Gonna List My General Goals, Not Necessarily What Theorems I Want To Learn But Rather Some Global "fix

7 III 2023

it's the second week of the semester and I must say that it's easier than I predicted

statistical data analysis is boring but easy, algebra 2 is easy but probably interesting, so is differential geometry

algebraic topology was funny because ⅓ of the group completed the algebraic methods course, so at first we told the professor to skip half of the lecture (we all know the required part of category theory) and then with every new piece of information he would say "ok maybe this will be the first thing today that you don't know", to which we would reply "naaah we've seen this" lmao. but the course overall will be fun and maybe it's even better that the level of difficulty won't be as high as I though, that would leave more time for my other stuff

the tutorial part of number theory was scary, because the professor wanted us to work in pairs. my autistic ass hates working in groups and the noise in the room was unbearable (everyone was talking about the exercises we were given to solve), so I was on the verge of a meltdown after 30 minutes of this despite ANC headphones. next time I will work by myself from the start. maybe without the requirement of communication it won't be as bad. the course itself will be easy, when it comes to the material. I know nothing about number theory, so the novelty will make it more enjoyable. a few people said that they would prefer the tutorial in the standard form, maybe I won't have to worry about surviving it if there are enough people who want to change it

my birthday is tomorrow and as a gift my parents gave me enough money to buy an ipad, I was saving for it since november. for a few days now I've been testing different apps for note taking, pdf readers and other tools useful for studying. I must say, this is a game changer, I absolutely love it

taking notes itself is less comfortable than on an e-ink tablet, which gives very paperlike experience, but it's better than traditional ones. the upside is that I can use different colors and the whole process is less rigid than on an e-ink

two apps that seem the best for now are MarginNote 3 and GoodNotes

the first one is good for studying something from multiple sources. the app allows to open many pdfs, take pieces from them and then arrange them in a mindmap. it's possible to add handwritten notes, typed notes, photos and probably more that I don't know yet. all of this seems to be particularly useful when studying for exams or in other situations when it's necessary to review a huge chunk of material

the second app is for regular handwritten notes. it doesn't have any special advantages other than I just like the interface lol what I like about taking notes on ipad is that I can take photos and insert them directly into the notebook, which I can't do on the e-ink. it's great for lectures and classes because I don't usually write everything down (otherwise I can't listen, too busy with writing) and even if I do, I don't trust myself with it so I take photos anyway. being able to merge the photos with notes reduces chaos

oh god this is going to be a long post! other news from life is that yesterday I had a meeting with my thesis advisor and we finally picked a topic. some time ago he sent me a paper to try and said, very mysteriously, to let him know if it's not too hard before he reveals more details about his idea. the paper is about symmetric bilinear forms on finite abelian groups, pure algebra, and I was supposed to write about algebraic topology, so I tried to search where this topics comes up, but didn't find anything. it turns out that it's used to define some knot invariant, which I would use to write about the classification of singularities of algebraic curves. in the meantime my advisor had another idea, which is an open problem in knot theory. we decided to try the second one, because there is less theory to learn before I could start writing the paper

to summarize what I'm about to do: there is a knot invariant called Jones polynomial, which then inspires a construction of a certain R-module on tangles and the question asks whether that module is free, if so, what is its rank. now I'm reading the book he gave me to learn the basics and I can't wait till I start working on the problem

when a pelican bites you there's no malice in their eyes. they aren't upset at you. they are just hungry and want to see if you fit in their mouths. and if you don't then it's no problem and everything is fine. and if you do then well i guess your fate is sealed but that's ok it's a beautiful animal

29 X 2022

another exhausting week finally over! fortunately I have two extra weekend days, so I can rest and do my homework without stressing over it

I found another promising youtube channel about learning. and "insanely difficult subjects" sounds about right when it comes to everything that's happening in math

I wish there was more content about learning math specifically. the tips I see, however good and useful for studying memory-based stuff such as biology or history, don't seem to work for math

for now my best method is to study the theory from the textbook, trying to prove everything on my own or if that fails, working through the proofs, coming up with examples of objects and asking (possibly dumb) questions that I then try to answer. afterwards I proceed to solving exercises

recently I've been studying mainly commutative algebra, in particular the localization

we didn't spend much time discussing local rings so I had to find some useful properties on my own. the whole idea of "local properties" is an interesting one and I definitely want to read more about it

I find it to be much more elegant to study localization through its universal property and exact sequences rather than through calculation on elements. it's funny how you can cheat so many of our homework problems by knowing basics of category theory and a little bit of homological algebra

I wonder if it's possible to learn math using mind maps, never actually tried. here is my attempt at doing that for one of the subjects in complex analysis:

other than studying I had to prepare a presentation for one of my courses

the topics were given to us by the professor so I thought it would be boring and technical, but I got lucky to discuss the possible generalizations of the Jordan theorem

now I'm gonna talk about something more personal

this week has been difficult because my brain doesn't enjoy existing. some days I had so many meltdowns and shutdowns, I could barely think and speak, let alone study difficult subjects in math. it's really disappointing, as I thought it got better after introducing new medication, but apparently I still can't handle time pressure and I break very easily when emotions become overwhelming (which they frequently do). one of the most discouraging parts of a neurodivergent brain is that you can't always say "alright then I'll just work harder" when you see that the situation requires it. you can't, because your brain has a certain threshold of "how much can you take before you snap" and no tips for studying when you're tired can change that. if you try, you'll just have a meltdown and your day is over, the rest of it must be spent regaining your strength and all you can do is hoping that tomorrow will be better

I wish I could always simply enjoy math and see it as an escape route from a confusing world of human interaction and unpredictable emotions, but whenever there is a deadline or grading criteria, I can hardly enjoy it anymore. I know that this is not what it's always gonna be, the further I go the less deadlines and exams we have, so I must wait and one day it might be okey

since june I've been trying to discuss accommodations regarding adhd and autism with my university but the process takes forever and I'm slowly losing hope that I will ever have it easier

nonetheless, I'm willing to do everything to achieve the goal of spending my days alone working on developing some new theory. just a few more years and I might start living the dream

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'm reblogging this to compare it later with 1.A from Hatcher's Algebraic Topology. in that chapter he defines the topology on a graph if anyone else wants to check it out

Intuitively, it seems to me that graphs should be some sort of finite topological space. I mean, topology studies "how spaces are connected to themselves", and a graph represents a finite space of points with all the internal connections mapped out. That sounds topological to me! And of course many people consider the Seven Bridges of Königsberg problem to be the "beginning" of topology, and that's a graph theory problem. So graphs should be topological spaces.

Now, I vaguely remember searching for this before and finding out that they aren't, but I decided to investigate for myself. After a bit of thought, it turns out that graphs can't be topological spaces while preserving properties that we would intuitively want. Here's (at least one of the reasons) why:

We want to put some topology on the vertices of our graph such that graph-theoretic properties and topological properties line up—of particular relevance here, we want graph-theoretic connectedness to line up with topological connectedness. But consider the following pair of graphs on four vertices:

On the left is the co-paw graph, and on the right is the cycle graph C_4.

Graph theoretically, the co-paw graph has two connected components, and C_4 has only one. Now consider the subgraph {A, D} of the co-paw graph. Graph theoretically, it is disconnected, and if we want it to also be topologically disconnected, it must by definition be the union of two disjoint open sets. Therefore, in whatever topology we put on this graph, {A} and {D} must be open. The same argument shows that {B} and {C} must be open as well. Therefore the topology on the co-paw graph must be the discrete topology.

Now consider the subgraph {B, D} of C_4. It is disconnected, so again {B} and {D} must be open. Since {A, C} is also disconnected, {A} and {C} must be open. So the topology on C_4 must again be the discrete topology.

But these graphs aren't isomorphic! So they definitely shouldn't have the same topology.

It is therefore impossible to put a topology on the points of a graph such that its graph-theoretic properties line up with its topological properties.

Kind of disappointing TBH.

free recall

here I am sitting and trying to learn something from a textbook by making notes and ugh I don't think this is gonna work

what I'm writing down will probably leave my head the second I switch tasks

today I found a cool video about taking notes during lectures and a method called free recall is mentioned there:

to summarize: taking notes during the lecture is ineffective, because it requires dividing attention into writing and processing the auditory input. instead of doing that one should just listen and then try to write down the contents of the lecture from memory. I can believe that – this is how I studied for my commutative algebra exam and the whole process went really fast. I highly recommens this guy's channel, he is a neuroscientist and bases his videos off of research findings

I will try to do this with textbooks and after a while I'll share how it felt and if I plan to keep doing it. the immediate advantage of this approach is that it gives raw information for what needs the most work and what can be skipped, which is often hard to see when trying to evaluate one's knowledge just by thinking about it. another thing that comes to mind is the accountability component – it is much easier to focus on the text while knowing that one is supposed to write down as much as possible after. kinda like the "gamify" trick I saw in the context of surviving boring tasks with adhd

I'll use this method to study differential geometry, algebraic topology, galois theory and statistics. let's see how it goes

Pick a point inside a triangle and drop perpendicular projections onto the sides. These define another triangle. Repeat, with the same point but within the new triangle. Do the same thing once more. The fourth triangle now has the same angles as the first one, although it’s much smaller and it’s rotated.

10-12 VIII 2021

finished the basics of the measure theory and god am i in love

sleep: ok

concentration: good

phone time: good

yeah so now i know what a measurable set and a measurable function is, i'm on my way to lebesgue integration. however, i don't have the intuition for measurable functions yet, just the basics. there are those two theorems that i merely vaguely understand and idk barely can touch them. one of them is lusin, the other one is frechet. they seem very important as they deal with continuity of a function in the context of measurability. and do we love continuous functions my dude yes we do

tomorrow i plan to solve some problems concerning measurable functions and then do topo. i must admit, measure theory devoured me entirely recently and i had a break from topo. gotta fix that. and possibly do some coding

10 IX 2022

today I need some extra motivation to study because I didn't sleep well these past few days and it has drastic effects on my productivity, energy, motivation and what have you

also I am struggling to make the choice as to what I should do today

yesterday I started solving some basic exercises from hatcher's textbook

Δ-complex structures are becoming more intuicitve with time. take my solutions with a grain of salt, I am just starting to learn about these things and won't vouch for them lmao

some more complicated objects (the last one is an example of a lense space)

I decided to study commutative algebra today

so far I'm enjoying it. not as much as algebraic topology (which will always be my number 1) but it has its beauty

right now I'm at hom and tensor functors, the structures are fairly complicated, but pretty, and they look like they need to be studied in stages, with repetition and breaks, to fully grasp what's going on

my sensory issues are terrible today and I'm exhausted and hyperactive at the same time uh

I'll try working through a lecture on commutative algebra and give an update on how it went later

update: I studied for a while, but it wasn't going great so I decided to take a nap instead. god knows I tried

tips for studying math part 2:

studying for an exam but the course is super boring and you don't care about it at all, you just want to pass

start by making a list of topics that were covered in lectures and classes. you can try to sort them by priority, maybe the professor said things like "this won't be on the exam" or "this is super important, you all must learn it", but that's not always possible, especially if you never showed up in class. instead, you can make a list of skills that you should acquire, based on what you did in classes and by looking at the past papers. for example, when I was studying for the statistics exam, my list of skills included things such as calculating the maximum likelihood estimators, confidence intervals, p-values, etc.

normally it is recommended to take studying the theory seriously, read the proofs, come up with examples, you name it, but we don't care about this course so obviously we are not going to do that. after familiarizing yourself with the definitions, skim through the lecture notes/slides/your friend's notes and try to classify the theorems into actionable vs non-actionable ones. the actionable ones tell you directly how to calculate something or at least that you can do it. the stokes theorem or the pappus centroid theorem – thore are really good examples of that. they are the most important, because chances are you used them a lot in class and they easily create exam problems. the non-actionable theorems tell you about properties of objects, but they don't really do anything if you don't care about the subject. you should know them of course, sometimes it is expected to say something like "we know that [...] because the assumptions of the theorem [...] are satisfied". but the general rule of thumb is that you should focus on the actionable theorems first.

now the problem practice. if you did a lot of problems in class and you have access to past papers, then it is pretty easy to determine how similar those two are. if the exercises covered in class are similar to those from past exam papers, then the next step is obvious: solve the exercises first, then work on the past papers, and you should be fine. but this is not always the case, sometimes the classes do not sufficiently prepare you for the exam and then what you do is google "[subject] exercises/problems with solutions pdf". there is a lof of stuff like this online, especially if the course is on something that everybody has to go through, for instance linear algebra, real and complex analysis, group theory, or general topology. if your university offers free access to textbooks (mine does, we have online access to some books from springer for example) then you can search again "[subject] exercises/problems with solutions". of course there is the unethical option, but I do not recommend stealing books from libgen by searching the same phrase there. once you got your pdfs and books, solve the problems that kinda look like those from the past papers.

if there is a topic that you just don't get and it would take you hours to go through it, skip it. learn the basics, study the solutions of some exercises related to it, but if it doesn't go well, you can go back to it after you finish the easy stuff. it is more efficient to learn five topics during that time than to get stuck on one. the same goes for topics that were covered in lectures but do not show up on the past papers. if you don't have access to the past papers you gotta trust your intuition on whether the topic looks examable or not. sometimes it can go wrong, in particular when you completely ignored the course's existence, but if you cannot find any exercises that would match that topic, then you can skip it and possibly come back later. always start with what comes up the most frequently on exams and go towards what seems the most obscure. if your professor is a nice person, you can ask them what you should focus on and what to do to prepare, that can save a lot of time and stress.

talk to the people who already took the course. ask them what to expect – does the professor expect your solutions to be super precise and cuts your points in half for computation errors or maybe saying that the answer follows from the theorem X gets the job done? normally this wouldn't be necessary (although it is always useful to know these things) because when you care about the course you are probably able to give very nice solutions to everything or at least that's your goal. but this time, if many people tell you that the professor accepts hand-wavey answers, during the exam your tactic is to write something for every question and maybe you'll score some extra points from the topics you didn't have time to study in depth.

alright, that should do it, this is the strategy that worked for me. of course some of those work also in courses that one does care about, but the key here is to reduce effort and time put into studying while still maximizing the chances of success. this is how I passed statistics and differential equations after studying for maybe two days before each exam and not attending any lectures before. hope this helps and of course, feel free to add yours!