I’m a graduate student in pure mathematics at UC Berkeley, and this is my math blog. They say that the best way to learn is to teach, and I figure blogging is close enough.
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Annoying Precision 
Hi!
Are you aware of some rendering issues on WordPress ?
Some formulas in your article are showing a red “Formula does not parse” warning (see https://qchu.wordpress.com/2017/11/26/the-representation-theory-of-the-additive-group-scheme/ for instance).
I think this is an issue with WP since https://terrytao.wordpress.com/2012/09/25/a-partial-converse-to-bezouts-theorem/ is also affected.
Thanks for the heads-up! Looks fine to me now; this happens every once in awhile and I’ve never been able to figure out why.
Hi Qiaochu. I noticed on your Berkeley webpage there are several interesting-looking unlinked talks. If you have copies would you mind sharing them please? isomorphisms (no e) at SDF dot org.
Pretty interested to know what you had to say about
Diagrammatic algebra and the representation theory of SU(2)
Derived stacks
Higher linear algebra
Thanks for the many blog posts and explainers over the years.
Also, I noticed you posted a paper in humancompatible.ai (although the authors listed in that paper don’t seem to match your name). Advice from someone who has spent too many years in the “Knightian uncertainty” + forward-only search + language-via-computer-science realms: tread carefully. Even David Mumford got sucked into this nonsense. (look up “the dawning age of stochasticity”). It would be a shame for your obvious talents to go to waste in the service of obsolete social-science dogmata. If you want more detail about why this is a great field to avoid, again feel free to ping me at SDF.org.
Thanks again for all your public work.
Thanks! The material in Higher linear algebra is more or less the contents of this blog post, but I don’t have any written material about the other two.
I don’t have any such paper, but in general I think the Center for Human-Compatible AI is doing important work, none of which I would describe as “obsolete social-science dogmata.”
Hi. I would like to know how do you make LaTex work in a post excerpt on your homepage. I had to put the whole post to make Latex work; if I chose to put an excerpt, Latex would not parse! Thank you
Hi, do you do also new mathematics? If yes where can I find your papers?
Hi, I am a sophomore in pure math. I want to apply for phd in pure math after graduation. I heard that research experience is very important in the application process. I have not done too many advanced courses yet. I really prefer to do some self-study in the summer instead of spending time on something beyond my ability. But I worry that I might fall behind since I only have one summer left if I do not do a research this summer. Do you think that undergraduate research experience really that important? Will it be better that if I try to learn more things myself this summer ,take some graduate courses next year, and try to do a research that I am really capable of next summer? will my background be considered weak if I only have one research experience?
My impression is that research experience is not very important and that strong recommendation letters will generally count for more. This is because most undergraduate research is bad, whereas recommendation letters come from professional mathematicians and can attest to your potential for producing good research in the future, which is more important than your ability to produce good research now.
Thank you very much! By the way, I enjoy reading your posts a lot : )
Thank you very much! 🙂
Hi, I’m an 18 year old in the UK and im working real hard to try and make the UK IMO team, but generally I do it because I love math, the IMO is just a motivation taht makes me work faster or harder. Could you give me ANY advice at all from your experience of doing math while at high school? I would love to hear any of what you have learnt throughout your journey and the things one should avoid and stuff (general things of course, I can’t end up trying to imitate someone else’s thinking patterns).
Hi! I am in the process of making my own blog. Content is in the making.
I really want to use your line “They say that the best way to learn is to teach, and I figure blogging is close enough.” with proper reference.
Would you please allow me to do the same?
Regards
Sure.
Like the professor in Good Will Hunting, knowing that people like you exist just sucks.
– Someone your age, a 4th year undergraduate struggling with the elementary
Hey, at least you can take solace in our lack of free will. Qiaochu is deterministically bound by the laws of physics to grow up the way he did and be where he is now, as are you. 😉
Hi Qiaochu. I was just wondering how you pronounce your first name (I assume your first/given name is Qiaochu). Essentially my doubt is on the “Q”: do you pronounce is like in “qi” and “Qing”, or like k? That would make the first syllable like italian ciao (hi/bye).
Thanks.
It’s roughly like a combination of ciao and tsiao. I’m not very good at describing Chinese pronunciation.
You should upload a recording of yourself saying your name; it would help a lot when I constantly run across your lovely answers on MO, and when I want to mention (aloud) that something I’m talking about was your idea.
Thanks for the compliment! My name can be pronounced “Chowchew.”
I’m guessing you’re a 3rd year graduate student now? Best of luck to you, and thanks for you’re teaching here and on MathOverflow!
That’s flattering! I’m actually about to enter my first year of graduate school.
How come you call that place “University of Berkeley” in your MathOverflow page?
Whoops. Fixed!
Hello, Qiaochu . I am also a math undergrad in Nigeria, but i will love to contact you on some admission requirements in MIT for Grad school. Kindly direct me on how i can reach you through e – mail. Thanks in advance.
Hi,
I hope this is not an inappropriate place to ask. But I noticed your interest in representation theory and combinatorics. I have to present a talk on the history of the representation theory of the symmetric group (upto say Knuths 1970 article on the RSK correspondence). Could you suggest some sources? (Curtis’ Pioneers is not detailed enough for my purposes). Thanks a lot.
Thank you for the answer, I just want to make another question.
How I said before, I start to learn superior mathematics at some months ago, If you could give me some advices about the main and most necessary subjects to learn at this moment so I can improve my background, and the best ways to understand and get a more profitable reasoning . I know that this is a question of taste too, so I have directed my study in what I prefer, that is abstract and linear algebra, calculus, number theory, I read before some things about functional analysis, and I think that is really interesting, but I’m afraid that I don’t have enough background to study at this moment.
Thank you very much, and I’m sorry to be so insistent
Dear Jorge,
Of course it depends heavily on what your interests are and what you want to do. My basic advice would be to learn everything very roughly just so you’re aware of the possibilities, then refine as desired. As for more detailed advice, I’m planning on writing a blog page about this sometime in the near-ish future.
Hello, I will enter this year to college, since I want to take investigation in mathematics, I decide to understand the advantages of using a blog where I can publish my work and another things related with mathematics. I read your advices, but I want to know if sustain such a blog is difficult to conciliate with your investigation work and how it helps in your career. Thank very much!
Dear Jorge,
I considered the time I spend maintaining my blog to be part of the time I spend thinking about mathematics as a whole. I don’t have a large enough audience that they have expectations about what I should be talking about, so I talk about whatever’s on my mind, and doing that helps me focus my thoughts and sort out details in a way that I find very valuable. As for how a blog helps in one’s career, I am not yet far enough in my career to offer even anecdotal evidence of this, but I have high hopes.
Just read this article to get a feel of how science blogging is good and can provide some very tangible results on one’s carrer: http://www.ams.org/notices/201003/rtx100300333p.pdf
hi, thanks for answering all my silly questions on math.SE 😀 also you are really smart!! did you go to IMO?
No problem. I never went to the IMO, although I did alright in the USAMO.
Very cool blog! I found it from AoPS, and I’ll definitely be stopping by more often. Hope you’re having a good time studying abroad!
hi luyi! wow what a small world
Thanks. I might actually try this method of learning; it seems like it would allow me to quickly get a basic feel for various subjects which I might not have time to study thoroughly.
This is probably off-topic but I think it is at least slighly relevant to mathematics: may I ask how many hours of sleep you usually get per day? To function properly and perform basic important tasks, experts usually recommend 8 hours of sleep per day (but some people can manage with less). However, with math, one needs to be able to think quite clearly at times and I have come to realize (over many years) that 8 hours of sleep is usually *not* enough. I would be curious to know whether you get more or less sleep than 8 hours and how you think it affects your ability to learn mathematics, and in general, do research.
I generally find that if I get around 7-7.5 hours of sleep, I simply can’t think very well so I would not be able to solve the harder problems in a text book whereas if I get 8 hours sleep I can think but can’t sustain concentration for long periods of time (say, if I was solving a problem that required this) and if I get 9 hours of sleep I really feel fresh and perform very well. Does any of this relate to you?
That sounds about right. I aim for 9 hours when I can, although I’ve been messing up lately.
I’m interested in math as well and do it all the time but I certainly don’t know as much as you do. May I ask how you managed to learn so much? For example, you appear to have done all the core areas of mathematics and have gone deeply into a number of them as well. Is this because you did advanced mathematics in high school or do you think you just use your time efficiently? For example, it would be interesting to know how many pages of material (that is new to you) you can absorb in a day. For me it’s usually 3-4 pages. If I go any faster, I can’t remember everything.
I don’t use my time particularly efficiently, but I did do a lot of mathematics in high school. The answer to your other question depends on what you mean by “absorb.” I don’t generally learn something by reading it once and understanding it. First I read it when I don’t have the background to understand it, then I might happen to read a blog post about it a few weeks later, then some things click, then a month or so later I have the background to understand it and when I read it again things click. Generally it’s a very piecemeal process like that, and for any given topic I talk about on the blog I might be anywhere in that progression.
Very interesting. That’s certainly different to the way I learn. You say that you might read a concept before you have the background to understand it; do you have a particularly good example of a concept that you studied in this way? What is your general feeling about learning mathematics by reading a textbook systematically and progressing to another textbook etc.? I’ve heard people saying that reading from textbook to textbook (like I do) is not the best way to learn once one reaches a certain stage in his learning (of math), but I’m not sure I understand the details completely.
I’ve actually been studying math for about three years and, without meaning to sound arrogant, have found that it’s possible for me to pick up concepts and get a “basic feel” for them rather quickly. My main problem though is that I quickly forget things; do you find that you forget some of the details of certain topics that you have studied, say, about six months later, or do you generally remember what you study for a reasonably long time? If so, it would be interesting to know what techniques you use to help your memory (e.g., I tend to forget proofs of theorems, even those that I worked out myself, quite quickly).
Apologies for the many questions; I’m just curious to find out what sorts of methods for learning math work out for different people.
The process I described is how I am learning category theory. I haven’t taken a class on it or gone through a textbook (although I’ve been meaning to read Mac Lane for ages now). Instead, I read a lot of random things: Wiki articles, nLab articles, John Baez’s blog, etc. I also slowly worked out some basic examples on this blog. At this point, while I wouldn’t say I know a great deal of theorems or proofs, I am very comfortable with the basic definitions and have absorbed, if you will, the basic philosophy of the subject.
Of course different things work for different people, and it’s not as if I don’t read textbooks (although I haven’t had time lately). But I generally don’t have the patience to do exercises, so when I’m learning something on my own my goal is to get a clear conceptual framework instead of to learn a lot of details. I also try to learn as many things simultaneously as I can so that I can more easily make connections between different subjects. Once I’ve built a conceptual framework and made enough connections, it’s generally easy for me to remember things. When I don’t remember the proof of a theorem, I try to work out an alternate proof, and if that doesn’t work I carefully scrutinize the original proof to figure out which step is hard to remember and see if I can conceptualize it so that it seems like a more natural thing to do. If something interesting comes out of this process I blog about it to ingrain it more thoroughly in my memory.
I notice that you post very frequently on mathoverflow and math.SE. Props for your work! I think these types of QA site may become very influential in the future and in the meantime, I intend to make heavy use of them for answering my questions.
That’s logical, I’d say physics and mathematics go hand in hand. Apparently Cambridge has the best mathematics department in the world too, so it’s a good move! Good luck with everything dude 😀
State is public. Private is for rich people. Hopefully I’ll be studying at Cambridge postgrad in a couple of years, though MIT would be my dream!….or CalTech 🙂
So do you do any other subjects apart from mathematics? And why did you choose Cambridge?
I’ve been trying to teach myself some physics, but other than that I haven’t seriously thought about anything besides mathematics for a few years now. I decided to come to Cambridge to get a different perspective and experience a different system from MIT’s, and that’s what happened, so I’m pretty satisfied.
I see. ‘Practice, practice, practice’ – a good mentality to have. Did you go to private schools or state schools? I’m very curious too and look up everything I don’t understand, but you’re on another level I think!
What I meant with the last question was, does the approach you use when learning mathematics transfer into how you grasp other subjects?
I can never remember which term means which – whichever one is free. I haven’t needed to grasp other subjects for awhile now. And I guess the specific approach I use to learn new mathematics is dependent on the mathematics I already know, so it doesn’t really generalize to a field about which I know relatively little (e.g. history).
Man, how are you so smart? Would you say it was nature or nurture? Mathematics is so complex at times, did you just used to practice all the time?
Does it help with learning other subjects?
It’s difficult to answer this question without sounding extremely arrogant, but: good resources, good teachers growing up, and curiosity. Both. Yes, I did used to practice all the time (and still do; that’s what this blog is for). No, it doesn’t really help with other subjects (depending on what you mean by this), except of course with the math in them.
Nice blog.
Keep on posting.
I like mathematics and I will visit the blog in regular basis.
I hope to see new posts every day. 🙂
Qiaochu Yuan,Your math blog is very good , my source of learning mathematics.
Hi,
I’m Brazilian and I would like talk with you about the process of admission on the MIT because I wish to attend the university.
And so,visit my blog for know about me and my goals:
(http://igortitara.wordpress.com/)
Do you have email?Hotmail or Yahoo?Anything.
And please contact me on my email igor_titara@hotmail.com.
Regards,
Igor Titara.
Hi Igor,
I’m not really the best person to talk to about MIT admissions; international admission is a little different and I don’t know anything about it. You should go to the website and contact someone at the admissions office; I’m sure they’d love to help.
[…] 19, 2010 in Matemática, Math, Putnam | Tags: Matemática, Math De ontem (via Qiaochu Yuan, Annoying […]
Hi,
I am bboypa of Mathlinks,congrats for your academic success and for your blog, it looks very nice; can i ask to modify some unparseable formula as the ones in the post prime and ideals (although they are intuitive), but for clarity; I know that latex in wordpress is damned, also additional spaces makes it unparseable xd
A tecnical question: how can i put a url link in this kind of blog? Have a nice day
Paolo Leonetti
That’s odd. WordPress seems to have modified its html parser.
Links are just the usual HTML syntax.
Is this the default requests page on this blog? If you do requests, I’d be interested in a post on how to calculate the genus of a curve using a Newton polygon.
I suppose it would make more sense to have a separate requests page. As for the Newton polygon, Danny Calegari briefly describes the process here but I don’t really have anything intelligent to say about genus myself.
Qiaochu, would you consider placing a link to a bibliography on this site – I see that you occasionally mention a specific text here and there, but it would be great to see what’s in your library.
That’s a good idea! I’ll get right on it.
Tom: I like this idea as well and will set one up for myself as well.
I like your blog and I’ll try to follow it. 🙂 Best regards!
Damnished MIT！
the URL is fine if you remove the last /. thx!
Fixed – there was an extra parenthesis at the end.
that URL doesn’t seem to work. gives me a 404 Error. thx
you cannot deny that you are a mathematical genius given the advanced nature of your blog posts here and your age, but how did you ‘become’ one?
Its really great that there is access to such wonderful and inspiring thinking. Don’t stop.
Hi Kim. Studies of successful academics show a high IQ (perhaps two standard deviations or so, e.g., around 130-140) is important but past some point offers no further success. Then what matters is social aptitude (and contrary to popular belief, math is a very social activity).
Environment and passion are about just as I important. My undergraduate advisor and I are amazing friends. We went to lunch most days and talked about everything. He’s one the most funny and caring people I know; we even exchange Christmas presents! Without this sort of nurture, I would likely be in Sillicon Valley right now working on a startup.
Thirdly, you have to realize that while it seems funny to think so, the math you learn in undergrad and grad school is likely to be by far the hardest to understand. This is because you are making about several dozen conceptual leaps in thinking that took mathematicians hundreds of years to make. After I learned basic commutative algebra and algebraic geometry, reading papers in these fields actually became fairly easy! It is like learning a language: it is more difficult to learn Japanese than to read and comprehend a very dense Japanese political text even though children do the former “so easily.”
It also helps to get started early. 🙂
The feed is at the same URL for every WordPress blog (https://qchu.wordpress.com/feed/ ), it just doesn’t show up on this theme.
hey there – no RSS feed on your new site? thx
Hey Qiaochu this is really interesting man. Props to you and your intense math skills.
Hi there. Happened upon your blog accidentally. Great! Could you tell me what software you use to write math symbols in you blog?
Thanks.
WordPress supports LaTeX natively, so there’s no need for extra software.