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tips for studying math

I thought I could share what I learned about studying math so far. it will be very subjective with no scientific sources, pure personal experience, hence one shouldn't expect all of this to work, I merely hope to give some ideas

1. note taking

some time ago I stopped caring about making my notes pretty and it was a great decision – they are supposed to be useful. moreover, I try to write as little as possible. this way my notes contain only crucial information and I might actually use them later because finding things becomes much easier. there is no point in writing down everything, a lot of the time it suffices to know where to find things in the textbook later. also, I noticed that taking notes doesn't actually help me remember, I use it to process information that I'm reading, and if I write down too many details it becomes very chaotic. when I'm trying to process as much as possible in the spot while reading I'm better at structuring the information. so my suggestion would be to stop caring about the aesthetics and try to write down only what is the most important (such as definitions, statements of theorems, useful facts)

2. active learning

do not write down the proof as is, instead write down general steps and then try to fill in the details. it would be perfect to prove everything from scratch, but that's rarely realistic, especially when the exam is in a few days. breaking the proof down into steps and describing the general idea of each step naturally raises questions such as "why is this part important, what is the goal of this calculation, how to describe this reasoning in one sentence, what are we actually doing here". sometimes it's possible to give the proof purely in words, that's also a good idea. it's also much more engaging and creative than passively writing things down. another thing that makes learning more active is trying to come up with examples for the definitions

3. exercises

many textbooks give exercises between definitions and theorem, doing them right away is generally a good idea, that's another way to make studying more active. I also like to take a look at the exercises at the end of the chapter (if that's the case) once in a while to see which ones I could do with what I already learned and try to do them. sometimes it's really hard to solve problems freshly after studying the theory and that's what worked out examples are for, it helps. mamy textbooks offer solutions of exercises, I like to compare the "official" ones with mine. it's obviously better than reading the solution before solving the problem on my own, but when I'm stuck for a long time I check if my idea for the solution at least makes sense. if it's similar to the solution from the book then I know I should just keep going

4. textbooks and other sources

finding the right book is so important. I don't even want to think about all the time I wasted trying to work with a book that just wasn't it. when I need a textbook for something I google "best textbooks for [topic]" and usually there is already a discussion on MSE where people recommend sources and explain why they think that source is a good one, which also gives the idea of how it's written and what to expect. a lot of professors share their lecture/class notes online, which contain user-friendly explenations, examples, exercises chosen by experienced teachers to do in their class, sometimes you can even find exercises with solutions. using the internet is such an important skill

5. studying for exams

do not study the material in a linear order, instead do it by layers. skim everything to get the general idea of which topics need the most work, which can be skipped, then study by priority. other than that it's usually better to know the sketch of every proof than to know a half of them in great detail and the rest not at all. it's similar when it comes to practice problems, do not spend half of your time on easy stuff that could easily be skipped, it's better to practice a bit of everything than to be an expert in half of the topics and unable to solve easy problems from the rest. if the past papers are available they can be a good tool to take a "mock exam" after studying for some time, it gives an opoortunity to see, again, which topics need the most work

6. examples and counterexamples

there are those theorems with statements that take up half of the page because there are just so many assumptions. finding counterexamples for each assumption usually helps with that. when I have a lot of definitions to learn, thinking of examples for them makes everything more specific therefore easier to remember

7. motivation

and by that I mean motivation of concepts. learning something new is much easier if it's motivated with an interesting example, a question, or application. it's easier to learn something when I know that it will be useful later, it's worth it to try to make things more interesting

8. studying for exams vs studying longterm

oftentimes it is the case that the exam itself requires learning some specific types of problems, which do not really matter in the long run. of course, preparing for exams is important, but keep in mind that what really matters is learning things that will be useful in the future especially when they are relevant to the field of choice. just because "this will not be on the test" doesn't always mean it can be skipped

ok I think that's all I have for now. I hope someone will find these helpful and feel free to share yours

rb this with ur opinion on this shade of pink:

7-9 VIII 2021

did math and coding nothing special really

sleep: good

concentration: good

phone time: good

reading about measure theory. here is a great book:

everything is so well explained here. i wish i could do more math than i have time for but i guess it's fine, it's holidays, i will wreck my brain completely anyway when october comes

tomorrow more measure theory and topo

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.

ah yes my boy tom cardy. everyone must listen to him, he's the best

I need everyone to see this ABSOLUTE MASTERPIECE

→ 30 VIII 2021

not much has happened really

concentration: 4

doing topo as usual, stopped doing as much analysis, just enjoying my break from coding with abstract ideas

reading books about math became sort of a comfort thing for me. i fell in love with just sitting there and trying to imagine everything. i wish i could be payed for studying math, i would be a fucking billionaire at this point

hey guys quick question

If a homomorphism is a structure preserving map and an isomorphism is a bijective homorphism, does that mean an isosexual is a bijective homosexual?

yes, this. taking photos of the blackboard and writing down only the "sketch" of the lecture usually does the trick for me: I have all the details I need but I'm able to actually listen

a thing that i didn’t understand as a student, that many of my students don’t understand, and that i still sometimes struggle to put into practice: taking the most detailed notes is not always the best way to learn the material. trying to write down every single thing a teacher (or other person who is presenting auditory information to you) says is not only slow but it also can easily stop you from being mentally present during the lesson, internalizing the main ideas and how everything fits together, which is what will actually help you learn the material.