TotientQuotient
@t0tientqu0tient
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Survived tripos somehow. All effort, no skill. Still trying to become a mathematician. Probably suffering from some form of the Algernon-Gordon effect.
a real vector space element
Joined July 2023
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Last Seen Profiles
Yeah it's kinda hard to do that when the exam questions are of this difficulty
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For those who love algebraic graph theory ✍️
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Markov chains play a central role in probability theory, and a particular area of interest to mathematicians is that of random walks on groups. Here's an interesting example where we can easily find the invariant distribution by associating the MC with a sequence of iidRVs.
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(Since this post is blowing up and I didn't make it obvious in the original post, these aren't real exam questions, they're made up for memeing purposes)
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Hey darling, are you the kernel of a ring homomorphism? Cuz damn, you're ideal.
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@Anthony_Bonato
A mathematician is someone who says "a set of people" instead of "a group of people" irl unless they can prove that there exists a well-defined binary operation on the set of people in question satisfying closure, associativity and existence of identity & inverse.
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@Mathinity_
X = set of unordered pairs of distinct subsets of {1,2,...,n} of size 2
Y = set of subsets of {1,2,...,n+1} of size 4
Define f: X → Y
{x₁, x₂} ↦ x₁ ∪ x₂ if |x₁ ∪ x₂| = 4
x₁ ∪ x₂ ∪ {n+1} if |x₁ ∪ x₂| = 3
Then f is 3-to-1.
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that's a pretty S∪S way to say that the set S generates the group G
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Mathematicians are the type of people who would unironically put parentheses in sentences for the purpose of disambiguating the parsing possibilities, regardless of how silly the resulting sentence looks.
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@Anthony_Bonato
Applied mathematicians be like: let's allow the definition of "linear combination" to include combinations of infinitely many vectors, so that the dimension is just \mathbb{N}
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How I published 0 single-author papers during my 3 year undergrad:
- Discovered a possibly novel result, LaTeXed it into a research article, complete w/ abstract & references
- Tried submitting to arXiv, got rejected
- Haven't found someone who could help me submit to a journal
How I published 13 first-author papers during my 2.5 year post-doc:
- All the data were already collected
- 5 papers were literature reviews
- I read incredibly fast
- I write even faster
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@anaghadoesmath
Thankfully these aren't real exam questions lol, I made them up for the memes. But the real deal is frankly not that far off. And yes, every course appears on every paper in part II so you can't just revise 1 course for each paper, you gotta revise everything at once.
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@MathMatize
Same energy as "trivial", "elementary", "obviously", "clearly", "follows directly from definition", "one can check that" & the like.
See also: "by some wicked sorcery" - taken from a real piece of homework I submitted
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@Anthony_Bonato
The question must've been:
Find the Fourier series solution to the heat equation
uₜ(x,t)=13uₓₓ(x,t)
on a rod of length π whose ends have fixed temperature zero, i.e. boundary conditions
u(0,t)=u(π,t)=0
and a triangle-wave shaped initial condition, i.e.
u(x,0)=|x-(π/2)|-(π/2).
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Late update, but I passed by the slightest of margins. Not a result I'm particularly proud of, but it's the culmination of 2 decades of hard work. On the bright side, things could've gone so much worse - had I been slightly unluckier, I would've left this place without a degree.
Results coming out tomorrow. I hope I can at least graduate from this place. Would really suck if 3 years of hard work amounted to nothing.
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#Math
Recently I've been thinking about functions that commute under composition with their derivative, i.e. solutions to the ODE
(†): y'(y(x))=y(y'(x)).
Obviously, if y'=y then LHS=y(y(x))=RHS, so (†) trivially holds. Thus
y(x)=Beˣ, B constant,
is a family of solutions.
(1/)
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@LongFormMath
And that's how mathematicians end up resorting to proofs by intimidation and proofs by cumbersome notation.
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@yiannis_entropy
And most algebraists view category theorists the same way said mathematicians view algebraists.
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Wrote a rant about identifying levels of nestedness in
#Math
and how this important skill lacks attention in its isolated form from a pedagogical standpoint. If you relate to it and/or agree with the existence and extent of this issue, please retweet and share to raise awareness.
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@TonyTheLion2500
"What is a tensor?"
"An object that transforms like a tensor."
Yeah, I understand that feeling.
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@miniapeur
Physicists strike me as the type of people that would allow infinite linear combinations. I wouldn't be surprised if they say that
<(1,0,0,0,...),(0,1,0,0,...),(0,0,1,0,...),...>
is a basis for \mathbb{R}^\mathbb{N}.
But hey, I know nothing about physics, so who am I to judge 🤷♂️
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Let S be a vector subspace of ℝⁿ of dimension d. Then ℝⁿ\S is:
• disconnected if d=n-1;
• path connected but not simply connected if d=n-2;
• simply connected if d=n-3.
For fixed k≥4, is there a generalisation of "simply connectedness" that holds iff d≤n-k?
#Math
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@WaltonStevenj
Worst part is that I've seen this 4 years ago when I was preparing for my undergrad admissions interview 💀
Having tried, they're mostly doable (way easier than IMO) w/ pen & paper, but it's designed to be done without, in a high-pressure interview in which I'd probably crumble.
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@miniapeur
me when I ask people to explain elementary results about groups/rings/fields/modules/vector spaces and they're like "follows from the yoneda lemma", "sequence is exact", "trivial because this diagram commutes", "[...] is contravariant"
did they seriously think this would help lol
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@yemeen
Even cooler that the function
p ↦ πₚ
has symmetry under
p ↦ p* = 1/(1-1/p).
It might seem like a trivial observation that if p and p* satisfy
1/p + 1/p* = 1,
then the unit circles in the two norms produce the same geometric shape when graphed...
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Forget graduate math, forget learning "by themselves". I struggled learning *under*graduate mathematics even though I had access to some of the best teachers in the world.
Do people actually learn graduate math successfully by themselves? I honestly feel like I'm just a hopeless cause. 😭🫠I really struggle with a lot of things about math. I tend to just change subject when I'm stuck somewhere, which has ended me here. 😭💀💀😳😳
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My younger son (12) isn't much of a reader. He craves Roblox, Youtube Shorts, Fortnite, anything but a book. So I gave him the most brutally addictive novel I could think of, ENDER'S GAME ... and it's working. I think. But I can't think what to give him next. Suggestions?
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Worst part is when they leave the proof of said theorem as an exercise. How do they expect us to reinvent the wheel (something that took dozens of mathematicians centuries of struggles to accomplish in the first place) by ourselves within a few hours?
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Uni made me lose touch with reality. Whenever I struggled academically & reached out to my peers, their only advice was to get a neurodivergence diagnosis (I'm neurotypical). When I was struggling w/ sleep, I've been told to (ab)use drugs. Luckily I didn't do any of that, but...
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I'm in London for the next few days. Any mathematicians from ICL/UCL/KCL willing to meet up with an aspiring postgrad student? (About myself: I just graduated from Cambridge with a maths BA. Interests include graph theory, Markov chains, groups, number theory, cryptography etc)
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You know you've gone too far down the abstract algebra rabbit hole when your first reaction to the word "factorisation" is to assume that it refers to ideals in a ring. Now I can't get this version of the Chinese remainder theorem out of my head and idk if that's a good thing.
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@MathMatize
However, when the individual sums in the sum fail to converge absolutely, the sum of the sum need not equal the sum of the sum
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Are people even seeing my tweets now? Has the algorithm gotten this bad?
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@scottba17371714
Normally I'd agree, but in this case, try the question first. You'd be surprised when none of the methods you've learned (separation of variables, integrating factor, exact ODE etc) seem to work. And that's for a good reason - I made the question nasty on purpose :)
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I bombed all my exams.
2 decades of hard work down the drain.
My dream of going into academia shattered.
Yes, I can sugarcoat my failure story as much as I want, but at the end of the day there's still some extent to which my lack of ability and potential is being reflected.
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@howie_hua
By this logic, if a reply to a reply to a reply to a comment equals the original comment, then reply chains of all periods exist by Sharkovsky's theorem.
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My problem isn't not understanding maths; it's the *rate* at which I understand new pieces of maths. What's the point being a tortoise who'll always reach the finish line when that isn't even the win condition? It's peak speed that matters, and the hares dominate in that regard.
Why you don't understand math — and what you can do about it. ⤵️
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After 4 years of learning group theory, I think I finally achieved the level of competence required to tackle my 1st year exams. Pretty demoralising that it took me 4 years to achieve what's expected from me in 4 months. Such is the struggle of a tortoise in a hare vs hare race.
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Now that I have more followers, here's a repost of a meme I posted last year
(This was a real exam question for 2nd year undergrads, just without the funny emojis)
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@mattecapu
My answer will be & has always been "I seriously don't care", because this isn't something that makes or breaks mathematics, unlike the axiom of choice discourse, in which case my answer would be "I *should* care, but I don't know enough maths yet to make a stance".
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In maths, it's 35 hours reading 6 papers. The main difficulty in reading papers in other fields is their verbosity (so skimming through them helps), but in maths, the problem is the exact opposite: brevity at the expense of confusing notation and hypercompression of ideas.
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As someone who studied mathematics at the undergraduate level, I know full well the feeling that the top x% mathematician feels a hundred times better at maths than the top 10x% for every real number x in (0,10].
One of my most closely-held beliefs is that the average person has no idea how much better a professional athlete is than they are
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Very nice illustrations of Cayley digraphs.
There is a marvellous group theory book on Quora from Roman Andronov, copiously illustrated, giving geometric intuition into what these abstract group theoretic objects track:
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And here I am, reading literally every paper that catches my attention, regardless of whether or not I understand the abstract. I do this purely out of passion and curiosity, as well as the desire to find out what research-level mathematicians do on a daily basis.
Advice I heard an old Fields medalist give to young mathematicians 25 years ago: "If there is even one word in the abstract you don't understand, don't read the paper"
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Had the opposite experience at Cambridge. Been self-studying uni-lvl maths yrs in advance, yet struggled cuz had to compete w/ ppl who knew little b4hand but had insanely high d(knowledge)/dt. Idk how but Cambridge is rly good at spotting such students in their admissions process
This was a huge realization for me in STEM classes at Harvard. Being from a rural area that didn’t offer the classes that other students had, I was taking computer science courses alongside people who had been programming since they were 10 years old. I was taking math courses
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I had a dream where this question came up in my 1st year undergraduate maths exam.
Thankfully it was just a dream.
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@miniapeur
I mean, if the Frobenius map
x ↦ x^p
is an endomorphism of a field with characteristic p, then surely we could just take p=1/2 and make it work... right? After all, we just need to find a field with half an element and prove that 1/2 is a prime, it can't be that difficult!
/s
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American mathematics is wild. You have people saying stuff like "f is one-to-one and onto" instead of "f is bijective". Also, wtf is a "trapezoid"? Why is "factor" also a verb? My British-pilled brain cannot comprehend.
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Before my undergrad: I don't know.
After my undergrad: I don't know and I probably never will.
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I've had both the pleasure and the misfortune of experiencing the good, the bad and the ugly of the mathematicians. Yes, some are very nice & open-minded, but there's also a lot of snarky, elitist assholes who spend all day complaining that people are too stupid to do maths.
@_onionesque
My experience is mathematicians are generally very open minded. OR maybe that's simply because I tend to interact with those who have bothered to interact outside their field!
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Algebraic graph theory is one of my biggest interests as of late, but why do we study it? Here's 1 reason: a graph G's adjacency matrix defines a quadratic form whose symmetry group is precisely Aut(G). So we can reduce a problem about graphs to one of pure algebra & vice versa.
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My exams start in 4 days.
2 decades of work (yes, been doing maths workbooks since age 1) comes down to this.
My uni never gives 2nd chances & I need a miracle to even pass. I want to pass and ideally do well because I want to be a research mathematician.
Wish me luck, X/Twitter.
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- Undergrads are meant to do exams, not research
- I was terrible at exams despite my best efforts, so my uni considered me a lost cause
- Thus no one at my uni was keen to help me w/ said paper; they all asked me to focus on exams
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I've never made a proper maths video before but I really really really want to spend the next month learning manim and use that to make a video about algebraic graph theory to submit to
#SomEpi
. Is that a good use of time?
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Found an answer. As a bonus, I stumbled upon this paper with a cute title. Hamilton would be proud.
#Math
Are there any examples of graded rings/modules/algebras where the grading isn't ℤ/2ℤ or ℕ?
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This is why I struggle to relate to others and why they struggle to relate to me. I tell people how much harder university level stuff is compared to everything before, yet they tell me that their experience is the opposite.
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My biggest regret is going to the University of Cambridge. It was simply not the place for me. I nearly worked myself to death (literally - I almost attempted s**cide) just to graduate with a third instead of a fail. Yet a third gets me nowhere, so what's the point?
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Mathematicians pride themselves on being rigorous and precise with their statements, but posts like this aren't doing us any favours.
The least you could do is add the phrase
"for x, y, z, w ∈ ℤ".
It's not like you're running out of characters - plus you have a blue check!
Here is the only number that cannot be written as x² + 2y² + 7z² + 11w²
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Every time I tell people that I'd like to think I'm capable of doing research despite doing poorly in exams, someone always tells me that having such confidence in the absence of research experience is the very definition of the Dunning-Kruger effect.
Thoughts?
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I know lots of "no math" people more than capable of doing a math PhD. One of my high school classmates who's now studying medicine humbled me in measure theory the other day.
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Today I learned about Hartogs's extension theorem: if n≥2, G is a connected open subset of ℂⁿ and K is a compact subset of G, then any function in n complex variables that's holomorphic on G\K can be extended to a unique holomorphic function on G.
(1/3)
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I have the utmost respect to anyone who can juggle commitments alongside any degree (let alone a PhD). I had the privilege of being able to spend 16 hours a day studying yet I still nearly failed my undergrad.
Yet I'm applying to grad schools with what's essentially a 2.0 GPA...
I used to be shy to publicly state that I had both of my kids during my PhD, but now I'm like: "Damn straight, I had two babies AND STILL managed to become a doctor in 5 years!" 👩🎓
@Momademia
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@TonyTheLion2500
These statements are equivalent. However,
(∀x)(∃δ) and (∃δ)(∀x)
are two different statements.
It's worth noting that ∀ commutes with ∀, ∃ commutes with ∃, but ∀ does not, in general, commute with ∃.
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@LucinaUddin
My undergrad professors like to say stuff like "I'm bounded by contract to not lie, and that includes lying by omission. As much as I'm happy to write about your contagious passion and unparalleled work ethic, I cannot omit your immense struggle and subpar grades."
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Has IMO gotten easier this year or have I just gotten better at maths? I've never been able to solve a single IMO question in the past, but this time I spotted a solution for Q1 and Q5 almost instantly. After a while I think I also have a solution for Q2 but I'm not sure.
International Math Olympiad (IMO) is the hardest math test for high schoolers.
—USA beat the favorite China with a mostly Chinese-American team
—India came 4th, 1pt short of Korea
—Haojia Shi [China] got a perfect score 2x in a row, the 7th to ever do that
6 questions, NINE hrs!
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- Hence I'm basically on my own, which is bad b/c validation from someone established is a near-necessary condition for 1st-time publishers to not get labelled as spam
- For the record, that paper was hardly a distraction from exam preparation
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I'm leaving the UK tomorrow. Part of me can't help thinking that it's about time I got out of this hellhole, but another part of me feels greatly indebted to all the amazing people I've met over the last few years.
Anyhow, it's going to be a new chapter of my life.
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If mathematics has taught me one thing, it's the ability to think and look before you ask. So much that I now get annoyed when someone asks another human a question that could be answered by either ChatGPT within 20 attempts or by Google search within the first 5 pages of results
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@MathMatize
Me personally, I have more trouble with the symbol for the Weierstrass elliptic function.
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Doing maths at the graduate level.
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I can't for the life of me remember whether it's the domain vector space or the codomain vector space whose change of basis acts contravariantly on linear maps. As such, I get the change of basis formula wrong half the time.
I’m a chess master and I can’t remember which month comes after another without saying all 12 out loud. My wife, a brain scientist, gets left and right mixed up close to 50% of the time when giving directions. Does everyone have some sort of basic knowledge blind spot like this?
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However, this example is only nice because the stochastic process on the associated sequence of iidRVs has a nice interpretation. Are there any other examples of such Markov chains (either another choice of µᵢ's or an entirely different family of transition matrices)?
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@tara_taylor
@Anthony_Bonato
Yeah, it becomes a habit. I struggle to communicate with non-mathmos because I always talk like this:
"there exists a non-empty subset of his friends who drink a non-zero amount of alcohol"
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what is the most deranged thing you’ve spent your adult money on? i’m trying to observe something
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Everyone keeps telling me that if I just keep trying different methods, I'll always solve it eventually. But more often than not, I pour hundreds of hours into an undergraduate-exercise-difficulty problem to no avail. Is this survivorship bias? Am I just not fit for maths?
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just the fact that such practices were so normalised in our uni makes me feel like I've been tainted with bad influences despite constantly telling myself that "this is bad and wrong".
Anyone else have a similar experience? I don't even know what's normal and what's not anymore.
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