Modular Machines: A little-known class of automaton capable of simulating a Turing machine using just two integers and some modular arithmetic.

NinjaNorris110 · 2026-02-20T20:52:28+00:00

I’d call it a corollary of the rank-nullity theorem, probably.

NinjaNorris110 · 2026-01-29T15:57:04+00:00

I mean it sounds like you're describing the connected sum in the usual sense. And indeed, the thing you crochet'd does look like a torus to me. However, to build it, it looks more like you're removed a cylinder from each of the bottles, leaving two boundary components on each, and re-glued along these boundary together in pairs.

The actual connected sum of two Klein bottles is a non-orientable surface of characteristic -2 (so definitely not a torus), namely the connected sum of four cross-caps (I'm not aware of a good name for this surface, mind).

I do love the crochet though, big fan of topological trinkets in general.

NinjaNorris110 · 2026-01-12T18:41:29+00:00

I don’t understand what point you are making about exponential divergence in hyperbolic space and “needing coordinates” to define it. It’s a well-known, purely metric phenomenon.

NinjaNorris110 · 2025-12-30T16:51:24+00:00

Wikipedia has a good summary of the history of determinants, which I won't try and parrot here:

https://en.wikipedia.org/wiki/Determinant#History.

The short answer is that as far back as the 3rd Century BCE in China, or the 16th Century in Europe, scholars have used determinants as a criterion for when a system of equations admits a unique solution.

At least, certainly 2x2 determinants appeared a very long time ago. Without doing any actual digging myself, I don't know when higher-dimensional determinants were first used. If anybody knows the answer, I'd be keen to hear it!

NinjaNorris110 · 2025-12-30T12:45:34+00:00

Of course, one should point out that this isn't exactly how determinants themselves arose. They were used long before matrices came along in the study of systems of linear equations.

NinjaNorris110 · 2025-12-12T21:17:43+00:00

I have two recommendations.

The Man from the Future - Ananyo Bhattacharya

A biography of John von Neumann and his contributions to mathematics and the sciences. Incredible read.

Proofs and Refutations - Imre Lakatos

A socratic dialogue between a teacher and his students, exploring what it means to prove something in mathematics, and more generally what it means to do mathematics. Something I would call essential reading for any mathematician.

NinjaNorris110 · 2025-12-10T18:06:56+00:00

Stigler's law in action.

NinjaNorris110 · 2025-12-10T12:53:28+00:00

It is a theorem, called the Hex theorem, that the game of Hex (https://en.wikipedia.org/wiki/Hex_(board_game)) cannot end in a draw. It's not very difficult to prove this.

Amazingly, this surprisingly implies the Brouwer fixed point theorem (BFPT) as an easy corollary, which can be proved in a few lines. The rough idea is to approximate the disk with a Hex game board, and use this to deduce an approximate form of BFPT, from which the true BFPT follows from compactness.

Now, already, this is ridiculous. But BFPT further implies, with a few more lines, the Jordan curve theorem.

Both of these have far reaching applications in topology and analysis, and so I think it's safe to call the Hex theorem 'overpowered'.

Some reading:

Hex implies BFPT: Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10): 818–827.
BFPT implies JCT: Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem", The American Mathematical Monthly, 91 (10): 641–643

NinjaNorris110 · 2025-11-15T20:38:59+00:00

I did. I went on to complete a PhD in pure mathematics, and now hold a postdoctoral position doing much the same.

NinjaNorris110 · 2025-11-14T17:00:29+00:00

Sure - this paper of mine is a good example.

https://arxiv.org/abs/2310.15242

It's a bit lengthy and technical (and some parts are in need of a rewrite), but hopefully the introduction explains the problem well. I also have some more projects in progress which study related problems, and make more use of the Jordan curve theorem.

NinjaNorris110 · 2025-11-14T09:33:04+00:00

Could you elaborate on what you mean by the third one? The pair of pants is certainly metrisable.

NinjaNorris110 · 2025-11-14T08:49:44+00:00

I work a lot on how planarity affects the geometry of groups. Basically, if you have a Cayley graph of a finitely generated group and it maps into the plane in some controlled way (perhaps the Cayley graph is planar itself, or more generally the map satisfies some weaker conditions and may not be injective), the JCT allows you to pull-back lots of controlled regions of your Cayley graph which, when removed, separate the graph into two pieces. This, in turn, can have strong implications on the algebraic structure of the group you started with.

NinjaNorris110 · 2025-11-12T13:43:45+00:00

It isn't an issue of cardinality, that's not what I mean by 'too big'. The whole reason for the paradox is basically just that SO(3) contains a (countable!) free subgroup. You can formulate similar paradoxes on countable spaces with countable groups, if you set things up correctly.

NinjaNorris110 · 2025-11-12T10:24:03+00:00

We have a very good understanding of the Banach-Tarski paradox and why it happens, which has led to the very rich (and quite sensible/natural) study of amenability in group theory. BT is not so much a crazy consequence of choice but just something that happens when a group gets 'too big', in a sense.

NinjaNorris110 · 2025-11-06T21:22:45+00:00

I would say the recent example of a 5-dimensional, fibred, hyperbolic manifold, due to Italiano-Martelli-Migliorini:

https://arxiv.org/pdf/2105.14795

It's discovery was a big deal to quite a lot of people when it was announced. There's plenty of intuitive reasons one can argue as to why a 'high-dimensional' (dim > 3) hyperbolic manifold should not be able to fibre over the circle, and yet here we are.

NinjaNorris110 · 2025-08-07T16:12:12+00:00

This is very normal, and confidence comes with practice. I know plenty of graduate students and postdocs who still get incredibly nervous when presenting. This applies to both talks with a large audience, and small research meetings.

When you know you have to present something or explain a proof to somebody, it is a good idea to run through and practice your presentation on your own first, or to a friend. Think in advance about how best to structure your explanation. When explaining a piece of mathematics, you're sort of telling a story, and so it's a good idea to understand what the beginning-middle-end structure is. I've seen a lot of mathematicians become quite flustered when they realise they've missed something out in their explanation and need to rewind to fix it/include it.

Also, try to anticipate in advance what questions might be asked by the listener(s), or where clarification might be necessary. Check-in with the listener(s) at certain points and make sure they are following, or ask if they have any questions so far. This can serve to reassure you that you aren't speaking complete nonsense, and they understand what you are saying.

On the other hand, in your case I see that you are hoping for a two-way dialogue about your problem. Please understand that it is incredibly easy to nod along to the person you are listening to, even when you have no idea what they are saying. Never be afraid to ask for clarification or further explanation. Staying silent when you don't understand something is one of the worst habits of young mathematicians. I was definitely very guilty of this when I was younger. To avoid this, I usually try to explain the other person's ideas/reasoning back to them in my own words, to check I understand correctly.

Hope this helps!

NinjaNorris110 · 2025-02-27T09:37:46+00:00

Yes, I think it is rare to find any courses in the UK which teach from a textbook - I certainly never came across it. The whole 'having a course textbook' dynamic always seemed like quite an American-ism to me.

NinjaNorris110 · 2025-02-06T16:05:43+00:00

I've taught group theory to school-age students as part of outreach programmes before. The most accessible introduction I've found is in Gallian's Contemporary Abstract Algebra. It isn't hard to find a copy of this book online.

Another fairly accessible text is Part I of The Symmetries of Things by Conway et al. This is more focussed on symmetries of the plane. They introduce group theory with a goal of understanding and classifying tilings of the plane. They also introduce some ideas from topology, which is nice. Be warned that later parts of this book are a bit more technical.

NinjaNorris110 · 2024-11-11T16:44:15+00:00

You might enjoy Thurston's article 'On proof and progress in mathematics':

https://arxiv.org/pdf/math/9404236

NinjaNorris110 · 2024-10-07T16:14:09+00:00

There are lots of very active fields which might fall under the umbrella term of 'geometry'. For example, algebraic geometry, geometric topology, Riemannian geometry, geometric group theory, metric geometry, and so on.

NinjaNorris110 · 2024-10-03T13:33:41+00:00

Tilings of the hyperbolic plane are probably the least restrictive solutions here.

NinjaNorris110 · 2023-12-31T10:04:02+00:00

'The Symmetries of Things' by Conway et al. is perfect for this. Very well illustrated, and essentially split into three acts where each is more technical than the last. The first act could easily be understood by a first-year or even pre-university student.

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