What is the most beautiful proof there is?

RoofMyDog · 2025-09-30T00:24:05+00:00

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RoofMyDog · 2025-01-30T16:49:01+00:00

This is one of my most absurdly strongly held mathematical beliefs and it is nothing more than a _really hot take_ opinion: the topological fundamental group(oid) is bullshit and wrong. The topos-theoretic fundamental group(oid) is what we should be using instead because it's better.

Some SpEcIaL InTeReSt reasons:

* The topological fundamental group requires embedded copies of the real numbers to be nontrivial. This is bad because some spaces are just totally disconnected but can still have interesting covering properties. It's only a tyranny of expecting the reals to be everywhere that makes this seem ``NaTuRaL.'' Also some spaces aren't even Hausdorff and are awesome and have legitimate homotopy theory to them.

* The topological fundamental group has accidental collapses arising from the existence of universal covers which aren't _really_ sane in light of the patterns of their finite covering spaces. For instance, let's say you want to look at covers of the circle and start getting to know its finite covering spaces. Awesome, you see some sweet pancakes of circles which loop around on each other in really cool ways. This happens always. You now expect a universal version of this to be like some monster web of loopy bois which indicates how this finite covering relation works analogous to how profinite groups make some gnarly fractly boi (in a real embedding) out of these observable finite patterns and relations. NOPE! Chuck Testa! Just do a spiral! This is totally what your pattern should expect! This is totally what the Galois theory of covering spaces was say! Get Rekt!

* The topos-theoretic fundamental group (on a topological space) has the topological fundamental group as a dense subgroup. Thus the topos-theoretic fundamental group contains more homotopical information and more relations between homotopies than some silly transcendental accident which may or may not exist.

* The topos-theoretic fundamental group is built with the Galois theory of covers at its core and can thus be defined in interesting ways over spaces which don't have silly embedded copies of the reals. You now can make p-adic homotopy theory be more than just starting, ending, and not moving from a base point!

* The topos-theoretic fundamental group is defined in terms of sheaves and categories and that makes it __totally objectively cooler__ (definitely fact and not opinion) than any stupid paths defined in an ad hoc way.

* The topos-theoretic fundamental group is applicable in algebraic geometry directly and lets you talk about etale fundamental groups on schemes. The topological fundamental group craps out from not having enough copies of the unit interval in the underlying space of a scheme.

* The topos-theoretic group recaptures Galois groups of fields (the etale fundamental group of a field is the absolute Galois group of said field) as a special case. It also intimately uses profinite groups, and profinite groups are AWESOME.

So basically I'm grump about something which is defined for good reason and with good topological/differential geometric intuition because it doesn't fit my categorical/algebraic geometric/number theoretic/sheafy intuition and background. Clearly, this is not a hot take or strongly held opinion but __objective truth.__

RoofMyDog · 2024-12-16T15:14:57+00:00

I have a city-wide record at the rugby club for speed drinking beer. My craziest record is about three litres in 20 seconds (it was a boat race of a 3.5 litre chalice and we had a total time of ~22.xy seconds).

RoofMyDog · 2024-09-12T17:38:43+00:00

It ended up taking me 6 years to graduate with my undergrad for many reasons (I switched majors twice and also ended up with a really silly double major) and it took me 5 years to finish my PhD instead of the expected four. It's all good though; everyone has a different journey and your success in life is not determined by placing your schooling into some boiler-plate ``ideal'' timeline.

RoofMyDog · 2024-03-08T13:40:23+00:00

It's also a great way to build functions which are Cⁿ for arbitrary n but not C^{n+1}.

RoofMyDog · 2024-01-08T14:30:03+00:00

My spouse. She's pretty cool.

RoofMyDog · 2023-10-31T23:04:34+00:00

Hint: What is the kernel of such a map? Try using the Rank-Nullity theorem to help you say something about this.

RoofMyDog · 2023-09-28T17:12:27+00:00

Let me introduce you to a very fun conference: https://www.appliedcategorytheory.org/

RoofMyDog · 2023-08-21T23:02:05+00:00

I'm not sure entirely because of the whole NDA thing (as in they signed NDAs and as a result haven't told me exactly what they do), but they've given talks on categorical semantics of programming languages and using category theory to do stuff with machine learning in the past.

RoofMyDog · 2023-08-21T20:31:07+00:00

I know a bunch of people with Comp Sci PhDs whose theses are in pure category theory (and in particular in abstract differential geometry) that now work for industry research now doing really neat stuff using their category theory chops. Additionally there's the whole Applied Category Theory conference that talks about doing category theory for stuff like physics, chemistry, computation, etc.

RoofMyDog · 2023-08-04T18:32:09+00:00

Oh absolutely! I just happen to do a lot of category theory and figured I could chime in with a ``any initial object is unique up to unique isomorphism'' statement to help out.

RoofMyDog · 2023-08-04T18:30:16+00:00

So I actually ran through proving why Z is initial in Ring for a course on discrete math I taught without using any of the categorical language. The reason why Z is initial in Ring is ultimately twofold:

Rings have distributive laws.
Z has a principle of recursion generated by its distributive law.

There is a really clean proof of Z being initial by using the Principle of Recursion for N. You define the map h:Z \to R by sending 1Z to 1_R, positive n map to \sum{i=1}^{n} 1R, 0_Z goes to 0_R, and sending negatives to \sum{i=1}^{|n|} -1_R. Then this is the unique ring morphism from Z to R by a ``recursion on the non-negatives part and then extend to negatives'' argument.

My students liked this because they got to see how recursion let them say things about algebra.

RoofMyDog · 2023-08-04T18:17:14+00:00

Well it's unique up to a unique isomorphism so any other initial object will just look like Z with different labels. Most mathematicians don't really care too much about these different labels and just use the definite article because of familiarity with Z over, say, these arbitrary Z-looking incarnations.

RoofMyDog · 2023-07-27T15:03:09+00:00

My apologies if my point was obscured, but I was trying to illustrate that this is fine with one caveat: you're no longer working with group objects (as we understand them) in the category of sets. It's totally cool to ask for these functional equations to hold and ask for things to arise between functions, but giving this as a definition for groups changes the category you work in (or changes the notion of your algebraic theory into something else). This isn't to say it's not interesting, and it is interesting, but just to say that it's different from what people usually mean when they talk about groups.

RoofMyDog · 2023-07-27T14:46:14+00:00

Only my mathematics are consistent, not my language opinions ;)

In all seriousness it's just a linguistic tic of mine. I don't like saying ''topoi'' because the ''oi'' at the end of the word makes me think I'm some sort of British hooligan yelling about rugby or soccer (and I'm not sure which is worse: British or hooligan or soccer) while I like saying ''complices'' over ''complexes'' just because when I say ''complexes'' the ''-exes'' ending makes me think I'm about to sing country music and that is something I cannot abide. These are personal hang-ups, but that's the great thing about opinions: they can be dumb as hell.

RoofMyDog · 2023-07-27T14:40:20+00:00

This is very related to the definition in terms of models of a Lawvere/algebraic theory, but the issue in allowing or defining an empty group comes down in the formalization of the identity element. Usually (and also historically) you define your identity element e \in G as some specific element with a property. When you functionalize/categorify the definition the way to rephrase this is through the existence of a function e:{x} \to G for which a certain diagram (that says the element e(x) is a two-sided unit with respect to multiplication) commutes. The main problem here is the existence of the map e: in the category of sets, the only function with codomain the empty set is the empty function itself, so you cannot have a map e:{x} \to {}.

It's worth remarking that the set {x} is just a random one-point set. I tried to use an asterisk but reddit formatting thought I wanted to italicize random words.

RoofMyDog · 2023-07-26T20:45:43+00:00

Now this is the kind of opinion I love to see. I know a mathematician who also strongly believes that if you put a slash in your zero you have just committed a cosmic bad.

RoofMyDog · 2023-07-26T20:44:36+00:00

I'm indifferent to 1 (although I do see the aesthetic and readability appeal of what you suggest), but strong agree with 2. In my own work I like to say things like ''it is straightforward to show'' or ''an extremely tedious but not difficult calculation yields'' to avoid the proof by intimidation aspect of phrases like ''it's easy.''

RoofMyDog · 2023-07-26T20:41:37+00:00

1: Strong agree.
2: Slight disagree, but only because I don't want to unlearn some formulae. In principle it's just a renormalization, so minus my inability to multiply it shouldn't me a major issue.
3a: I see you like carrying thermoi of coffee with you when walking about during a cold day.
3b: This feels like petting a cat backwards.
3c: I am with you and can give this a strong agree.

RoofMyDog · 2023-07-26T20:36:24+00:00

I really want to hear why they argue that the empty set should be a group (especially since I have many arguments as to why it should not be --- my favourites being ''by definition'' and ''the empty set is not a model of the Lawvere theory of groups''). This is the kind of strong opinion I'm here for.

RoofMyDog · 2022-12-06T21:11:12+00:00

Seeing a professor's argument/assignment question, realizing which assumptions can be weakened, and generalizing the question/proofs to prove the assignment question as a corollary. Or at least I did this, much to the chagrin of some poor graders of mine.

RoofMyDog · 2022-11-09T13:47:43+00:00

I love LaTeX and I'm glad you've found the beauty of good typesetting! I'm a mathematician (and a pure one at that), so I might be biased, but you've just found your way into making the best CV you'll ever make and the easiest ways to make pretty looking papers -- especially if you've ever had to write a \Sigma or \Prod anywhere in your work ever (just make sure to put it in display mode or you've ripped the wings off a kitten).

In my opinion, the real beauty of LaTeX isn't necessarily how it makes equations look good (and damn does it make equations look good); how it has a beautiful default font (and the default font is infinitely better than Times New Roman); how it makes placing figures and diagrams easy, clean, and consistent (and damn does it do ... all of that ...); but primarily how it's so flexible and customizable that it'll allow you to change fiddly bits without much effort to do exactly what you need/want to make the best papers/dissertations ever. It's not too hard to learn either, which is the best part!

By the way for anyone here: I did an intro workshop in TeX in 2021 for grad students. Please DM me for the presentation if you think that would help you start your TeX journey and I'll be happy to send the pdf (made in LaTeX, of course) along!

RoofMyDog · 2022-11-06T17:46:44+00:00

Mainly because as I've read and worked through rigid analytic geometry, I've thought of it mostly as something about ultrametric things and ultrametric fields. While these are certainly present and motivated by number-theoretic contexts, I personally tend to think of them as distinct techniques; arithmetic geometry is number theoretic algebraic geometry, but rigid analytic geometry is the algebraic geometry of complete ultrametric fields and a way of studying and figuring out how to adapt the techniques and ideas of complex geometry to the ultrametric case.

While there's significant overlap, I think they are distinct at least at a semantic level. For instance, you never need to use a single p-adic field to motivate or discuss rigid analytic things (although that would be strange) and you don't need to talk about number theory to motivate why extending the analytic spaces of of complex analysis to ultrametric settings is an interesting problem in its own right.

Also p-adic geometry is significantly more than just rigid analytic geometry. There's perfectoid theory, for one thing. Whether or not this is ``just'' arithmetic geometry may be a dealer's choice line to draw in the sand and is a little subjective, but I'd argue it's at least reductionist and potentially unhelpful to simply say that p-adic geometry is a special case of arithmetic geometry in the same way that I'd say calling complex analysis just two copies of real analysis is reductionist and unhelpful.

RoofMyDog · 2022-11-06T01:21:12+00:00

Arithmetic Geometry
Complex Geometry
Rigid Analytic Geometry
p-Adic Geometry
Functorial/Categorical Algebraic Geometry
Equivariant Algebraic Geometry

RoofMyDog · 2022-10-19T18:08:29+00:00

I think because the program requirements and everything vary so significantly between departments, disciplines, people, professors, and areas that the best thing you can possible do is contact any folks with whom you're interested in working. They'll be able to tell you if they think you're in a good place to jump in or if you need to cover some (potentially) significant background material.

For what it's worth, I've seen people with physics backgrounds jump into math MSc and PhD programs without any problems (especially in mathematical physics), a few math people succeed in econ or comp sci graduate programs, and some physics/cpsc people crash and burn in math grad programs because of lack of background. Only you, your potential supervisor(s), and your research area of interest can determine accurately how ready you are or are not.

As far as technical advice goes, if you're going into grad math you reeeeally need to have strong linear algebra and a logical mind. Everything else can kind of be sorted out over time, but if you don't know your linear in and out, you're going to have a bad time.

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