:)

a vaguely confused maths student who has no idea if he's a mathematical physicist or a geometer or a topologist

[hamiltonianhorsey]

People of Mathblr, what field would you say best describes what you’re doing, or what you’d like to do in the future? (Choose what your heart tells you is the correct option, I couldn’t get more specific.)

Algebra

Analysis

Combinatorics

Discrete Mathematics

Geometry

Number Theory

Logic

Probability / Statistics

Topology

Applied

Computational

Something outside of mathematics

Voting ended onApr 9

[tensorboi]

honestly kinda funny that Z is so often called "infinite cyclic", even though the entire point is that Z has no cycles. like we defined cyclic groups as generated by one element, saw that Z was included in this, and we all sorta just went "huh i guess that's fine, infinity do be kinda weird"

[tensorboi]

Why is that the definition of a group?

if you’re anything like me, you were probably a little mystified when you first started reading about groups. groups are quite possibly the nicest algebraic objects; they're simple enough that their axioms fit on a postcard, but they're rich enough to have interesting structure theorems (cfsg, for instance). but there's an important question that never seems to get answered in intro resources:

why those axioms over anything else? why is it that a(bc) is equal to (ab)c, and not something like (ba)c?

when you start to develop some mathematical maturity, the answer starts to come into clearer focus. but i don't think you should need to wait! the underlying idea is actually fairly simple, and it teaches some great lessons about how modern mathematics is organised. so that's the point of this post: to explain the point of the group axioms in the purest sense possible.

Groups as symmetries — but not really

alright, so the textbook motivation for groups is that they model the concept of symmetry. but what does this mean, really? the problem is that mathematicians mean something much broader than what we usually mean by symmetry in this context. here are some examples:

the textbook example is a polygon of some kind. the symmetries of an equilateral triangle are 120° rotations and flips, for instance. when we say symmetry, we are really talking about a function which takes the polygon to itself, and preserves lengths and angles.

now take a line of dots (not necessarily evenly spaced), extending infinitely in either direction. the relevant "symmetries" here are the functions taking dots to dots which preserve the order of the dots.

take a finite collection of objects of any kind whatsoever; i'll call it X. i can define a "symmetry" of this collection to be a function f: X -> X which doesn't take any two objects to the same object. in this situation, the "symmetries" aren't preserving anything but the size of X: by this i mean that f(X) is the same size as X.

for a more complicated example, take a vector space V and consider all the bijective linear maps from V to V. the "symmetries" are preserving not just the size of V, but the sum and product operations in V as well.

with these examples in mind, i'll define the word "symmetry" in the way it's meant here. given some object X, a symmetry of X is a function f: X -> X which "keeps X the same" in a way we don't care to specify. in each of the examples above, the meaning of "keeps X the same" changed to "preserves lengths and angles", "preserves order", "preserves size", and "preserves operations".

so, this is the point of groups! groups model the things you can do to something while "keeping it the same" in some sense. and from this perspective, it might make sense why we'd want to codify and systematically understand this idea; sameness appears a lot in maths, so if we can prove something about sameness that's always true then that should be really useful.

Defining groups (the nonstandard way)

now that we have a proper motivation, a definition of groups is not too far away. however, it turns out to be different to the usual one. it basically follows from these 3 observations:

in all of the examples above, the symmetries took the form of functions from a set to itself. (in the case of polygons, the function took the points of the polygon to itself.) in addition, if a function keeps any set X the same, it should at the very least preserve the size of the set. therefore, a group should be a collection of bijections from a set to itself. the collection of all such bijections is usually denoted Sym(X), so a group G is a subset of Sym(X).

if you have two functions which keep X the same, then obviously doing one after the other should also keep X the same. this means a group should be closed under composition: if f and g are in G, then f o g should also be in G.

finally, whenever you do something that keeps X the same, you should always be able to undo it. this means a group should contain all its inverses: if f is in G, then f^-1 should also be in G.

so that leads us immediately to a definition of a group!

A group is a subset G of Sym(X) for some set X, which is closed under composition and contains its inverses.

...how is that a group?

if you know anything about groups, you'll know that this doesn't really look like the textbook definition at all. in fact, the group definition doesn't even make mention of a set X in the first place, just the set G! it turns out that we can massage this definition and show that it's equivalent to the usual one; but first, let’s talk a bit about why you might want a different definition.

put simply, this definition has too much baggage due to the set X. to elaborate, consider the symmetries of a square and of a square pyramid: these correspond to different sets X, so functions will look very different on them, but the underlying geometric operations are the same! eliminating the set X from the definition would "purify" things: symmetries which were genuinely the same could be treated on equal footing.

ideally, then, we'd like to define a group as something independent of a set on which it acts. but how could we do this? well, it comes down to the 3 observations we made when defining them the first time:

"a group is a set of bijections G from a set X to itself." if we want to forget the set X, we can just take G to be a set.

"if two functions in G are composed, then that should also be in G." if G is just a set, then its elements aren't necessarily functions and composition might not make sense. how to fix this? the answer is basically inventing our way out of the solution: we just give G an operation which takes two elements and returns another element (which is exactly what conjugation does for functions). such an operation is called a binary operation.

"if a function is in G then so is its inverse." again, if G is just a set, inverses don't make sense. so we fix this in the exact same way as before: we just give G an operation which takes an element and outputs another element (like inversion does). this is called a unary operation.

so there we go! a group is just a set G equipped with a binary operation (which is composition) and a unary operation (which is inversion). done!

That's still not a group!

...except not quite. the problem with our definition is that not all binary operations behave like composition, and not all unary operations behave like inversion. let's think through some examples to show how this might be the case:

consider a rock-paper-scissors binary operation: the set is {r, p, s}, and the binary operation is defined to take in two elements and return the winner element. for example, r*p = p because paper beats rock, p*s = s because scissors beats paper, r*r = r because that's a tie, and so on. now imagine that r, p, s could be interpreted as functions from some set to itself, and that * could be interpreted as composition. if that were true, then r(p(s(x))) would need to be equal to both (r*p)(s(x)) and r((p*s)(x)), meaning (r*p)*s would need to equal r*(p*s). but this isn't the case: the former is equal to s, while the latter is equal to r! this means the rock-paper-scissors operation * can never be interpreted as composition.

consider another three-element set {0, 120, 240} corresponding to rotations by a given angle, and give it the usual operation of composition (so 0 o 120 = 120, 120 o 120 = 240, and so on). now, give this set a unary operation which takes each element and gives back the same element: 0^-1 = 0, 120^-1 = 120, 240^-1 = 240. this doesn't properly represent the "inverse" concept, because it's meant to be the same as undoing; however, doing 120 and then 120^-1 is the same as doing 240, which isn't doing nothing!

luckily, these examples give us ideas on what we should add to our definition to make it work.

the problem with the rps-operation is that function composition is always associative, meaning the bracketing doesn't matter, while the rps-operation isn't. so we'll add to our definition that the binary operation needs to be associative.

the problem with the {0, 120, 240} inversion was that it didn't correspond to undoing. we can make this precise by saying that x composed with x^-1 should be the same as doing nothing. of course, since G is just a set, it's not clear what "doing nothing" is supposed to mean; we can fix this by choosing a particular element of G, say e, to be the symmetry which does nothing. we codify its "doing nothing" property by saying that x composed with e is the same as just x.

and that's it! finally, we have arrived at the textbook definition:

A group is a set G equipped with an associative binary operation *, an identity element e for which x*e = x, and a unary operation ^-1 for which x*x^-1 = e.

all well and good! well, except...

How do we know that's enough axioms?

after all, we just looked at two examples and said "huh, functions do this but they don't do that, so we'll just add it to the definition". how do we know there aren't any examples which fit our new definition, but don't correspond to functions?

the answer is in the form of a beautiful result:

Cayley's theorem: Every group G can be viewed as a subset of Sym(X) with the usual composition and inversion operations, for some set X.

the proof? just take the set X to be G itself! then interpret each element g of G as a function from G to itself by taking any other element h to g*h. these functions have inverses given by g^-1*h, meaning they are bijections. also no two elements are represented by the same function, since e gets taken to a different element by each g. (it is an instructive exercise to figure out where exactly this argument breaks if you relax associativity!)

in any case, this theorem tells us that our search is done. we have removed the set X from our definition, but we can always put it back and still get nice functions if we like. we've shown that this can be done when X is taken to be G, but the general concept is so important that it has been formalised:

Let X be a set and G a group. A G-action on X is a function φ: G -> Sym(X) such that φ(g*h) is the same as φ(g) o φ(h) in the output.

in other words, a G-action on a set is just an interpretation of the elements of G as bijections on X, in such a way that composition looks the same as the binary operation. in essence, this is why my initial definition is never presented anymore: we don't want to deal with the extra baggage of X, and we don't even need to because it can be encapsulated as a G-action instead, which we know is always possible by cayley's theorem!

What was the point of all that?

now that that's done, the natural question is this: why present a worse definition first to just arrive on the usual one anyway? and here's the answer: because this kind of thing happens all the time in maths, it only becomes clear to you when you've been dealing with it for years.

rings? they're just collections of endomorphisms of abelian groups, and their abstract definition allows them to be thought of that way.

banach algebras? they're just collections of bounded linear operators, and their abstract definition allows them to be thought of that way.

lattices can be defined completely algebraically, but once again, this definition essentially serves to abstract away the concrete notion of a lattice in terms of a partial order.

even categories themselves fall on this list! the definition of a category is bluntly abstract, but it's essentially an abstraction of a place where functions from one thing to another can be chained together.

if you are a beginner then you might not understand a whole lot of that list, and that's okay because you'll get to it eventually. but if you're not quite a beginner then i'm sure you'll see what i mean! all of this can be viewed through the lens of the yoneda lemma from category theory if you're interested in that kind of thing.

tl;dr: if you defined groups purely in terms of what they're used for, then they'd be subsets of Sym(X) for some set X. but since the notion of a group shouldn't depend on the set that it acts on, we can just insist that composition is associative with identity and inverses. these axioms are necessary and sufficient to prove cayley's theorem, which tells us that we can always think of a group in terms of some Sym(X) if we want to. this pattern of definition-making is all over maths.

[aninfinitedelphinium]

One day, a vector space was involved in a hit-and-run in a carpark and, while not fatally injured, was sent to the hospital for surgery. Due to its poor healthcare plan, a bill of thousands was incurred. The vector space was not by any means in poverty, but thousands is a lot for anyone to pay out of pocket, and so it decided to try to track down whomever had hit it.

After returning to the carpark and reviewing the security camera footage with the police, it was discovered that the one who hit the vector space was none other than the axiom of choice itself. Now, it just so happened that the axiom of choice was one of the vector space's employees, and was notorious for not showing up to events on time — if at all. The vector space decided to speak to its lawyers, and soon a court date was arranged — the 12th of September, 10am sharp.

On the day of the court appearance, the vector space and its lawyer showed up to the courthouse early and waited patiently for all involved parties to show up. One by one, people started filling up the benches in the courtroom, but someone was conspicuously missing — the defendant itself, the axiom of choice. As the clock slowly ticked up to 10am, the vector space grew more and more restless: why wasn't it showing up? Where could it be? What did this mean for its case?

10am quickly turned into 10:15am, and then 10:30am, and by 11 the axiom of choice had still not made an appearance. At this point, the vector space's lawyer motioned for a recess. As people started leaving the room, the vector space turned to its lawyer with a puzzled look and said, "What's going on? If the defendant doesn't show up, does that mean I automatically win the case?"

The lawyer responded, "I'm sorry, but without the axiom of choice, you might not have a basis."

[literallymechanical]

This is a contender for my new favorite fusion paper. How does it feel to be the realest god damn scientist on the planet Dr. Smiet

[adventures-in-mathematics]

math that I don't know has always seemed like magic to me (looks pointedly at analytic number theory with its L-functions and modular forms), but nothing makes me feel more like a wizard than casting "long exact sequence of cohomology" and my desired result falling out

[confusedcanaries]

I love maths, Wdym you can define crazy infinities and bonkers spaces and examine what truth and provability mean, but then you do some combinatorics and suddenly counting is difficult

[tensorboi]

hot take: the number e isn't special

the exponential function? sure, it's very pretty! does so many cool things, has so many nice properties, very natural little guy

but the *number* itself? ehhhhh it's okay

[spectracled-eider]

It would be interesting to see the link between how mathematicians think about tensors and how physicists think about tensors. Of course physicists usually mean tensor fields which is all well and good but I've still not seen the relationship between how the formal definition as smooth sections of tensor bundles and the whole "a tensor is something that transforms like a tensor" thing

[tensorboi]

this is a struggle that all mathematical physicists have to deal with eventually, and i actually asked a question on mse about it quite some time ago! i'll link it here, but the upshot is this: the "physics" tensor is just what you get when you take the components of a "math" tensor in a coordinate chart, and the transformation laws correspond to change-of-basis rules. in this light it's even more baffling that the "transforms like a tensor" thing is the dominant attitude towards tensors in physics

How are the different definitions of a tensor the same definition?

I'm mostly self taught in undergraduate physics and maths, so I haven't had much of an education in this stuff. Basically, I know how the th

[tensorboi]

thinking about how the real reason 1 isn't prime is because it's an empty product of primes

[algebraic-dumbass]

reasons to do math

because it's beautiful

because it's cool

because it will scratch my brain right when i understand the thing

because i haven't understood the thing, and that deeply bothers me

the thing is looking at me from my whiteboard. it's mocking me. i can feel it

i'm gonna go back to the thing

[tensorboi]

to those who know anything about characteristic classes: i am LOSING my MIND rn

like ok there are about 20 different definitions of chern classes and stiefel-whitney classes. and none of them make sense, they just EMERGE from nothing (yes it is very nice that you can construct cohomology classes from symmetric polynomials in the curvature of a connection but that tells me nothing about what it IS)

well... except for one definition i found recently. turns out, the integral cohomology of BU(n) is just the unital ring generated by the chern classes of the universal bundle; that means you can just define the chern classes over an arbitrary U(n)-bundle to be the pullback of the generators of this cohomology (under the corresponding map f: X -> BU(n)). same thing for BO(n), but for Z_2-valued cohomology and the stiefel-whitney classes instead.

but then, huh? when were y'all going to tell me??? i feel like i stumbled over this by accident, why is this not just what the chern classes are DEFINED to be???????

[tensorboi]

idk if anyone else will find this interesting but i JUST HAVE TO SHARE

ok so you know all the different R^n's? the wonderfully different spaces that give rise to so much maths?

turns out that as abelian groups, they're all ISOMORPHIC

why? well they're all vector spaces over Q, which are classified up to dimension, and they all have the same dimension over Q (namely 2^ω)

so the only thing distinguishing these infinitely many spaces is the way real scalars behave on them

isn't that freaking AWESOME???

[mathsuggestions]

Invent yet another kind of cohomology.

[tensorboi]

my new cohomology theory is multiple cohomology. it's like singular cohomology, but. there's a lot of them

[spectracled-eider]

For the math ask game: 8, 23, 47

8. Least favorite notation you’ve ever seen?

As answered here it's Einstein Summation Notation

23. Will P=NP? Why or why not?

From everything I understand about the problem, I don't think so. I also hope it isn't (though even if it is, that doesn't imply that finding the algorithms which solve NP problems in polynomial time is in anyway easy)

47. Just how big is a big number?

As my friend would say: at least 3. More seriously, it'd be one that can be defined using a sequence that can't be computed in finite time. (There's a fancy word that I've forgotten now)

Thanks for the ask!

[voodootortoise]

Can I ask why the index notation hate?

[spectracled-eider]

First and foremost it is ambiguous. Sure it might reduce the number of summations you have to write but for that you sacrifice ambiguity. There were times in my mathematical physics course where the lecturer had to state that we weren't using ESN. If you have to state when/when you're not using it, it's not good notation.

There's another thing from my mathematical physics course that really confused me. We defined what Lorentz invariance was but then ended up just saying anything that's written in the form [something with a lower index]•[something with an upper index] is Lorentz invariant. So I could just write down any old expression and claim it's Lorentz invariant cause of the way I wrote it. Seems a bit dodgy to me. Also the up down stuff doesn't make any sense. I thought it was to do with whether something is a vector or a covector but then we could just write them in any order and it didn't matter? That's not good notation

Also somewhat related is the objects that physicists usually do this with, they call "tensors" are actually tensor fields. And even that's not 100% correct because it's actually the components of the tensor fields in local coordinates. This isn't a mark against ESN but it's certainly a related gripe I have

[tensorboi]

i'm not sure whether or not you're aware of this, but here's something i think more people should know: einstein summation notation can almost always be thought of in a more abstract and cleaner way.

here's what i mean by that. instead of thinking of a tensor like T^i_j as a collection of n² components indexed by i and j, think about that symbol as telling you what the tensor can do. in this case, T can take in a vector (indexed by j), and output a vector (indexed by i, or perhaps input a covector indexed by i but whatever). this framework is called abstract index notation, because the indices no longer refer to unspecified components but rather the algebraic shape of your tensor; each index represents an abstract input/output.

immediately, many of the formulas from physics which look daunting become much easier to parse. given a vector v and a covector ω, the expression ω_i v^i doesn't necessarily mean summing over the components of ω and v (even though it can be interpreted that way); all it means is that ω has one vector input which is being filled by v. an equation like v_i = g_ij v^j is much less weird to look at, since the summation approach leaves a variable unsummed; instead, it just says that the metric tensor g has two vector inputs, and filling one of them with a vecto4 v gives you a covector with very similar properties. the definition of the riemann curvature tensor as the unique tensor R for which R^i_jkl u^j v_k w^l = (D_u D_v - D_v D_u - D_[u, v])w^i is exceptionally complicated in terms of components, but reading the indices abstractly elucidates matters: R takes in two vectors u and v along which we directionally differentiate, and another vector w which we parallel transport in this differentiation. lorentz invariance also makes much more sense here: if all slots of a tensor are filled properly, the result is a scalar because that's how tensors work, and of course scalars are lorentz invariants!

i should note that this reading doesn't work if the objects you're looking at aren't tensors, for instance the christoffel symbols Γ^i_jk and tuples of tensors T^α (and well that makes sense, these things are born of component considerations while tensors really aren't). but as long as you're aware of that, the notation becomes much friendlier than it was.

[marigold-22]

Math enthusiasts of tumblr. What math subjects have you studied and which ones were your favorite? Which ones were your least favorite? Which ones were the hardest?

[tensorboi]

surprised nobody seems to have mentioned algebraic geometry! that was both the hardest and most interesting maths course i have ever taken, just because it's such a bizarre way of thinking about geometry (something i'm very comfortable with in the manifold setting).

least favourite would probably have to be tree theory; i hate discreteness and had successfully avoided it until i had to learn about geometric group theory lmao

[quinn-fucks-shit-up]

personally I love when maths stops including any numbers at all, that bit's my favourite part

[tensorboi]

the scary part is when random numbers start popping up where there definitely shouldn't be any, and now you have to deal with those. like ok i suppose 4 is a cosmically important number when discussing differentiable structures? and there are exactly five exceptional lie algebras?? awesome let's just try our best to sidestep that