30 VII 2021

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.

refseek.com

www.worldcat.org/

link.springer.com

http://bioline.org.br/

repec.org

science.gov

pdfdrive.com

september

I decided to start posting monthly, I hope it will help me keep it regular during the semester, it may also bring more structure into my posts

I gave my talk at the conference, I was surprised with the engagement I received, people asked a lot of questions even after the lecture was over. it seemed to be very successful in a sense that so many people found the topic interesting

what I need to do the most in the next 3 weeks is learn the damn geometry. sometimes I take breaks to study algebraic tolology, I did that yesterday

you guys seem to enjoy homology so here is me computing the simplicial homology groups of the projective plane. I tried to take one of these aesthetic photos I sometimes see on other studyblrs but unfortunately this is the best I can do lmao

my idea for mainly reading and taking notes only when it's for something really complicated seems to be working. I focus especially on the problem-solving side of things, because as I learned the hard way, I need to learn the theory and problem-solving separately. what I found is that sitting down and genuinely trying to prove the theorems stated in the textbook is a good way to get a grasp of how the problems related to that topic are generally treated. sometimes making one's own proof is too difficult, well, no wonder, experienced mathematicians spend months trying to get the result, so why would I expect myself to do that in one sitting. then I try to put a lot of effort into reading the proof, so that later I can at least describe how it's done. I find this quite effective when it comes to learning a particular subject. I will never skip the proof again lmao

in a month I'll try to post about the main things I will have managed to do, what I learned, what I solved, and hopefully more art projects

Quatrefoil Knot

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.

i ended up not doing topo yesterday and made an animation of a lipschitz function instead:

i'm kinda proud of it even though it's messy. had so much fun with it. maybe i'll do some more today

but i want to do that topo content i planned to do so we'll see. i also need to do some coding and finish an art comission

yea right in some parallel universe

All that I understand about algebraic geometry at my present stage of learning.

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.

13 III 2023

I remember putting it in my bio a while ago that I dream of doing actual research one day. well this is already happening, as I mentioned in some post, my advisor found an open question for me to write my thesis about

the progress for now is that I'm done with most of the reading I need to do to tackle it and I'm slowly moving forward with thinking of ideas for the solution (or at least a partial one)

this is what I want to do for the rest of my life: reading papers and trying to write my own ones

ofc I don't know if I manage to solve the problem or achieve anything at all with it but the process itself is fun

other than that I've been catching up with homeworks and assignments from work. fortunately I found an MIT lecture recordings for statistics so hopefully I might not die from boredom

watching probability and stats lectures from MIT has been my relationship's idea of netflix and chill for a while now, gotta cultivate the tradition

the algtop professor asked us to write down a full detailed solution for an exercises we did in class, because the person presenting was unable to explain it so I sent him mine

I don't know yet if it's correct but I'm pretty sure it is. I wrote this down partly because who doesn't want extra points and partly because I didn't have a chance to present it, the person who did was faster

I like how my life is right now, I want to keep it that way

℘²