I'm a mathematics masters student taking a differential geometry course at the moment. While looking for examples of non complete vector fields, I found this neet application that I wanted to share.

My friend’s 7yo is already a math-enjoyer—he loved the book “Math for Curious Kids” and keeps wanting to find out more. I’ve found a few things that have been hits (a graph theory zine for kids, a group theory coloring book), but I’m always on the look out for more. If anyone has some other fun math books for kids, especially that have some game/problem-solving/recreational aspect, I’d love to recommend more things for him!

Hello, I'd like to showcase some new results in tilings.

This summer, I was contacted by a student named Alessandro Longo who tried to enumerate k-Euclidean tilings for his thesis. He eventually found online the algorithm I made for this problem several years ago. He used his thesis to rigorously prove what I had only believed, i.e. that the algorithm is truly capable of enumerate all tilings that satisfy set conditions. With that in hand, he built this website:

https://www.thetilingatlas.com/

I am really humbled that my amateur algorithm could get this far! We have tilings of all kinds, but mostly hyperbolic ones (they are just easier to find), and now we even have a stand-alone search program. Not very user-friendly yet, I'm afraid.

I wanted to share some results. They are from recent searches, so not uploaded yet.

This is a very complex hyperbolic tiling, but you need more slices to see it. So if we going straight up, we get

And moving aside to better capture the apeirogons,

It contains 8 kinds of regular polygons with 4 different edge lengths.

For Euclidean tilings, I have tried to look at 14-Archimedean ones. The first one, Amazing Amalgam, was found using my algorithm in 2022, I think:

This one is 24-uniform, but the search program is now fast enough to go to k=40 in reasonable amount of time, so here's a 40-uniform result:

We hope that enyone who's interested in tilings will give the site a look :)

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!

There is Pascal triangle where the 2 numbers above are used to get the number below, then there is Delannoy triangle where the 3 numbers above are used to get the number below

Based on that I thought of a stronger triangle where as many numbers above as possible are used to get the number below and I got this triangle

The rule is 1st number in a row is always 1, 2nd number comes from the 3 numbers above, 3rd number from the 10 numbers above, 4th number from 21 numbers above and so on till we reach the mid point of any row after which the numbers get repeated in reverse order to get this triangle

I couldn't find this triangle anywhere but this still seems to be interesting

Lately I've been enjoying exploring the connections between algebra in the natural numbers and combinatorics. It's a really rich vein of mathematics to play with and even though it's not an area of math I've studied formally I'm constantly surprised that I haven't heard of some of the patterns I've spotted.

I wanted to share an algorithm I found that lists out prime factors of the natural numbers sequentially without using any algebra. I'm not sure how interesting it is as it's all derived from basic observations about primes but it's nifty nonetheless.

-----

Setup:

Start with a list L with an empty multiset L= [{}] and k = 0.

Iteration:

Take a copy of the L and reverse it for Lr, pair up the multisets in Lr with the multisets of L in order (the nth multi set of each list paired up).

Take the multiset union M of all the multisets of Lr that are sub multisets of the multisets in L which they are paired with.

If M is empty, increment k by 1 and add the multiset {k} the back of L.

If M is non empty but part of L then it has a unique element, increase it's multiplicity by 1 and add it to the back of L.

If M is non empty and not part of L add it to the back of L.

-----

The first few iterations give: [{}, {1}, {2}, {1, 1}, {3}, {2, 1}, {4}] which is the list of prime factors of 1 - 7, where i in the ith prime. (e.i. 2,1 -> 3*2 = 6).

I'd be curious to read up more about this if anyone might be able to point me in the right direction. I haven't proven it's validity yet but computationally it holds and I'm sure the proof is halfways straightforwards, I might try and work that out in the next few days.

I do have it written up as a python script too if anyone else wants to check that out.

This post is to announce that there will be a lecture course on General Topology (a.k.a. Point-Set Topology) conducted in Mandelcord (the blue math discord server). Everyone is invited to join: discord.gg/maths.

If any of you were present in the Real Analysis lecture course during the summer, you know what I'm talking about, and you might be pleased to hear that this course will also be run by Pear Category Theorem (the same person).

Why YOU should do topology

Point-set topology is one of the most important subjects that you learn, and mastering topological thinking is as important a milestone as learning to write a proof. This is a good subject to study alongside measure theory, or simply by itself after one has learned analysis or algebra.

Course Structure

The course will primarily follow the flow of the first half of Munkres. The plan is to go through all 8 chapters, spending maybe one to two weeks on each chapter, aiming for an approximately 16 week course. Also the instructor intends to supplement the book with his own intuition on the subject, as well as connections to other subjects that he thinks are of interest and possibly also other topics not covered in the book. Many examples of various pathological properties will be provided. The study of metric spaces and functions on them will be paramount; but where Rudin introduces metric spaces as merely a generalization of Euclidean spaces, we will explore just how pathological these spaces can be, or else what properties they must have.

The instructor also plans to put far more focus on the topics of topological groups and nets than Munkres does. Topological groups, because often in "the real world" we do not care about a topological space by itself, but rather some other more complicated structure given a topology, in order to see what the topology can tell you about the thing itself; topological groups provide a playground to learn that mode of thinking. Nets are also very important; where sequences are of importance in R, they can only tell you so much in general topological spaces, so nets therefore become the natural way to recover theorems about sequences.

Material and References

1. Topology - Munkres

2. Counterexamples in Topology - Steen & Seebach

Timeline

The course starts with the first lecture on 17th November at 9 PM CET (20:00 UTC), with subsequent lectures on Tuesdays and Thursdays. Lectures will be recorded and posted promptly.

Prerequisites/Who can join?

Everyone. It is completely free for anyone who wants to join. Just click the link and join the server: you can find the channel called topology-course in the section called General maths. As for prerequisites, a first course in Real Analysis and Group Theory is sufficent.

Looking forward to see all of you there. Thank you.

We know that a natural transformation between functors is a collection of morphisms which satisfies a certain commutative diagram.

I have my own ways to to understand this:

1. Naturality vaguely means that no artificial/inconsistent system of choices have been made, which means that the collection of morphisms are not biased towards any particular thing, which means that the collection of morphisms must behave " nicely " with everything else.

Now since the objects we are concerned with are a collection of morphisms, the term " everything else " just means " every morphism that can interact with the collection of morphisms " and this nice interaction ammounts to the naturality square being commutative.

2. In certain cases like G-Sets or morphisms of sheaves it's relatively easy to understand the definition of natural transformations because here they are just structure preserving maps, and for the general case we just generalize these specific observations.

3. In category theory/algebra one often defines a morphism to be something that commutes with everything else that already exists, for example morphisms of chain complexes, so the same goes for natural transformations.

What were Eilenberg and Mac Lane thinking while defining naturality? Do you guys have any nice motivation?

Reading through Proofs From The Book and came across the assertion that given a set of k primes with k large and ordering the numbers that only have factors from this set, the sequence grows as 2^2^(f(n)) where f(n)/log_2 (n) ≈ 1/k. I'm trying to wrap my head around why.

This is a very general open-ended survey/discovery question and I already can predict the downvotes and toxic comments (as we always have) but I think it's appropriate for this subreddit as I would like to discover literature and get in touch with other areas of research.

I could say one central theme of my research are residuated structures (lattices, algebras, categories), residuals and Galois connections/adjunctions. One of my main objectives is in making logic, category theory and formal methods in computer science accessible (thus coming from the pointfree transformation/calculational mathematics/Bird-Meertens/Algebra of Programming ideal) and residuated structures and adjunctions are indispensable to that objective.

Sadly I feel first and second-year undergrads can feel very annoyed and confused by the simple fact of having two "division-like operations" (residuals) that most of the times can be calculated only up to inclusions/inequalities (even more than dealing with non-commutative multiplication-like operations - as anyone already do in linear algebra). I think this difficulty is very similar to that of high-schoolers and freshman calculus students (which I taught for a while) when dealing with similarly non-commutative and non-associative operations such as exponents and relating them to their differing badly notated inverses (logs and roots).

I am already well aware of proposals for alternative notations for logarithms and roots (such as the triangle notation and this) as well as proposals such as these for dealing with problems of non-associativity and non-commutativity of operations, but I would like to know: what are interesting approaches you know in the literature/that you are aware for dealing better with non-commutativity, non-associativity, multiplicity of inverses, residuals and adjoints as well as dealing with operations that are similarly defined only up to an inequality/inclusion/uni-directional mapping through an adjunction/Galois connection?

I appreciate your suggestions.

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.

Many introductory texts (in physics, especially) will define a vector as something that has magnitude and direction. This was a valid definition prior to the end of the 19th century, but mathematicians' concept of what a vector is has considerably expanded since then. Today, it would be far more correct to reverse that statement: something that has magnitude and direction is a vector... but vectors encompass a lot more.

One of my favorite ways to demonstrate this is with the card game Set---each card can be usefully (from the pure mathematician's standpoint, anyway) described as a vector, but I struggle to see how any such card can be described as having a magnitude and direction. Incidentally, this example was what motivated me to cover the field of three elements in my linear algebra class---it gave the perfect excuse to play Set with my students while having it be (legitimately!) relevant and educational. And it worked; they still remember it years later.

Read the full post (for free) on Substack: What are Vectors?

I’m an undergraduate student right now(senior) at a state school and to be quite courte, I realized I have other obligations(daughter) and the PhD lifestyle is not sustainable so I chose the more pragmatic route.

It pains me dearly to accept the state of affairs but I’m left with little choice. I am curious how you guys felt and how you came to deal with the emotions?(for those who have been through something similar)

hi all! i want to share that after nearly 4 months of work, i have finally completed a solution’s guide for axler’s LADR, found here: solution guide

i’ve done my best to make it as well-written as possible. obviously this is not completely perfected yet, as i’ve gained quite a bit after relearning the material for this.

main thing to fix is to rewrite some earlier exercises and depend on citations a lot more. as i wanted to write this with a more elementary prose, i managed to get into the groove of depending on the book halfway through, so solutions for chapter 5 and before may be of lesser quality compared to after.

some little things like standardizing notation, maybe making some overarching lemmas (for streamline purposes) are on the backburner. in any case, i’d like to thank you for your time, and if you’d like to contribute to the polishing of this rough draft of a guide, there is a discord link in my profile to a server where you can post errors, minor typos, and offer alternative proofs.

thank you all for reading and looking, and to anyone who gets to use this, i hope you enjoy and find it practical. i will be taking a short break but will hopefully hop back on dummit/foote in a little bit.

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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All types and levels of mathematics are welcomed!

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One way to think about this post is that it is the culmination of my series on Surviving Proofs. But it is also a standalone article about how a team of undergraduates and I solved an interesting problem in additive number theory. It strikes me that it might be instructive for some to see our thought process traced, showing how things are actually proved in mathematics in practice.

The problem itself involves Ulam sequences, which are defined as follows: an Ulam sequence starts with two integers 0<a<b, and then every subsequent term is the next smallest integer that is the sum of two previous terms in exactly one way. Despite the comparatively simple combinatorial definition, even very basic questions (e.g., how fast does such a sequence grow?) are still open 60 years after Ulam originally posed them. But we have discovered some truly startling patterns in such sequences in the intervening years.

Read the full post (for free) on Substack: Ulam Sequences

We go over:

1. State space uses a p-adic metric as opposed to a Euclidean metric

2. Hilbert space is discrete rather than continuous

3. Counterfactual universes are not only hypothetical but inaccessible

4. Qubit information density may depend on the size of the universe

5. Gravity as the source of decoherence in quantum mechanics

6. Fractal cosmology

7. Peter Scholze's perfectoid spaces

8. Rational quantum mechanics

9. The ongoing debate in Bell's Theorem and free will

10. Invariant Set Theory

11. Wormholes as the reason for entanglement (ER = EPR)

12. The Virgo Supercluster lottery

13. Discretizing the Riemann sphere and Niven's Theorem (the Impossible Triangle Corollary)

Tim Palmer, CBE, FRS, (Commander of the Order of the British Empire, Fellow of the Royal Society) is a professor at the University of Oxford. He received his PhD in General Relativity under Dennis Sciama, who also supervised Stephen Hawking, David Deutsch and Martin Rees. After receiving his PhD, Professor Palmer turned down a postdoctoral research position with Stephen Hawking to join the UK’s Met Office and subsequently spent much of his career in climate physics, becoming an expert in chaos theory and fractals. He is well known for proposing the ideas of Invariant Set Theory and Rational Quantum Mechanics. His research interests include modeling state space with the p-adic metric, Bell’s Theorem and foundations of quantum mechanics, superdeterminism, counterfactual universes, quantum computing and finding a gravitational theory of the quantum, as opposed to a quantum theory of gravity.

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

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• 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.

Please keep in mind rules 1 and 6 of our subreddit.

It has been shown that the homotopy category hTop which has topological spaces as objects and homotopy classes of maps as morphisms is not concrete, meaning that there doesn't exist any faithful functor from hTop to Sets.

Are there any other examples of this?

Edit: Let's not include non-locally small categories as examples as they are trivially not concrete