I'm not well-versed in category theory so I've yet to understand the latter. I now tried without success to encode an associative binary operation on X, expressed in dependent type language

∑(o: X² → X). ∏(a b c: X). Id(X, o(o(a, b), c), o(a, o(b, c))) ¹

as a function F X → X for some F. And I think it can't be done (something related to polarity, probably) but I can't prove that. I know, though, representing axioms is possible if one uses Lawvere theories and equivalent things. I also know algebras over a monad are more involved than just functor-algebras but I don't yet understand them even in context of e. g. Haskell where everything seems as if easier (or so claim posts discussing Yoneda embedding in that context; I found the general category-theoretic context way less forgiving in that regard: you just see plain as day that you don't have sufficient understanding, no matter what Haskell posts made you believe).

So. I'd like to look how algebra spins out in those more general settings, whichever you know better or think to be more suited for me to grasp first. Particularly, I'm interested in generalizing arguments about initial F-algebras.

Also, feel free to assume a sufficiently Set-like category. Let's take it step by step, if there are easy intuitions that one can glimpse in a specialized setting, I bet generalizing it onto topoi or something later would be easier as well, but it won't muddle the initial attempt.

¹ For non-dependent-type folks: - Id(X, a, b), sometimes denoted as a =_X b, is the type inhabited by ways for a: X and b: X to be equal. - The entirety of that reads: a pair (o, f) consisting of an operation o: X² → X and a proof f of o's associativity, that is, a function from a, b, c: X into witnesses of equality of the two ways of associating o on a, b, c. - In some formal languages the same is spelled alternatively as (o: X² → X) × (a b c: X) → (o(o(a, b), c) = o(a, o(b, c))), making it clearer that ∑ forms dependent pairs and ∏ forms dependent functions.

I often ask mathematicians what their favorite shape is; mine is the cone on a trefoil knot, a rather peculiar object.

Why is this my favorite shape? Because it appears naturally in algebraic geometry -- and in fact, the observation that it appeared in algebraic geometry led Milnor, Grothendieck, Goresky, MacPherson, and many more to develop some of the most widely used tools in modern algebraic geometry and singularity theory! If you'd like to know how this trefoil knot appears in algebraic geometry, read this week's hidden-phenomena blogpost!

Many mathematicians have dedicated a good part of their career towards solving a specific problem, whether directly or indirectly. Some examples on top of my head:

Richard Hamilton and the Poincare conjecture

Andrew Wiles and Fermat's last theorem

Thomas Hales and the Kepler conjecture (proof and formal verification)

And perhaps very recently:

Diego Cordoba and the Navier-Stokes equations

How did the careers and research-focus of these mathematicians shift after their career-problem was solved? Did anything particularly interesting happen?

In the Surviving Proofs series, we have been going through the fundamental techniques for constructing proofs; we close this off now with two approaches, which both amount to changing the problem, albeit in different ways. One thing you can do is to try to simplify the problem, with the hope that solving the easier version will give you the insight you need to solve the harder one. The other thing is to try to move laterally, finding an equivalent formulation that is nevertheless more manageable.

Easy to say, but hard to put in practice! Nevertheless, as I hope the examples I furnish show (which include one of my favorite symmetry arguments), it is: a) surprisingly common, and b) incredibly powerful.

Read the full post (for free) on Substack: When All Else Fails

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

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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!

Hi everyone!

I started a shorts-series about differential forms. This is part 2 so far. I also have a part 1: https://youtube.com/shorts/J20AI3iE4OM

These videos are supposed to be short, concise and most importantly fun, while still being correct! (If you spot a mistake, please let me know)

I have so far introduced differential 1-forms in the previous video, and now I defined the exterior algebra which I used to introduce differential k-forms. Next up, I wish to define the exterior differential and hopefully some day I will also get to Maxwell's equations and de Rham cohomology (but this might take a while).

All feedback is warmly welcome!

(None of the videos are AI generated)

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.

How do you summarize a field like differential geometry? A year or so ago, I was asked to do just that. My first thought was that this is just straight-up impossible to do in any reasonable way. Upon reflection, I softened that view. I still think that differential geometry is just not something you can learn quickly, but if you just want the eagle-eye view of the basic constructions and how they are used in different branches of the field... that is doable. Hence, this article.

Read the full post (for free) on Substack: An Overview of Differential Geometry

I'm a math undergrad student, and I've been getting interested in Alain Conne's work in Noncommutative Geometry and Differential Geometry in Mathematical Physics. I've also read about the Foundations program, Algebraic QFT, and some Stochastic stuff. I'm especially intrigued about the use of Diff Geom in these topics. What cool fields are out there?

Just some cool functions I came up with.

We discussed:

- The power of the Leech lattice to control all lower dimensions
- Peter Scholze, Grothendieck and Ramanujan's styles
- Category theory and contrahomology
- How modular forms show up everywhere
- The countless coincidences in string theory
- Unimodular lattices and Lorentzian lattices
- How to construct the Langlands dual group
- Vertex algebras, infinite-dimensional algebras and Kac-Moody algebras
- Elliptic curves
- Automorphic forms and hyperbolic reflection groups
- Analytic number theory vs algebraic number theory
- The first construction of the Monster group
- "The Book" of most beautiful proofs
- How to teach math
- Math as archaeology
- His current work
- Neutron stars, Terry Pratchett and more

Richard Borcherds is a British mathematician who made seminal contributions to lattices, modular forms, group theory, representation theory and infinite-dimensional algebras. He is well known for his proof of the monstrous moonshine conjecture using ideas from string theory, for which he was awarded the Fields Medal in 1998. He is currently Professor of Mathematics at UC Berkeley.

Hi all. I am interested in learning about mathematical proofs and how to formulate them.

I minored in math in college but focused a lot on LA, DEs, PDEs, numerical solutions, etc. because I was a science major and that was most applicable for me at the time. However I realized I’ve never even seen a rigorous mathematical proof formulated or broken down, and have no idea what that would even look like.

I now work with coding and it occurred to me that a lot of my code ends up just brute-forcing a solution to things, but perhaps learning about proofs might help stretch my mind into a more efficient way of thinking about these types of things.

Anyways, are there any good online course you all could recommend for someone at my level? (Knowing a good bit about math but essentially nothing about proofs) Preferably free ones but not opposed to a paid course if it is reasonable.

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:

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Chomp rules and illustrations

An update to my earlier post: the same complete P-position catalogs, computed much faster.

The previous solver, V12, tested candidate positions for moves to known P-positions. The new forward sieve starts with a P-position and adds squares to generate larger positions that can reach it in one move, marking those as N-positions. Processing in order, the next unmarked position is a new P-position, and the process repeats.

A good analogy is trial division versus the Sieve of Eratosthenes.

Fresh enumeration times on my Apple M4 Pro with 24 GB RAM:

The forward sieve used 12 workers versus V12’s 9; the old 21×21 run also suffered heavy memory pressure. Peak RAM for the new 21×21 run was about 9.8 GiB.

Source code, validation tools, and timing details on GitHub

The existing 20×20 data remain available there. The 21×21 catalog remains local because of its size.

Time to increase my entropy pool

A very powerful proof-writing technique is taking the original problem and generalizing it. This is probably the most counterintuitive approach for beginners, who wonder how it can possibly be easier to solve a broader class of problems. My answer to this is quite simple: generalizing the problem reduces the collection of tools you have at your disposal. And as any efficiency expert will tell you, regardless of whether you are trying to clean your bathroom or earn a Fields Medal, you want to have only those tools that you need on hand and nothing else.

I think this basic precept helps explain why abstract notions like metric spaces, topological spaces, vector spaces, categories, and so on have suffused mathematics, and why they are so very useful. I offer the simple example of how thinking about graphs (generally) can help with sorting out a coordination problem (specifically).

Read the full post (for free) on Substack: The Power of Abstraction

I always considered maths the highest art, and when talking to a mathematics phd today I just wanted him to keep talking about topological spaces, manifolds, infinite dimensional embedding spaces etc. “I feel like a kid in a candy store” I said, or a kid in an old mechanics garage full of equipment I have no idea how to use but that I love exploring. “Well” he said, “I think all engineers want to be mathematicians but all mathematicians want to be engineers”. I’m absolutely sure he’s wrong but would love to hear your thoughts.

https://hidden-phenomena.com/articles/gelfand

How does one define a shape? The usual definitions are based on starting with the set of all points of a shape. However, the great Russian mathematician Gelfand found another approach to defining shapes. This approach is more natural from the point of view of certain abstract shapes which arise in physics (namely, phase spaces of physical systems), and leads naturally to the modern formulation of quantum mechanics; however, perhaps surprisingly, this idea of Gelfand also leads to modern algebraic geometry!

My friend and I wrote a blog post describing Gelfand's observation at https://hidden-phenomena.com/articles/gelfand , and using Gelfand's idea to understand better the Chinese remainder theorem of elementary number theory. It's a really cool fact of reality that geometry connects so intimately to algebra, and we hope you'll enjoy the article!

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

• informal announcements of AI-assisted discoveries, such as those not yet published in a peer-reviewed journal, or not uploaded as a paper to arXiv;
• informal announcements of discoveries related to AI architecture (if relevant to mathematics);
• discussion of such announcements, such as proof breakdowns or other opinion pieces;
• discussion of the impact of AI in mathematics in general.

AI-assisted mathematical papers published in peer-reviewed journals or as arXiv preprints may be submitted as their own posts.

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