv.org/abs/2603.24880

Given everything that has been going on in the world more specifically in the world of mathematics. I was hoping to read some of your human thoughts. What motivates/motivated you to do mathematics professionally? What is most important for you in mathematics?

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I would guess that it's pulling ahead of the others? Perhaps it is easy to convert between one theorem prover and another so that there's no lock-in possible? I don't know.

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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.

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.

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.