Omg I Want This So Much, I Could Share My Ideas And Things I Learned

So the exponential function is given by

which evaluated at a real number x gives you the value eˣ, hence the name. There are various ways of extending the above definition, such as to complex numbers, or matrices, or really any structure in which you have multiplication, summation, and division by the values of the factorial function at whatever your standin for the natural numbers is.

For a set A we can do some of these quite naturally. The product of two sets is their Cartesian product, the sum of two sets is their disjoint union. Division and factorial get a little tricky, but in this case they happen to coexist naturally. Given a natural number n, a set that has n! elements may be given by Sym(n), the symmetric group on n points. This is the set of all permutations of {1,...,n}, i.e. invertible functions from {1,...,n} to itself. How do we divide Aⁿ, the set of all n-tuples of elements of A, by Sym(n) in a natural way?

Often when a division-like thing with sets is written like A/E, it is the case that E is an equivalence relation on A. The set of equivalence classes of A under E is then denoted A/E, and called the quotient set of A by E. Another common occurence is when G is a group that acts on A. In this case A/G denotes the set of orbits of elements of A under G. This is a special case of the earlier one, where the equivalence relation is given by 'having the same orbit'. It just so happens that the group Sym(n) acts on naturally on any Aⁿ.

An element of Aⁿ looks like (a[1],a[2],...,a[n]), and a permutation σ: {1,...,n} -> {1,...,n} acts on this tuple by mapping it onto (a[σ(1)],a[σ(2)],...,a[σ(n)]). That is, it changes the order of the entries according to σ. An orbit of such a tuple under the action of Sym(n) is therefore the set of all tuples that have the same elements with multiplicity. We can identify this with the multiset of those elements.

We find that Aⁿ/Sym(n) is the set of all multisubsets of A with exactly n elements with multiplicity. So,

is the set of all finite multisubsets of A. Interestingly, some of the identities that the exponential function satisfies in other contexts still hold. For example, exp ∅ is the set of all finite multisubsets of ∅, so it's {∅}. This is because ∅⁰ has an element, but ∅ⁿ does not for any n > 0. In other words, exp 0 = 1 for sets. Additionally, consider exp(A ⊕ B). Any finite multisubset of A ⊕ B can be uniquely identified with an ordered pair consisting of a multisubset of A and a multisubset of B. So, exp(A + B) = exp(A) ⨯ exp(B) holds as well.

For A = {∗} being any one point set, the set Aⁿ will always have one element: the n-tuple (∗,...,∗). Sym(n) acts trivially on this, so exp({∗}) = {∅} ⊕ {{∗}} ⊕ {{∗∗}} ⊕ {{∗∗∗}} ⊕ ... may be naturally identified with the set of natural numbers. This is the set equivalent of the real number e.

I hate it when I am blessed with a new tumblr feature without my consent

I can relate to your undergrad experience! And I think it might be a good sign looking forward, because you've developed insights and ways of thinking and motivation to go beyond undergrad and seek out new spaces where you can do your own work. That's by no means common, I know many fellow undergrads who are a) as mystified when they retake a class as they were the first time round, and b) feel accomplished enough to have passed eg Introductory Analysis and have no drive to look onward. You seem to know very much what you're good at, what interests you and which areas you'd like to grow in. I'd argue that undergrad studies, which give you an introduction and overview of the field and teach basic reasoning skills while not expecting any really original problem solving aren't exactly made for people like you. Talking to your professors or Masters or even PhD students is a really good idea!

thank you for your input, it brings a huge relief!

I already talked to two of my professors and they said that there is nothing to worry about. my advisor said that in his opinion learning new concepts while working on some problem is the right way to learn and from his experience this is way more rewarding than learning for school or even "just to learn". he also said that if I'm interested in working more on open stuff then he will let me know when he finds some questions I could ponder. the other professor said that it's a good thing, because from his experience a lot of people tend to get discouraged when there is no way of knowing how long solving the problem will take or how much new theory is needed, and I seem to be the other way around, so the work I'll be doing in the future probably won't scare me as much

I talked to some of my friends who are about to finish undergrad like me, and there are people who feel the same way as I do. coincidentally, those are the people who had the same situation as mine, that is, they were lucky enough to find an advisor who gave them an open question to work on. other people I talked to seem to be fairly content with studying for the classes and completing homework assignments, and they didn't get to work on something open yet, so maybe it has something to do with getting the taste of the good stuff haha

I can see now that the future looks good and I'm motivated to go exploring. I am aware that I have so much more to learn, but having got the reassurance that I'm probably doing it right, it doesn't sound as scary anymore

in a way. over the last two years or so. mathematics has become the altar at which I pour out my private grief, and transmute it to something like solace. it does not particularly matter to me if I am ever any good at it. what matters is that the effort I apply to it is rewarded by understanding. I have no natural aptitude for it; I am climbing this hill because it was the steepest and least hospitable to me. there is less agony in the gentler slope, but less valor

"based and purple pilled" with deleted vowels. the first adhd medication I tried was life changing, I could finally study and function (half-)properly, and the pills are purple, hence my version of "based and red pilled", which I probably don't have to explain

Guys please reply to this with what your url means or references I’m really curious

maybe a littel late for Real’s Math Ask Meme 18, 6 and 3, please?

hi, thanks for the questions!

3: what math classes did you like the most?

tough choice! for the content itself I'd say abstract algebra, commutative algebra, analytic functions and algebraic topology. for the way the class was taught, a course on galois theory I took last semester was probably the best. the pace of the lecture allowed me to learn everything on the spot, not too fast, but not so slow that my mind would wander. the tutorials were also great, because the teacher found the perfect balance between explaining and showing the solutions, and engaging us to think about what should happen next. the courses I mentioned above were also taught well, but the galois theory one was absolutely perfect

6: why do you learn math?

I enjoy the feeling of math in my brain. I can spend hours thinking about a problem and not get bored, which doesn't usually happen with other things. when I finish a study session I feel tired in a good way, like I spent my time and energy doing something valuable and it's very satisfying

18: can you share a good math problem you've solved recently?

given a holomorphic line bundle L over a compact complex manifold, prove that L is trivial iff L and the dual of L both admit a non-zero section

this problem is quite basic, in a sense that you work on it right after getting started with line bundles, but I believe it to be a good problem, because it forces you to analyze the difference between trivial holomorphic bundles and trivial smooth bundles, so it's great for building some intuition

omg that's the most beautiful thing I've seen today

Chapter 2 of commutative algebra!

omg this + bonus points if this is yet another "autistic genius" representation. don't even get me started on how harmful both of those things are for various reasons

Fuck the way media talks about “child prodigies” and “geniuses” especially in fields like music and mathematics.

Like they are gods whose level of understanding we could never reach.

How come we rarely hear about all the people who started young and then fizzled out? How come we never hear the stories of people who started late in life and made a huge difference.

Why do we only hear about their natural aptitude and not the hard work and misteps they took to get there.

For gods sake…

Terry is just a guy!

oh i just saw, congrats on the bachelors!! im still in calc 3, i thought itd be less mundane but it is actually killing now to the point where i cant even open our stewart text. all my friends in decent math programs are doing more fun and general versions this course. i just cant wait to not use this awful book anymore (all our work is based on the books problems and methodology). all this is to say your progress is inspiring. hopefully i get to a point where i can also be having fun around structures and such, i just have to finish grinding through the filter of "do a bunch of this and don't worry about what it really means, btw good luck problem solving on your exams with 0 neither provided intuition nor rigor". i hope blogs like this stick around!

thank you for the nice message!

I'm so sorry to hear that this is how they teach you math, something like this takes away all pleasure and satisfaction. I didn't have calc 3 as such at my university, we would generally focus on theory and understanding from the start. however, we did have some courses where the mindset was like you just described and it was torture. I hope it changes for you soon so that you can finally enjoy some beautiful math!

Thinking about how when my oldest brother took Japanese classes his professor was like your pronunciation is really good 😊 but you need to watch movies that aren't about the Yakuza because you sound like a criminal