Mathblr - Blog Posts

Chill notes

With a pen that’s as old as I am (I really need new stationary :-| )

🎧 Oldschool Rap - Lofi Mix #1 by VoidGalaxy

I’m starting to feel like I’m getting this studying thing down.

They discovered Busy Beaver Five!

BB(5) = 47,176,870

https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237

A lot of people absolutely fail to distinguish between 'I hate this because I was taught this badly' and 'I hate this because this thing is inherently bad', and I hate it.

you know I absolutely agree with the fact that the education system is fucked and traumatises students. I absolutely agree with the fact classes like english and maths and history can be piss boring and teach you nothing of worth and could be a trigger depending on how hellish of a teacher you were given.

Hell, the only reason I love maths is because I primarily engaged with it outside of class and found it fun on my own terms. I'm still bitter towards my history teachers for having clear favourites and not engaging half the class.

but like. if you're railing against the *subjects* themselves, refusing to learn any of the basics as an adult, and bringing up how much you hate them at every opportunity? At some point that becomes a problem you have to deal with yourself.

and if you don't, you're going to be the reason this happens again for the next generation.

Day 1

Assignment: Classwork of Vector Calculus

[11.32 am]I have ODE lab viva soon and I am terrified but hopefully, it will be over soon. I have given myself the task of completing 20 problems of vector calculus.

***

[3.20 pm] I did start but there has been a dip in my motivation.So I guess I will just take a nice bath, cook some snacks for myself and my parents and then sit and study again.

***

[9.45 pm] I am done for the day.Today I did some problems on vector calculus.I don't know why but I felt tired in the evening and just could not sit and solve stuff so I did not force myself.There is some kind of peace in calculating the angle between two very complex surfaces don't you think? Only if I had more energy to go through it.But now I remember also loving the concept of directional derivative.It is not done yet so gotta continue tomorrow.

222 days left to go..

223 Days, 50 assignments, 24 courses and 1 tiny room...

The Exam:

Joint Admission test for Masters (JAM) is a common admission test conducted every year for admission into M.Sc and other post-graduate science programs at Indian Institute of Technology (IITs), Indian Institute of Science (IISc) and National Institutes of Technology(NITs), organized by alternating institutes every year. 

The Contender :

A brave and a somewhat kind soul is trying to make a place for herself in this world.Managing bullies while trying to make new friends is proving to be a challenge.Having clumsy habits is not making her life any better.Self doubt is killing whatever is remaining and all she has left with is hope.Hope that she will get better and her life with it.She intends to do this by appearing for the IIT JAM Mathematics exam.An exam where 15,000 students compete for 500 seats in the best colleges around the country.

Will she make it? How far will she go? We can only wait. And wait. And wait…..

The Mayuriit Project. Coming soon to a Tumblr blog near you

Theorem: Take a circle, the area of this circle is the same as area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle..

A long time ago in a faraway land people led simple lives by the means of agriculture. There was plenty of food to eat and to be merry. They had roofs on their head and fresh river water flowing nearby. Life seemed perfect but it was not. Every year when the rains 🌂 began, the river nearby would flood into the village and destroy their lands and homes. The people in the village would move to a nearby village for shelter with their cattle. When the rain stopped they used to come back and each time they came back they found their place in destruction. Their houses had to be rebuilt and their lands had to be outlined again. Fights were a common scene on how the outline was before the water washed it away. So the concept of the area came to maintain peace.

Finding areas of lands made with straight lines was easy but how to find the area of a land that is a curve ?

Firstly instead of taking a land made up of crazy curves let's take a land made up of the simplest curve, the circle.Draw a circle and fill in its area. Then divide it into large equal parts and arrange them in a rectangle.

It's not yet a perfect rectangle.So divide the circle in parts and try arranging these sections into a rectangle.You'll get a thing that starts looking like rectangle.

Now as you divide the circle more and more and try to arrange those parts you'll get a more nice rectangle. This more and more is nothing but the concept of limits in calculus.

So the area of the circle is the area of the rectangle.The area of the rectangle is Base×Height. Here the height of the rectangle is the radius of the circle and Base is equal to twice the area of the circle. So cut up the rectangle diagonally and you'll get a right-angled triangle with a Base as the radius of the circle and height as the circumference of a circle.

What does FTC say?

It says that if a person takes the derivative of a function and then integrates it over a region on the number line say [a, b] then this is the same as evaluating the function on its endpoints.

What does the Green's Theorem say?

Green's Theorem is the fundamental theorem of calculus in 2 dimensions.Instead of taking the derivative of a single variable function we take the curl of a 2 variable function.Instead of integrating this over a number line we integrate it on the xy plane.Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve in a counterclockwise direction.

What does the stokes' theorem say?

Stokes' theorem is the fundamental theorem of calculus in 3 dimensions. Instead of taking the derivative of a single variable function, we take the three-dimensional curl. Instead of integrating this over a number line, we integrate it on the surface (To evaluate the surface integral one has to dot the vector field with unit normal vectors). Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve on a surface in a counterclockwise direction just like in Green's Theorem.

Why do we use the symbol for partial derivatives as the symbol for boundaries of manifolds?

After the math pope, we have a math president

From the Wiki page of Nicușor Dan, the new president of Romania:

He won first prizes in the International Mathematical Olympiads in 1987 and 1988 with perfect scores.[3] Dan moved to Bucharest at the age of 18 and began studying mathematics at the University of Bucharest.[4] In 1992, he moved to France to continue studying mathematics: he followed the courses of the École Normale Supérieure, one of the most prestigious French grandes écoles, where he gained a master's degree. In 1998 Dan completed a PhD in mathematics at Paris 13 University, with thesis "Courants de Green et prolongement méromorphe" written under the direction of Christophe Soulé and Daniel Barsky [de].[5][6]

Dan's 1988 gold medal also means he was one of eleven contestants to get full marks on the infamous Problem 6, a question so difficult that nobody on the IMO problem committee could solve it.

His personal website lists his primary area of research as Arakelov geometry, a method of studying Diophantine equations from a geometric point of view.

His thesis, in the same field.

His arXiv.

So I recently stumbled on the Wikipedia article for the Grothendieck-Riemann-Roch theorem, which is an algebraic geometry thing that I'll hopefully learn some day once I actually have the prerequisite knowledge =w= But at the top of the article was this letter, which I thought was a wild thing to have at the top of a Wikipedia article about a niche abstract math thing - here's a translation:

Witches' Kitchen 1971 Riemann-Rochian Theorem: the latest craze*: the diagram

is commutatif**! To give this statement about f: X->Y some approximative meaning, I had to abuse the listeners' patience for nearly two hours. In black and white (in Springer's Lecture Notes) it seems like it will take up to about 400, 500 pages. A gripping example of how our thirst for knowledge and discovery indulges itself more and more in a(n il?)logical delirium far removed from life, while life itself is going to hell in thousandfold ways - and is threatened with absolute annihilation. High time to change our course! (6.12.1971) Alexander Grothendiek

* "der letzte Schrei" is a reasonably common German idiom meaning "the latest craze", but here it could alternatively be translated non-idiomatically as something like "the last cry". I think its more fun to imagine he means the idiom. ** I'm assuming this is a weird old-timey spelling probably taken from french but googling it I can find no examples of anybody using this spelling in German besides this letter

Note that this is 20 years before all of this happens:

Explaining math jokes

So yeah I said I was gonna do it and now is the time I think (if I wait any longer I'm gonna have too much jokes to explain) IMPORTANT: I had a lot of trouble writing this because I don't really know what background to assume the reader has. So at each new explanation, assumed background changes. Difficulty of the concepts explained is in no particular order. So if there is something you don't understand, that's fine, just go read something else. Dually, if there is something you already know well, don't throw away the whole post. Though it is very possible you already know everything I'm gonna rant about lol Anyways let's get to it

geometric group theory talk but on the speaker’s slide instead of the Cayley graph of the free group on two generators there’s just loss

(link) Geometric group theory is a subfield of math that studies groups using geometry. A particular geometric thing that is often interesting to study for a given group is its Cayley graph, which roughly speaking is a graph that reflects in a way the structure of the group. My neurodivergent brain thought the Calyey graph of the free group on two generators lowkey looks like the abstracted loss meme in the original post:

sooo, turns out the #latex tag is not for typography enthusiasts

(link) LaTeX is a typesetting engine and the industry standard for math. It's the thing almost everyone uses to typeset beautiful math equations and stuff on computers. If you're seriously interested in math I'd recommend you learn it. A good place to start would be Overleaf which is a free online LaTeX editor and has some tutorials on how to get started, though eventually you may want to switch to doing LaTeX on your computer directly

so a homological algebraist goes to see their therapist and says “doc, i’ve got complexes”

(link) Homological algebra is a branch of algebra that was born from algebraic topology. It has become widely used in many parts of math because of the computational power it brings. The gist of it is that people define a thing called a chain complex, that is a sequence of abelian groups (or modules, or maybe something even more general, look up abelian categories), with homomorphisms from one abelian group to the next called differentials, such that doing one differential then the next always gives you zero:

If you're more comfortable with linear algebra, you can replace "abelian group" with "vector space" and "homomorphism" by "linear map". The fact that doing one differential then the next gives you zero means that the image of one differential is contained in the kernel of the next. Homological algebra is about finding ways to calculate exactly how far away we are from the image of a differential being exactly the kernel of the next. This is made precise when one defines homology groups, which are the quotient Ker(d_n)/Im(d_{n+1}). What happens in practice when we apply homological algebra is that we try to define an interesting chain complex related to what we are doing, so that the homology groups tell us something interesting about what's going on with whatever math thing interests us, and then we apply methods from homological algebra to calculate them. Any serious example of homological algebra being used is going to require a bit of math background, but two examples I can give are singular homology, from algebraic topology, and de Rham cohomology, from differential geometry (don't worry about the co-, it just means the indices go up instead of down). So yeah a homological algebraist would have complexes

If you’re not careful and you noclip out of reality in the wrong areas, you’ll end up in Hilbert’s Hotel

(link) Hilbert's Hotel is an imaginary hotel with infinitely many rooms, that is one room labeled zero, one room labeled one, one room labeled two, and so on. One room for every natural number. It reminds me of the backrooms, hence the joke. Hilbert's hotel is commonly used as a metaphor to think about how infinity behaves and how bijections work. For instance, if the hotel was full, but one new guest showed up, you could still get them a room: simply tell the person occupying room number n to move to room number n+1. Then room 0 will be empty and the new guest can stay in it. However, quite interestingly, it is possible for too many guests to show up and the hotel to be unable to give a room to all of them.

If you speak French, excellent math youtuber El jj (I heavily recommend you subscribe to his channel!) has a very good video on Hilbert's Hotel.

If you don't speak French but still speak English, here's a Veritasium video on Hilbert's hotel, and a Ted-Ed video on it.

“cats are liquid” factoid actually formalized by mathematicians as saying a cat is only truly defined up to homeomorphism

(link) Topological spaces are mathematical objects that abstract away the concept of nearness. What do I mean? Well, a topological space is a set X, together with a collection of subsets called "a topology", that in way specifies which points are "near" each other. This allows us to generalize a lot of concepts from real analysis, for instance limits: if you have a sequence of points (x_n), and they get "nearer" and "nearer" to some point x, well that point can be called the limit of the sequence. But topology also turns out to be massively useful to geometry: if I only gave you the set of points of a sphere, you wouldn't know they make up a sphere because you wouldn't know how to assemble them. But if I give you the set of points of a sphere and the correct topology on it, then you can actually know it is a sphere and do stuff with it. But as always, in math, we only consider things up to some notion of being "the same". This notion of "the same" for topological spaces is called "homeomorphism", and two things being homeomorphic corresponds to the more intuitive geometric intuition of "I can continuously deform one thing into the other without cutting or gluing stuff". For instance, a cube is homeomorphic to a sphere:

Or more famously, a mug is homeomorphic to a donut:

So cats only truly being defined up to homeomorphism kinda works to say they're liquid. Not math, but physicist Marc-Antoine Fardin did actual physics on cats being liquid and was award the 2017 IgNobel prize in physics for it.

i would describe my body type as only defined up to homotopy equivalence

(link) Homotopy equivalence is a weaker notion of two topological spaces "being the same". I won't go into details but I have seen it being describe as kind of like a homeomorphism, but you are also allowed to inflate/deflate objects. For instance, a filled cube is a "3d object" in a way (when you are inside the cube, you can move in 3 directions). This means it will never be homeomorphic to a 2d square because a property of homeomorphisms is that they preserve dimensions. But, the cube is homotopy equivalent to the square, because you can "deflate" the cube and squish it to make it a square. In fact, it is even homotopy equivalent to a point (you can just deflate it completely). Homotopy equivalence is weaker (more permissive) than homeomorphism, that is if two things are homeomorphic, then they must be homotopy equivalent, but not the other way around. You may ask yourself why we would care about a notion weaker than homeomorphisms that can't even tell apart points and cubes, and that's a fair question. I will provide one answer but there are definitely many more I haven't even learned yet. In algebraic topology, we are concerned with studying spaces by attaching algebraic thingies to them. Why do we do that? Because telling apart spaces is hard. Think about it: how do you prove a donut is not homeomorphic to a sphere? You'd have to consider all possible deformations of a donut and show none of them is a sphere. This is mathematically hopeless. Algebraic topology solves this by attaching algebraic invariants to spaces. What do I mean? Well we have a way of saying that a donut has "one hole" and a sphere has "zero holes", and we have a theorem saying that if two things are homeorphic they must have the same number of holes (the number of holes is an invariant). Therefore we know that a donut cannot be homeomorphic to a sphere. Usually we have more sophisticated invariants (homotopy groups, homology groups, the cohomology ring, and other stuff) that are not just numbers but algebraic structures, but the same principle remains. It turns out a lot of these invariants are actually invariants for homotopy equivalence, that is, they will not be able to tell apart homotopy equivalent spaces. This is useful to know: for instance, a band and a Möbius band are both homotopic to a circle, so you know that if you want to tell them apart, you're going to need more than the classical algebraic invariants (if you know a bit of algebraic topology and you're curious about that, this can be done by thinking of them as vector bundles, but also through more elementary methods, see this stackexchange post). Also, if you want to calculate some invariants for a complicated space, a good place to start can be to try to find a less complicated space that is homotopy equivalent to the original space (and this is often doable since homotopy equivalence is a kind of weak notion).

in ‘Murica land of the free every module is born with a basis

(link) In 1st-year linear algebra, we study vector spaces over fields. But in more advanced linear algebra, we study modules over rings, which are basically vector spaces, but over rings instead of fields. It turns out dropping the condition that every non-zero scalar must be invertible makes the algebra much more complicated (and interesting!). When a module has a basis, we say it is free, hence the joke. If this basis is finite, we say that the module has finite rank, and the length of the basis is the rank of the module (exactly like dimension for vector spaces!), hence the tag "not every module ranks the same though".

testicular torsion? this wouldn’t happen over a field

(link) Continuing on modules, modules can sometimes have what is called torsion. Let's take Z-modules, or as you may know them, abelian groups! Indeed, a "vector space over Z" is actually the same as an abelian group: any module has an underlying abelian group (just forget you know how to scale elements) and conversely, if you take an abelian group, you know that any element a is supposed to be 1a, so 2a must be (1+1)a = 1a + 1a = a + a. More generally, for any positive integer n, n.a = a + ... + a, n times, and if n is negative, n.a = (-a) + ... + (-a), n times. So knowing how addition works actually tells us how Z must scale elements. With that out of the way, take the Z-module formed by the integers mod n, Z/nZ. It is an abelian group, so a Z-module, but something weird happens here that doesn't happen in a vector space: n.1 = 0. You can scale something, by a non-zero scalar (in fact a non-zero-divisor scalar), and still end up with 0. This is known as torsion, and vector spaces (modules over fields) don't have that. So yeah, testicular torsion? that wouldn't happen over a field. Also, watch out: the notion of torsion for a module over a ring is not necessarily the same as the notion of torsion for the underlying abelian group. Z/4Z doesn't have torsion, when seen as a Z/4Z-module.

Mathematical band names

(link) For these posts, I'll be quickly explaining each band name, and i'll be including good additions from other peeps! (with proper credit of course, you can't expect a wannabe-academic to not cite their sources) (also plagiarism is bad) (if no one is credit that means I thought of the band name)

Algebrasmith, The Smathing Pumpkins, System of an Equations, My Mathematical Romance, I Don't Know How But They Found X, Will Wood and the Tape Measures (by @dorothytheexplorothy), DECO*3^3 (by @associativeglassdesert), The Teach Boys, Dire Straight Lines, n Directions, XYZ Top, Mathallica: I have nothing to explain here

Rage against the Module: if you've read the parts of this post about modules, you get it (partly inspired by the commutative algebra class I'm taking right now, I love it, but I've been stuck on a problem for some time)

The pRofinite Stones: a profinite space is a topological space obtained by some process involving finite, discrete spaces. They are usually called Stone spaces, hence the joke

Mariah Cayley: Arthur Cayley was a mathematician. It's the same Cayley from the Cayley graphs! (also Cayley-Hamilton, if you've heard of the theorem)

Billie Eigen: eigenvalues and eigenvectors are linear algebra concepts. For a given operator on a space, its eigenvalues are scalars that tell us a lot about the operator. This is not my field but I have heard in quantum physics physical quantities like mass, speed, etc are replaced by operators, and eigenvalues correspond to states the physical system can be in

Smash Product: in algebraic topology/homotopy theory, the smash product is an operation on pointed topological spaces that is interesting for categorical purposes (it gives a symmetric monoidal category structure to the category of compactly generated pointed topological spaces, if you know what that means) somebody once told me, the world is categories

FOIL out boy (by @mathsbian): FOIL is a way of remembering how to expand products that some people learn. It means First, Outer, Inner, Last. So if you expand (a+b)(c+d) using FOIL, you get ac + ad + bc + bd.

Sheaf in a Birdcage (by @dorothytheexplorothy): a (pre)sheaf is a way of assigning algebraic data to a topological space (or a generalized notion of space). A presheaf is a sheaf if the data respects some locality condition. (pre)sheaves were introduced by Jean Leray but really used by Grothendieck to completely transform algebraic geometry, and are now widely used in modern geometry (they show up to abstract the notion of "a geometric thing"). I can't explain much more as I am still learning about sheaf theory!

The Curry-Howard correspondents (by @dorothytheexplorothy): the Curry-Howard correspondence, in logic/theoretical computer science, essentially says that algorithms (computer programs) correspond to mathematical (constructive) proofs. I'm no computer scientist or logician so I'll avoid saying dumb stuff by not trying to explain more, but I know it can be made more precise using lambda calculus.

Le(ast com)mon Demoninator (by @dorothytheexplorothy): I don't think I have to explain anything here (let's be honest, if you're reading this, you probably already know what a least common denominator is), but I will say that the band name being spoofed here is Lemon Demon (Neil Cicierega's musical project) and I love his music go listen to it. Also I love the word demoninator thank you for that dottie

Taylor Serieswift (by @associativeglassdesert): a Taylor series is an infinite sum that approximates a nice-enough (analytic) function around a point. This is useful because the Taylor series only depends on the derivatives of the function at one point but can approximate its behavior on more that one point, and also because the Taylor series is a power series, so a more tractable kind of function. In particular if we truncate it, that is stop at some term, we get a polynomial that approximates our function well around a point, and polynomials are very nice to work with (this is where kinda cursed stuff you may have seen in physics like sin(x) = x or tan(x) = x comes from!)

mxmmatrix (by @associativeglassdesert): you may have heard a matrix is a table of numbers. Actually, it's much more than that. Matrices are secretly functions! In fact, very special kind of functions (linear maps) between very special kind of objects (finite-dimensional vector spaces). And if you've seen how to multiply matrices before but have not been told why we do it that way, be not afraid, there is actually an answer. The answer is that when we take some x, do one linear map f to get f(x), then another linear map g to get g(f(x)), we actually end up with a new linear map, gf. And if you take a matrix representing f and multiply it (left) by a matrix representing g, you get a matrix representing gf. This is why the matrix product is done like that: it's actually composition of functions! If this interests you, consider reading more about abstract linear algebra.

Ring Starr (by @associativeglassdesert): a ring is an algebraic structure. Take the integers. What can we do with them? We can add them together, addition is associative (when adding a bunch of stuff we don't need parentheses), commutative (a+b = b+a), we have zero that doesn't do anything when adding (a+0 = a), and we have opposites: for any integer a, we have another integer -a such that (a + (-a) = 0). But we also have multiplication: multiplication is associative (no need for parentheses again), commutative, we have 1 and multiplying by 1 doesn't do anything, and multiplication distributes over addition. Now, re-read what I just said but replace "integer" by "real number". Or "complex number". When seeing such similar behavior by different things (there are in fact many more examples that those I just gave), mathematicians are compelled to abstract away and imagine rings. A ring is a set of stuff, with some way to add the stuff and some way to multiply the stuff that satisfies the properties I talked about above. Sometimes we also drop some properties, for instance we allow multiplication to not be commutative (ab =/= ba). By allowing this, square matrices of a given dimension form a ring! Quaternions, if you know what they are, also form a ring. A lot of things are rings. Rings are cool. Learn about rings.

WLOGic (by @associativeglassdesert): WLOG is mathematician speak for "without loss of generality".

Alice and Bob Cooper: in many math problems, people are called Alice and Bob. Because A and B. Yes there is a Wikipedia page for this

The four Toposes: a topos is a kind of category meant to resemble a topological space. Grothendieck toposes are used in algebraic geometry and elementary topoi are used in logic. I can't explain more since I don't really know anything about topoises besides that they are kinda scary and that people really like to argue about what the plural of "topos" should be

Green-Tao Day: the Green-Tao theorem says that if you have a positive integer n, then you can find prime numbers p1, p2, ..., pn, such that they are evenly spaced (or equivalently, in an arithmetic progression). It's pretty neat. I have no idea how the proof goes, though. It must be pretty complicated, since it was proven in 2004.

Aut(Kast)/Inn(Kast): I'm really proud of that one. So if you have a group G, you can look at bijective group homomorphisms from G to G, or as they are more well-known, automorphisms of G. Together with composition, they form a group, called Aut(G). Now we already know of some automorphisms of G: if g is any element of G, then x ↦ gxg^{-1} is an automorphism of G (proof is left as an exercise to the tumblr). These automorphisms are called inner automorphisms of G, and they form a normal subgroup of Aut(G). The quotient group Aut(G)/Inn(G) is called the outer automorphisms of G and denoted Out(G), which is reason behind the band name.

Depeche modulo: modulo is a math word that means "up to [some notion of being the same]". For instance the integers modulo 7 are the integers but we declare that two integers a and b are the same if 7 divides a-b. From there we get modular arithmetic which you may have heard of. This kind of operation is called a quotient and is insanely useful in all branches of mathematics.

Phew! We're done with the band names. For now.

"oh you like math? what's 1975 times 7869?" well that's a great question Jimmy but to answer it first I need to construct the natural numbers. [...]

(link) So this is a post about a type of response math people get when they say they do math which is that people automatically assume this give us insane mental math power. It does not. The rest of the post is about constructing the natural numbers in the ZFC axiomatic system. I'm kinda lazy and don't want to get into all that but here's a good video by certified good math channel Another Roof about it: what IS a number? The same channel has several other videos on that same topic, go watch em if you're interested

1957 times 7869 (IF IT EVEN EXISTS) is the universal object with morphisms into 1957 and 7869

This is a joke by @dorothytheexplorothy in the notes of the previous post. The joke here comes from interpreting "times" are referring to the categorical notion of product. I'm actually not gonna explain anything here because 1) this post is taking forever to write and 2) I will probably rant about category theory in the future. Here are two videos by Oliver Lugg you can watch:

27 Unhelpful Facts About Category Theory (funni video)

A Sensible Introduction to Category Theory (serius video)

and here are two books you can use to learn more if you're interested:

Seven Sketches in Compositionality (very applied, very nice, I think easy to read)

Basic Category Theory (less applied, is a typical math book)

she overfull on my \hbox till i (5.40884pt too wide)

(link) This she on my till i joke is based on a LaTeX warning you get when it can't figure out how to typeset your document well and that leads to a margin being exceeded.

Time for the math battle reblog chain

(half of the posts are by @dorothytheexplorothy)

fuck you *forgets your group is a group and only remembers it's a set now*

So any group has an underlying set, and any group homomorphism is actually a map between these underlying sets. This means that the operation of "forgetting a group is a group and only remembering it's a set" is a functor. This is less useless than you might think, because of adjunctions.

two can play at that game *constructs a free group over this set, even bigger and better than the one I had*

So basically in lots of cases the functor that forgets some structure is (right) adjoint to some other functor. You do not need to know exactly what this means to read the rest, don't worry. What it means is basically that from the operation of forgetting some structure, we can get another operation, which adds structure, in a "natural" way. In the case of forgetting a group is a group and only remembering it's a set, the adjoint functor is the "free group" functor, that takes a set and constructs the free group on it. This idea of free objects works not just for groups but for a whole lotta stuff. See this part of the Wikipedia page on forgetful functors for some information.

oh don't get me started *abelianizes your free group, now it's just a big direct sum of Z's*

A non-abelian group can be turned into one through abelianization, which is quotienting out by the commutator subgroup. This makes sense: commutativity is asking ab = ba for all a, b, which is asking aba^{-1} b^{-1} = 1 for all a, b, which is precisely what we get when quotienting by the subgroup generated by words of the form aba^{-1} b^{-1}. Abelianization is also a functor, so it fits the theme. The abelianization of a free group is a free abelian group, and a free abelian group is a direct sum of a bunch of copies of Z.

big mistake, friend *moves over to the endomorphism group over that group and treating composition as multiplication, thus replacing it with a unital ring*

The endomorphism group of an abelian group is actually a ring (like in linear algebra, endomorphisms form a unital ring with composition as multiplication). I don't think this construction is functorial, though? (correct me if I'm wrong on that. correct me if I'm wrong on anything, really. if i'm wrong about stuff send me an ask and i'll fix it)

you fool, you fell right into my trap! *takes the field of fractions of your ring* have fun working in the category of fields! now you only have monomorphisms and your eyes to shed tears

So I thought I had the advantage here because fields are, categorically speaking, very bad. This is (I think) mainly because a homormorphism of fields is always injective (so is a monomorphism, that is left-cancellable). In fact, products of fields don't exist, direct sum of fields don't exist, a lot of categorical constructions we usually like don't exist in this category. We basically only have inclusions. I will elaborate on why I was wrong in my post here in a bit

fuckkk idk enough about schemes or whatever to get out of this! you've bested me X(

Schemes are the main objects of study of algebraic geometry. I won't being to try and explain what they are because it is very abstract and I don't even really understand the definition (yet). I just know they're algebraic geometer's analogue of a "geometric object", like how smooth manifolds are to a differential geometer.

wait actually I just realized the ring of endomorphisms of a free abelian group has no business being an integral domain, or even commutative. so I think taking the field of fractions makes no sense, and I actually lost the battle.

So the field of fractions construction only makes sense for integral domains. The name of this construction is really explicit: passing from an integral domain to its field of fractions is the same thing as passing from Z to Q, or from k[X] to k(X) if you know what that is. However I made a mistake, since the ring we were talking about is almost never commutative (much like matrices).

WON ON A TECHNICALITY LET'S GOOOOO

well played, dottie

yeah, uh, we oidified your boyfriend. yeah we took his core concept and horizontally categorified it. yeah he's (or they're?) a many-object version of himself now. sorry about your one-object boyfriendoid

(link) Oidification (also known as horizontal categorification but "oidification" sounds funnier) is a way of categorifying a concept, by turning it in a "many-object" version of itself. For instance, a one-object category is precisely a monoid, so the concept of category is the oidification of the concept of a monoid. A category where every morphism is invertible is called a groupoid, and a one-object groupoid is precisely a group. The name "oidification" probably comes from the fact that after being oidified, the name of the concept gets added the suffix -oid. So a category is a monoidoid. In fact, you can even have monoidal monoidoids. Category theory really is well-suited to shitposting huh

My advisor [...] stared into my soul and noticed I liked categories. It's over for me, i am going to end up a homotopy theorist, or worse, a youtuber

(link) Category theory has the reputation of being abstract nonsense. I don't disagree. I guess I have a slightly-above-average tolerance to category theory and algebra. This has led to a not-insignificant amount of people in my life telling me I'm gonna end up in one of the abstract-nonsense-related fields like homotopy theory, infinity-category theory, etc. The "or worse, a youtuber" part was stolen from the following quote

Research shows that when someone becomes personally invested in an idea, they can become very close-minded. Or worse, a youtuber.

-hbomberguy, Vaccines and Autism: A Measured Response (4:12)

(this video is incredible, if you haven't seen it yet, go watch it)

PHEW.

I'm done. For now. This took multiple hours to write. I hope you enjoyed this post! If you enjoyed it, please let me know! If you have any questions or want to tell me "youre doing good lad" or want to yell at me, my asks are open! Thank you for reading this far! If there is a post I talked about here you found funny, you can click on the (link) to look at the original post. Give me those sweet sweet statistics. I crave them. I NEED that dopamine hit of knowing someone interacted with my blog. ok bye

I have a new post up on my blog, continuing the Fictional History of Numbers series. In part 1 we built on the natural numbers using algebraic operations, and got the algebraic numbers. In part 2 and part 3 we used geometric and analytic arguments to build up the real numbers.

These two sets of numbers overlap, but aren't the same; there are real numbers that aren't algebraic (as we saw in part 3) but also algebraic numbers that aren't real. So what happens if we combine the two? We get the complex numbers, which are complete and also algebraically closed. But proving this is a little tricky, and touches on the deep strangeness of complex analysis.

And in the process of adding algebraic closure to the real numbers, we lose the ability to order them, which has its own consequences.

I have a new post up on my blog, continuing the Fictional History of Numbers series. In part 1 we started with the natural numbers and built up the algebraics, which let us solve equations. In part 2 we started asking geometric questions, and constructed the real numbers.

But the real numbers are weird and hard to define. In part 3 we see one way they're extremely strange, and then talk about why we want them anyway. In the end, we shouldn't worry about the definition of the reals; we should worry about what they allow us to do. And it turns out they're exactly what we need to make calculus function as it should.

A new post up on my blog!  Last time we talked about the algebraic numbers, and how just wanting to solve simple equations can create a ton of different numbers.  But they don’t get us everything.

So this time we start off with the idea of measurement, and wind up inventing the real numbers.  The real numbers are weird.  Real weird.  But they show up when we start asking questions about size or measurement.  And in part 3, we’ll see they’re exactly the right way to do calculus.

Today on the blog I start a new project: where do numbers come from?

By which I mean, mathematicians deal with lots of weird kinds of numbers. Real numbers, complex numbers, p-adic numbers, quaternions, surreal numbers, and more. And if you try to describe the more abstract types of "numbers" you sound completely incomprehensible.

But these numbers all come from somewhere. So I'm going to take you through a fictional history of numbers. Not the real history of the actual people who developed these concepts, but the way they could have developed them, cleaned up and organized. So in the end you can see how you, too, could have developed all these seemingly strange and abstract concepts.

This week in part 1, we cover the most sensible numbers. We start with the basic ability to count, and invent negative numbers, fractions, square roots, and more.

But that will still leave some important questions open—like, what is π? So we'll have to come back for that in part 2.

fields of mathematics

number theory: The Queen of Mathematics, in that it takes a lot from other fields and provides little in return, and people are weirdly sentimental about it.

combinatorics: Somehow simultaneously the kind of people who get really excited about Martin Gardner puzzles and very serious no-nonsense types who don’t care about understanding why something is true as long as they can prove that it’s true.

algebraic geometry: Here’s an interesting metaphor, and here’s several thousand pages of work fleshing it out.

differential geometry: There’s a lot of really cool stuff built on top of a lot of boring technical details, but they frequently fill entire textbooks or courses full of just the boring stuff, and they seem to think students will find this interesting in itself rather than as a necessary prerequisite to something better. So there’s definitely something wrong with them.

category theory: They don’t really seem to understand that the point of generalizing a result is so that you can apply it to other situations.

differential equations: physicists

real analysis: What if we took the most boring parts of a proof and just spent all our time studying those?

point-set topology: See real analysis, but less relevant to the real world.

complex analysis: Sorcery. I thought it seemed like sorcery because I didn’t know much about it, but then I learned more, and now the stuff I learned just seems like sorcery that I know how to do.

algebraic topology: Some of them are part of a conspiracy with category theorists to take over mathematics. I’m pretty sure that most algebraic topologists aren’t involved in that, but I don’t really know what else they’re up to.

functional analysis: Like real analysis but with category theorists’ generalization fetish.

group theory: Probably masochists? It’s hard to imagine how else someone could be motivated to read a thousand-page paper, let alone write one.

operator algebras: Seems cool but I can’t understand a word of it, so I can’t be sure they’re not just bullshitting the whole thing.

commutative/homological algebra: Diagram chases are of the devil, and these people are his worshipers.

Word rhyming is an equivalence relation

Take the definition that two words rhyme if and only if they end with the same sound.

Reflexive: Every word rhymes with itself.

Well, if two words are the same, all their sounds have to match, including the final one, so this point holds.

Symmetric: If A rhymes with B, B rhymes with A.

This one’s really hard to prove, because it’s so obvious. If A rhymes with B, then the final sounds of A and B are the same. They will still be the same if we swap the words around. Please don’t make me explain it more, I’ll cry.

Transitive: If A rhymes with B and B rhymes with C, then A rhymes with C.

Call the sound at the end of word A ‘&’. If A rhymes with B, then B also has to end with ‘&’. If B rhymes with C, and B ends with ‘&’, then C also has to end with ‘&’. This means that both A and C end with ‘&’, and so A rhymes with C.

There we go. The argument no one cares about but me has been made. Rhyme is an equivalence relation. You can all go home.

Suggested Alternatives to the One China Policy

Currently, the policy of the United States on the Taiwan question is that the US recognizes that polities on both sides of the Taiwan Strait hold that there is only one China and that Taiwan is part of China. In the current tense international climate, it may be useful to considers alternatives to that policy.

Two Chinas Policy: The United States recognizes the independence of Taiwan as a sovereign state, separate from the People's Republic of China.

Three Chinas Policy: The US recognizes Taiwan, Hong Kong, and the mainland as independent states.

Four Chinas Policy: The US recognizes Taiwan, Hong Kong, Macau, and the mainland as independent states.

One China Policy (Retro 1978): The US switches its diplomatic recognition back from the PRC to the ROC.

One China Policy (Retro 1911): The US recognizes the Qing Dynasty as the legitimate government of China and finds some schmuck to play Emperor-in-Exile.

Many Chinas Policy: The US recognizes the sovereign independence of every Chinese province.

Too Many Chinas Policy: Hong Kong makes a perfectly fine city-state, so why not let everyone do that? The US recognizes every Chinese municipality as its own independent state.

1436506450 Chinas Policy: The US recognizes the sovereign independence of every Chinese person.

2^1436506450 Chinas Policy: The US recognizes the sovereign independence of every subset of of the set of all Chinese persons.

2^1436506450-1 Chinas Policy: Same as above, but not including the empty set, because that doesn't even make sense because it's already claimed by Germany.

Infinite Chinas Policy (Countable): The US recognizes that (1) The PRC is a China and (2) for every China c, the successor S(c) is also a China, and (3) for every China c, c != S(c).

Infinite Chinas Policy (Uncountable): The US recognizes that the set C of all Chinas is an ordered field, and that every non-empty subset of C with an upper bound in C has a least upper bound in C.

No Chinas Policy: The United States embraces mereological nihilism and recognizes only atoms and the void.

Generalized Golf

I have looked up nothing about golf to write this.

Let C be any topological space. We will call this the ‘course’. For any two points x,y ∈ C we have a collection S_xy of ‘shots from x to y’, where each ‘shot’ s ∈ S_xy is a path in C from x to y, which is to say a continuous function s: [0,1] → C with s(0) = x and s(1) = y. For a shot s ∈ S_xy we call x its ‘start’ and y its ‘end’. Let S denote the collection of all shots in C between any two points.

A ‘hole’ on C is a triple (t,h,p) where t ∈ C is a point called the ‘tee’, h ⊂ C is a subset called (confusingly) the ‘hole’, and p is an ordinal number called the ‘par’. For any cardinal number κ we define a ‘golf’ of length κ to be a function g: κ → H, where H is a set of holes on C. A golf g is called ‘finite’ if κ is finite and the par of every hole in the image of g is finite. We define the par of a finite golf as the sum of the pars of its constituent holes.

A quintuple (C,S,κ,H,g) defined like above is called a ‘game of (generalized) golf’.

Take a hole (t,h,p), a successor ordinal ω+1. Let F: ω+1 → S be a function such that F(0) is a shot from t, for every i < ω the end of F(i) equals the start of F(i+1), the end of F(ω) is an element of h, and no F(i) ends in h before this. Such an F is called a ‘play’. We call ω the ‘score’ of F.

A ‘golfer’ is a collection of probability spaces, which for any shot s ∈ S with start x and end y gives a probability space on the set of shots from x. This is to be interpreted as the ways in which a shot can deviate from the golfer’s intent.

…

Now to define the real numbers by way of games of golf on ℚ.

Thanks

she circle my unit until i circumference

just truly bonkers how much i love lying down..........like being horizontal? unparalleled

AAAAAAHHHH I HAVE A MATH TEST TMRW

Normally, I’m not so stressed, but I missed class this week (I was skipping shame on me 😭) and I have no clue what’s going on… Me and the math teacher are besties because he’s the coach of the golf team, so I’m scared of disappointing him with my bad test scores.

I also had a long lab in chem this week. It’s really sad because we got bad results even though we ran our trial for over and hour lol. I think it’s my lab partner’s fault because he keeps on putting his crusty fingers all over the cuvettes which is messing up the spectrophotometer. Hopefully he actually writes his part so it’s not another repeat of last year.

- Practice midterms for matrices

- Grade practice midterms

- Write lab report

- Debug binary system simulation

- Work on astro research paper

- Work on program applications

- Finish new deal research

- Finish college assessment

- Email people back 😬

Don’t ask me why we have a midterm in January… I have no clue.

Pretty winter sunset but I’m stuck inside 🌅 Just cleaned my room though so I don’t mind that much.

3rd picture is my desk— should I put something on the wall? I don’t want to drill anything in.

Today was pretty frustrating for a variety of reasons, so I didn’t think I would get much done today but then I hopped on to Tumblr and saw a bunch of people’s study posts so I guess I’m trying this again.

- Study for math test

- Work on research program applications

- Upload a new software onto my computer

Good luck 👍

i think i’m cooked for this midterm. and i haven’t even studied for hyperbolic paraboloids or hyperboloids or ... i don’t even remember their names

my test is in two days lol