My name is Akhil Mathew. I’m an undergraduate mathematics student.  I blog about the subject here, at Climbing Mount Bourbaki.

For me, blogging is a useful way for me to better understand the subject; I’ve long enjoyed writing notes to myself, and the internet medium allows me to communicate mathematics with others. It also gives me additional motivation to check for errors.

My interests are rather amorphous and frequently change.  The varying topics discussed here will reflect that. Right now, however, they are centered on algebraic geometry.

My email address is amathew (at) college (dot) harvard (dot) edu. My academic webpage is here.

60 Responses to “About”

  1. Hello Akhil Mathew:

    My name is Jim Spinosa. I first heard of you from

    an article published in the Daily Record when you won

    the Davidson scholarship last year. Do you have a

    regular email address that I can send things to?

    I’m working on an essay about one of theorem that

    you’ve discussed on this blog.

  2. I’ve read two of your postings from your blog,the one
    entitled ” A crash course in one complex variable” and
    the other entitled “Flatness and the local and infinitesimal criteria”. I have a very limited understanding of the topics you write about,even after
    consulting the Mathematics Dictionary by Glenn & Robert James. Interestingly the dictionary contradicts
    some of your statements. For example,in the complex variable posting you wrote,”It might be objected that
    Stokes’ theorem is just Green’s theorem for n=2.” But
    in the Mathematics Dictionary under the George Green
    entry it says,”Green’s theorem is the special case of
    Stokes’ theorem when the surface lies in the (x,y)-plane.” I can’t be sure if the two statements are contradictory because your blog may be beyond the scope of the Mathematics Dictionary.

    1. Akhil Mathew Says:

      I can’t really see much of the dictionary on Google Books, but the version of Stokes I am referring to is the general n-dimensional space: if \omega is a compactly supported n-1-form on an n-dimensional manifold-with-boundary M, then \int_{\partial M} \omega = \int_M d \omega, where \partial M is the boundary of M. (Cf. e.g., Spivak’s _Calculus on Manifolds._) The dictionary may be referring to a more restricted version of Stokes’ theorem, possibly the one that typically appears without a proper proof near the end of multivariable calculus textbooks.

  3. I have recently been reading Don S. Lemons book “Perfect Form: Variational Principles,Methods and
    Applications in Elementary Physics. As I was reading the first chapter, I came to suspect that there were significant errors in the mathematical analysis he was
    presenting to the reader. In section 2.2 of Chapter 2,
    Lemons gives the reader an explanation of the Euler-LaGrange equation that is four pages in length. I believe the explanation is a mix of reasonable mathematical procedures and highly whimisical mathematical procedures. I will post the full essay
    on my website shortly. I hope you read it.

  4. I have just posted the full essay on my website:

    I would be very interested in your interpretation.


    Jim Spinosa

    1. Akhil Mathew Says:


      Thanks for your questions.

      -Differentiation under the integral sign is certainly permissible if the integrands are reasonably smooth, which is probably assumed. This can be checked simply by looking at difference quotients; cf. this post of John Armstrong, for instance.

      -The objection with integration by parts that you make is symbolic, since the formula in question is simply
      \int fg dx = Fg - \int F g' dx, where F is an antiderivative of f and the integrals are indefinite. The proof in Lemons’ book follows this. Also, if you write dv  = v'(x) dx, you should find that the calculation given by Lemons is indeed correct. (The notation dv=v'(x) dx can be taken as a statement about 1-forms or a convenient but purely formal shorthand, whichever is preferable. However, the statement that I made earlier in terms of F,f,g may be easier for avoiding confusion.)

      -As far as the Euler-Lagrange equations are concerned, there are doubtless other sources that may be more helpful; unfortunately I don’t really have a great reference, but googling should turn up something. For an example of this sort of business, cf. my posts on the first and second variation formulas. This is material that should be in any textbook on Riemannian geometry.

  5. […] Notícia | Tags: Ciências, Divulgação, Notícia (via Return to Mount Bourbaki,  Akhil Mathew,  Climbing Mount […]

  6. If you are interested in and have not read it yet, you can find in The Mathematical Intelligencer, Vol. 15, No. 1, p.27-35 1963 the following article on the historical side of Bourbaki:

    A Parisian Café and Ten Proto-Bourbaki Meetings (1934-1035) by Liliane Beaulieu.

    1. Akhil Mathew Says:

      Thanks! I’ll take a look at the article – it sounds interesting.

  7. Correction: The Mathematical Intelligencer, Vol. 15, No. 1, p. 27 – 35 1993

  8. Jon Says:

    Hi Akhil,

    After reading your blog, I wanted to send you a beta invite to our startup called OpenStudy. We are located in Atlanta, Georgia (on Georgia Tech’s campus) and founded by Dr. Ashwin Ram. (

    We just left Alpha and have built a community of over 450 students from the top programs nationwide. Now we are moving into Beta and expanding our community.

    When you get a chance will you let me know if you’d like an invite?


    1. Akhil Mathew Says:

      Thanks! Unfortunately I doubt I’d be helpful, though.

  9. whoami Says:

    Hey Akhil ,
    I wanted to know whether u r student ? I found ur blog very useful.

    1. Akhil Mathew Says:

      Yes, I’m a student (see the “About” page).

  10. Henry Says:

    Hi Akhil. Just out of curiousity, how old are you and what age did you start learning university-level math? What branches of math have you learnt so far?

    1. Akhil Mathew Says:

      I’m 18, though I can’t really answer your other questions – I don’t like to divide mathematics into “university-level” and “non-university-level.” I certainly can’t claim to have “learnt” any area of mathematics, but the topics covered here generally reflect what I currently like to think about.

      1. Henry Says:

        I guess I should’ve been more specific. Have you done general topology, abstract algebra, analysis etc.? If so what books have you studied? When did you start learning analysis or algebra and when did you start learning multivariable calculus? I wish you well Akhil! I hope you continue to pursue math in the future.

        1. Akhil Mathew Says:

          Sure, to varying extents – the bibliography on the blog indicates the books I’ve used as of late (well, I guess I left out some of the more elementary textbooks I read like Herstein’s _Topics in Algebra_).

  11. Henry Says:

    Akhil, how do you learn mathematics? Do you learn several different areas at one time (as is done in university) or do you learn one area at a time? When you read a book, do you typically try to read it from front to back (in an orderly manner) or do you just scan through the book and see what interests you and read that.

    Also, on an average, how many pages of math per day can you read?

    I ask these questions because I’d be interested to know how different people learn mathematics. You have a broad knowledge by the looks of things so it seems reasonable to believe that you have a “secret strategy” to learning math though I could be wrong; I’m not a mathematician.

    1. Akhil Mathew Says:

      No, I have no secret strategy (and I’ve never heard of anyone having one). I usually look at a few different things at the same time, but usually with one focus (currently alg. geo.). Also, I typically skip around in and re-read books, but that depends on the book. The number of pages is *very* nonconstant and depends on the book, how much time I spend, and my energy level. I’d imagine that’s the case for most students.

  12. Pierre Deligne Says:

    I just wanted to congratulate you on your paper that you submitted to arXiv math Akhil. Well done!

  13. David Says:

    Akhil, have you tried the book entitled “The Geometry of Schemes” for picking up the basics of schemes? It’s more to the classical side of things and not “EGA style” so to speak. What book are you currently following on algebraic geometry? There are plenty of excellent books on the subject but it’s not always obvious which one to choose. Please don’t hesitate to contact me if you need any advice. (I’d prefer to remain “annonymous” so to speak on this forum, so if you recognize who I am by my email, please don’t refer to me by my full name but just by “David”.)

    1. Akhil Mathew Says:

      Thanks for the recommendation. Right now it’s been a mix of Hartshorne and EGA; I actually really enjoy the EGA style a lot so have been giving into the temptation to read it (even though I’m not fully solid on Hartshorne yet). At least for me, it was difficult to see some of the bigger picture that Grothendieck seems to paint in Hartshorne.

      I saw the older version _Schemes: The Language of Algebraic Geometry,_ but I’ll try looking at the newer one soon. (I am, incidentally, planning to take a purely classical algebraic geometry course next fall, and ideally a scheme-theoretic reading course sometime in the spring.)

      1. David Says:

        Yes, I think that you should probably go the Hartshorne and EGA route if you are already doing so. Harvard is very flexible when it comes to reading based courses and so you should be in an excellent position to study algebraic geometry. Have you spoken with some of the faculty there regarding doing some graduate courses (e.g., Theory of Schemes I and II)? Of course, the reading based courses probably don’t differ too much from the graduate courses depending on what you do, but it’s always a good idea to talk with the faculty about these things. On a different note, have you considered doing physics courses? Specifically, down the road you may find that some aspects of string theory are intimately connected to some of the mathematics you are doing …

        1. Akhil Mathew Says:

          Unfortunately Theory of Schemes isn’t offered next year, but yes, I’m hoping to take a few graduate (or, at least, 200-level) courses in the fall (namely classical alg. geo, commutative algebra, and algebraic topology), assuming the administrative procedure works out and (which it should, since I talked to the faculty members of the courses and they seemed to be fine with my taking it). This is why I was interested in doing the independent study on schemes in the spring; also, depending on how much independent reading I get done in the fall, perhaps we could talk about things like stacks or descent, which I don’t think are in the schemes course.

          I do intend to take several physics courses over the years, but I know very little right now (having never been able to understand physics books as well), so I’ll only take the standard honors freshman course (Physics 16) this fall. Hopefully I can take quantum sophomore year or something like that.

  14. Joe Says:

    Hi Akhil! I came across your website looking for information on how to get the right math advice for my son. You see my son is a 11 year old and he’s already been exposed to lots of advanced math. E.g. I showed him the qual. exam for math in Harvard and he could answer most (not all – say 80% he could answer) correctly. I’m under the impression that Harvard has the best math department in the world, though people have informed me that Chicago is better in the mathematics department (do you know which one is better in which areas?). But I don’t know (or haven’t seen) the qual. exams that Chicago offers.

    Now my son has just entered high school. That he has a similar background to students who are doing a PhD in math worries me since I’m wondering what is best to do with him. I mean Harvard is Harvard – it’s the top univeristy in the world so would it be that everyone there has a very high background in math upon entrance? If so, then why are the PhD exams so easy there? Being a student there do you know how much background the typical math student enters there with? I mean are most students who go there exposed to advanced mathematics topics like topology and functional analysis already or not? The reason I ask is that I’m wondering whether my son needs to get more math background before going there and whether he should stay in school until he can.

    Thanks for any advice you can offer me. I heard about you through a friend who said that you seemed to be in a similar boat to my son – that you did advanced math before going to university. Would you be able to offer me an advice? Thanks a lot.


    1. Akhil Mathew Says:

      Dear Joe,

      I don’t think the typical entering mathematics student at Harvard knows what a Hilbert space is, or what paracompactness means, though there are certainly some that do. I think this is true even if you restrict to the subset of students who take a fairly advanced course (Math 55 or beyond) when they enter in the fall, which probably coincides with the subset of those who want to graduate with honors (i.e. write a senior thesis). Needless to say, my hypothetical honors student very quickly absorbs a lot of mathematics, and writes a long paper on highly advanced material at the end of her college years. If your son is able to answer a large proportion of the qualifying exam for graduate students, then he certainly is better prepared than most of the freshmen here.

      On the other hand, I would also say that the qualifying exams probably do not pose a problem for *graduate* students here. I am merely speculating, because the graduate students here are all extremely, extremely talented, although I know very little how things work for them.

      I regard myself as wholly unqualified–as a beginning mathematics student—to offer any further career advice, so I’d strongly recommend contacting an actual mathematician! At Harvard, Joe Harris is both brilliant and one of the nicest people I’ve met here (he’s my freshman advisor), so I recommend emailing or calling him, for instance.

      Good luck to your son!

      P.S. Both Harvard and Chicago are great places. Harvard has a heavy focus on number theory and algebraic geometry; I don’t know about Chicago.

      1. Joe Says:

        Dear Akhil,

        Thanks a lot for your much appreciated advice. I can understand that the qual. exams won’t be so hard for PhD students. Though I’ve never gone anywhere remotely near to research math, I’ve read up on math before for my own interest. So I have kind of a rudimentary feeling for things like top. vector spaces and paracompactness though it’s been a long time since I’ve done this.

        I’ll definitely consider contacting the people at the math department there. One of my worries is that if my son goes to Harvard for his undergrad., we wouldn’t be able to do is PhD there since I’m well aware of the math departments encouraging people to go elsewhere for their PhD’s. Do you know anything about this and whether it’s possible to do both an undergrad. degree and a PhD at Harvard in math? (which places have you considered for your PhD?)

        The other thing I’m worried about is the tough admissions at Harvard. It’s extremely difficult to get in there I know, so I’m worried about that. You see I’m not sure how universities look at young (very young) people entering since I’ve heard of people who couldn’t attend uni. (in the UK mainly) since they were too young and so had to wait for a couple of years. I know Harvard certainly doesn’t have prejudice against young people but I’m kind of worried that they may worry about my son’s age and reject him for that. As a student there, do you know of the youngest students that get admitted there?

        The second thing I’m worried about is my son’s grades and SAT scores. I know his teachers will give him excellent letters of recommendation but my son isn’t so good at the SAT, and his grades in other subjects aren’t very good as math. Would it hurt him if he had lots of B’s and C’s in other subjects even if he could show that he had a lot of ability in math? If you know how the admissions committee works at Harvard would you know how the reject and accept students? (maybe if you’ve read up on this when you applied there.)

        Sorry for asking so many questions. I’m just interested to hear your experiences with all of this. It’s funny you should mention Joe Harris since my son actually tried to read his textbook on representation theory and actually got interested in that subject with that book. I’ll definitely contact Joe Harris but what I’m worried about is how much of a say Joe Harris has in accepting or rejecting students there. I’ve heard that there are specialized people for this kind of thing mainly non-math, and it’s generally difficult to explain to non-math people how much math someone knows. (I’m thinking of accelerating my son through high school – he’s only in the 8th grade, though he really doesn’t know his other subjects that well and that might hurt his grades – what is your advice on acceleration?)

        Thanks again,


        1. Akhil Mathew Says:

          I don’t know much about how Harvard’s undergraduate admissions process works; it always seemed a mystery to me. So I do not know any one way to get into Harvard. I certainly know people more qualified and intelligent than I who were not accepted.

          I know a student in my class who’s fifteen (and graduated high school) and one who’s sixteen (who dropped out of high school). So it is possible to get in at a young age, but it may indeed be more difficult; I do not know the details. In any case, if you email me (my email address is on the about page), I can put you in contact with the people that I do know, who may be able to help more than I.

          I don’t know how much of a say the mathematics department has in accepting students, but I have heard that sometimes faculty members write letters to the admissions office (if they know the student) asking them to accept her.

          I believe that it is extremely rare (and, in any case, strongly discouraged) for an individual to get a PhD and bachelors degrees from Harvard, because the math department frowns upon such things (probably with good reason), and does not usually re-accept their undergraduates. I have not thought at all seriously about where I would like to go for grad school, but MIT sounds fun. (I don’t know anything about how selective the admissions process is, though.)

          In any case, Harvard is not the Center of the Universe, even if I’m supposed to say otherwise :-). There are other equally strong mathematics departments (like MIT, for instance). Moreover, making a college decision based upon distant graduate school possibilities doesn’t seem like a good idea to me.

          I’m afraid that I can’t offer any advice on acceleration, because at 11 I was a normal fifth grade student with fifth grade intellectual sophistication and fifth grade maturity. (I was taking mathematics a year or two above my grade, though.) There are many successful mathematicians who were in fact child prodigies, however, and perhaps you should talk to one of them. It is important, in any case, to consider his social and emotional well-being in addition to his intellectual capabilities.

          1. David Says:

            Admission to the MIT mathematics graduate program is extremely selective. I have even heard of applicants who had mathematics research papers published to high quality journals being rejected. That said, judging from the mathematics in your blog, you will certainly be one of the stronger applicants when you apply. However, I would encourage you to consider other graduate programs in mathematics, particularly Princeton, Chicago and Stanford (not to mention UC Berkeley :)), since it is often not a good idea to be fixed on one particular school. Moreover, Chicago, Princeton and Stanford, have been, and are, some of the top mathematics graduate programs in the world. It depends on what you want to pursue, but if you wish to specialize in any subject within modern algebra, Chicago is the choice. Likewise, for analysis or number theory, I would recommend Princeton and Harvard. (Harvard does sometimes give admissions to the undergraduates into their graduate program, but this is rare. Nonetheless, I would encourage you to apply, since there is no better place for algebraic geometry and number theory as Harvard. If you do get in at another good school, you should probably not stay at Harvard, however, since it is good to be exposed to different mathematics cultures.)

            1. Akhil Mathew Says:

              Dear David, thanks! I confess to having given rather little thought to graduate school so far, and my mention of MIT was primarily because I would have probably chosen it for college if not Harvard. I don’t have any particular fixation on MIT other than that I know several people there and have some sense of the campus, but I was actually thinking that Berkeley or Stanford might be fun since I haven’t really been on the west coast much. (In addition to the fact that these are fine places, of course.) I realize that getting in to graduate school at a given institution is much more difficult than getting as an undergraduate.

              It hadn’t even occurred to me to apply to Harvard, actually. I know people who’ve gone to MIT and worked with Harvard faculty as their advisors, though (I assume this happens at other pairwise close institutions). I was under the impression that Harvard automatically rejected applicants from its undergraduate program.

              It’s nice to know the specific strengths of each program–I’ll bear that in mind when I apply/decide (for now, I think I want to study algebraic geometry, but I am not sure at all).

              1. David Says:

                You are definitely doing the right thing by not worrying about grad. school at the moment. Moreover, I think that if you continue doing the mathematics you are doing, you will have a very strong chance of being accepted. It is early days for you anyway.

                One thing to keep in mind is that it’s very easy for one’s interests to change over the years. By all means continue with algebraic geometry; I can guarantee that no whatever field you choose to pursue, algebraic geometry will be very useful. You’ve already been exposed to quite a broad spectrum of mathematics, which I think is great. However, I would nonetheless encourage you to consider getting at least 3 fields which you like close to research standard in your undergraduate years. This will give you a lot of freedom when you do your PhD. For one thing, if you suddenly decide that you do not wish to pursue algebraic geometry in the middle of grad. school, you’ll be well-prepared. (However, unlikely this is, of course, it’s always good to be prepared.)

                I think that for someone like you, algebraic geometry is a very good choice in terms of direction of research. It incorporates mathematics from many areas, and you’ll definitely find that you use most of the mathematics you learn outside algebraic geometry, in algebraic geometry, and vice-versa. So by all means continue doing algebraic geometry!

                Regarding Harvard’s policy on accepting their undergraduates … it does happen, but as I said, only very rarely. Competition into Harvard’s mathematics graduate program is extremely fierce as it is, but if you apply as an undergraduate from Harvard, it could certainly lower your chances. However, some exceptional students do get accepted. Ultimately, I think it’s for one’s own good to be rejected. There’s a world outside Harvard, and if you never get to see it until you finish your PhD, then you’ll be missing out on a lot. 🙂

                Anyway, may I ask what courses you are doing this semester in mathematics? I think you should definitely consider doing most of the grad. math courses at Harvard throughout your undergraduate years. If you do, you’ll have more time for research in grad. school.

                Good luck!

            2. Akhil Mathew Says:

              (I’m replying to this comment to avoid making the margins too narrow.)

              Thanks once again for your advice. One thing I do like about algebraic geometry is the breadth of the field, though I’m most interested in the scheme/stacks-theoretic side of things. I might also consider studying something like algebraic topology or algebraic number theory or representationtheory. I don’t know.

              This semester, I am taking introductory algebraic geometry, algebraic topology, and commutative algebra (the last one unofficially for administrativereasons). I certainly do intend to take as many of the core graduate courses as I can while I’m here (the ones I’m in count, though most of them are primarily populated by undergraduates). Algebraic topology and algebraic geometry are two-semester courses, so I’ll probably take the second halves in the spring along with perhaps a course on Lie theory and/or a reading course. Schemes isn’t offered this year, so I can take it next year.

              But yes, it will be fun to explore algebraic geometry in more detail, as well as other fields. I was actually hoping to get far into a series of posts on homotopy theory on this blog, but that is looking less likely until perhaps next semester .

              1. David Says:

                I would encourage you to take a course in several complex variables and arithmetic geometry if you haven’t already. These are useful for algebraic geometry. Also do as much commutative algebra as you can!

                For example, have you heard of the book “Cohen-Macauley Rings” in the Cambridge mathematics series? It covers some very useful commutative algebra in algebraic geometry. I don’t think Harvard offers a course on this sort of algebra, but it’s worth taking a reading course in it.

            3. Akhil Mathew Says:

              Yes, I would indeed like to learn several complex variables, and complex algebraic geometry as in Griffiths and Harris. Harvard doesn’t have a course on it, but I might take a reading course sometime. At least not this year–though the topics for graduate complex analysis could vary.

              I hadn’t heard of the book you mentioned, but I’ll take a look. (I realize that they come up in Serre duality.) I don’t think we’ll go very far into Cohen-Macaulay rings in the course I’m in right now, but I can always take a reading course. I remember starting Lang’s _Algebraic Number Theory_ a few years ago thinking that I would learn something and getting absolutely nowhere–I eventually realized that familiarity with Lang’s _Algebra_ (or otherwise some commutative algebra, more generally) was absolutely necessary, for that and for Hartshorne.

              1. David Says:

                Sorry, I should’ve been more clear. Please see:

       for the book. The prerequisites for reading it are basic comm. algebra (i.e., in Atiyah and Macdonald) and some basic knowledge of homological algebra (i.e., Tor and Ext functors, for example)

              2. Akhil Mathew Says:

                Thanks! I’ll take a look — it sounds like it would be at a good level for me, and it’s material that will be useful and fun

                1. David Says:

                  Thread (getting too skinny!)

                  I should add that one nice feature of the book is that it has a flavor of algebraic combinatorics. This is excellent if you haven’t done much combinatorics before. Even if you have, it’s nice to see the subject of algebraic combinatorics from the point of view of Choen-Macauley rings. The other great feature of the book is that it covers the topics of Gorenstein rings and the like, which is often neglected in more basic texts such as Atiyah and Macdonald. So in some sense, the book could be the perfect follow-up to Atiyah and Macdonald if you’ve read it. (I assume the comm. algebra course you’re taking this semester has Atiyah and Macdonald as the text. Is this right?)

            4. Akhil Mathew Says:

              I got a copy of the book, and it looks nice. I saw very little combinatorics in high school (I didn’t participate too actively in math olympiads) and Harvard does not offer many courses on the subject. Also I have seen nothing at all about Gorenstein rings and very little about Cohen-Macaulay rings–essentially, whatever was necessary for Serre duality in Hartshorne, which I’ve mostly forgotten now.

              We are actually covering a selection of topics from _Commutative Algebra: with a view towards algebraic geometry_ in the course I’m in. So far, we just talked about Picard groups and invertible ideals (up to a version of Serre’s criterion, as in Ch. 11), and have moved to completions now. There is still quite a bit of time left in the semester, so I don’t know where we’ll go next, as we have exhausted the syllabus (primary decomposition,completions, Noetherian rings, the Nullstellensatz).

              1. David Says:

                Ah right. Yes, I was under the impression that the course text was Atiyah and Macdonald since that’s how it has been for a long time (at Harvard); I knew it had changed, but I guess that escaped my mind.

                As I said, you’re doing everything right if you want to specialize in algebraic geometry. However, I would encourage you to try other branches of mathematics that you do not know very well. In Harvard’s grad. school, they require you to write a “minor thesis”. Usually you are given a few weeks to research as much as you can about a topic which you are ignorant about, and present it in a seminar or a report on the topic. The idea is that a mathematician often needs to be able to learn important concepts quickly (but properly!) and writing a minor thesis helps you in this skill.

                It’s good practice, in this sense, to try to learn more about topics that are active areas of research but aren’t necessarily too connected to algebraic geometry. For example, at the moment you are learning commutative rings. Have you considered looking at areas of algebra like noncommutative ring theory, for example? Noncommutative ring theory has beautiful structure theory (at the basic level, the Artin-Wedderburn theorem, Brauer groups, polynomial identity rings etc.) which is really worth learning. Noncommutative geometry is an area of mathematics where noncommutative rings do turn up. In addition, you might find that this happens in operator theory as well. So it’s these kinds of topics (if you don’t know them already) that might be worth spending time learning as well. Algebraic group theory is another example.
                Of course, you should spend most of your time learning algebraic geometry and algebraic topology at the moment, but consider trying to pick up mathematics which you might not have thought of doing before. This may be anything really – for example, you might want to learn about the representation theory of automorphic forms (let’s say you did …) and then you might need to consult the excellent text by Daniel Bump on the subject. Basically, I would encourage you to try to pick up some more specialized topics from a number of areas, as these could help you some day. You never really know.

            5. Akhil Mathew Says:

              Dear David, thanks again! I do plan to learn about other things as much as I can while I’m here. My current courses are currently focused on algebraic geometry and algebraic topology, and probably will be next term because algebraic geometry and algebraic topology have second semesters, but I am looking into the Lie groups course as well. Next year I’m thinking of complex analysis, algebraic number theory, and schemes.

              I do enjoy trying to learn areas outside algebraic geometry and algebraic topology, of course–it’s just that I’ve had very little time this semester to do that! I will have time over the winter to do more outside reading on topics like noncommutative algebra. I have interests in theoretical computer science and analysis as well, which I will probably end up pursuing further (there are courses on algorithms and pseudorandomness next term, of which I may take one).

              I have studied some facts about semisimplicity and noncommutativerings, but never too far (mostly in the context of representationtheory). Also I would like to learn more pure category theory sometime (e.g. read Lurie’s _Higher Topos Theory_). But I do not think I am mathematically mature enough for that yet. I don’t know anything about algebraic groups, and Harvard offers no course on it, but I hope to pick it up sometime (I’ve tried unsuccessfully a couple of times).

              I have heard of Harvard’s minor thesis, and it sounds like an interesting exercise.

  15. Sudarshan Says:


    This question pertains to Measure theory, more importantly the importance on the applicability of sigma algebras to a probability theory. When you do have a sigma algebra we know that it contains a collection of subsets of the the Set with a certain properties associated with it namely the union of the sets being present within the sigma algebra, complement being present as well, note that a sigma algebra always contains the set and the empty set. So my picture of a sigma algebra is that a it is that is bigger than the set itself( You could prove me wrong on this). When we define say a triple P(omega, F, P˜), we know that the range of the probability function is [0,1] so we have that the the sigma algebra of the sample space omega call it F say is a set larger than the sample space itself, so knowing that the range is [0,1] is there an easy way to visualize the concept of a measure. An example to visualize a measure would be say the Lebesgue Measure where, we can see that the a finite interval on the sigma algebra can be split up into finite closed intervals(boxes) and that the measure equals the continuous product of these the successive intervals. My personal view on this subject would be if you know that the range of the measure is a finite interval( note that I might be wrong on this aspect ), the concept of measurability becomes a lot easier since we consider the domain of the measure a certain geometric figure corresponding to each finite decomposition of the interval ie partition the interval based on a set geometric shape.

    1. Akhil Mathew Says:

      I never tried to “visualize” a measure other than Lebesgue or related measures, but if you find finite measures easier, you could consider \sigma-finite measure spaces, which are unions (probably inductive limits in some appropriate category) of finite measure spaces.
      In any case, a measure doesn’t have to correspond to some sort of nice geometric intuition (e.g. the measure on a set A that assigns zero to a countable subset and infinity to a co-countable subset).

      A \sigma-algebra on a set A is a collection of sets that contains A as an element, not (in general) as a subset.

      1. Sudarshan Says:


        Thank you for the reply. On a related note, given a sigma algebra how would you comment on the size of the sigma algebra. When I mean size I don’t mean with regard to the number of elements rather I refer to the size of the elements present within the algebra. Is it a good argument or rather a valid justification to see a sigma algebra as a set larger than the set the sigma algebra is defined on.


        1. Akhil Mathew Says:

          Sorry, but I’m not sure what you’re asking. The sets which are *elements* of the \sigma-algebra can have any size you want, which is unconnected with the size of the \sigma-algebra (as a set) itself.

  16. Hi Akhil,

    I’ve just published my first book entitled
    Nuts & Bolts: Taking Apart Special Relativity as both an ebook and a POD paperback. Full details about where it’s
    availible are on the front page of my
    webpage I hope you
    check it out. Thanks.

  17. MathHelpRequired Says:

    Hi Akhil!

    I’m stuck on a problem and wondering if you could help me.

    Prove that for every finite group G and every set of primes S there is a Hall S-subgroup of G if and only if G is solvable.

    I know the “if” part but I’m stuck on the “only if” part. Any advice?


    1. Akhil Mathew Says:

      I’m sorry I can’t help; you might try asking on math.SE.

      1. Fikreab Solomon Says:

        H. Lenstra in : The if-part is algebra and is fairly straightforward, the only if-part is harder : it is number theory!

  18. MathHelpRequired Says:

    Akhil I also want to ask how you’ve managed to learn so much math in so little time! I mean there’s no such thing as “genius” or anything, but you seem to be very talented. Learning math is something of a committment. I’ve always thought that if one spends enough time and perseveres at it, it isn’t something beyond one’s reach. But that’s easier said than done!

    One of my problems is that I always forget the math I learn. I learn it and then 3 months later I forget it. This worries me. Do you get something similar? That suddenly all the math you knew “evaporates” after a year or so? What would be some advice you have to avoid this?

    I mean I can learn math super slow and I might remember, but if I learn at my natural pace (which isn’t super fast) then I forget stuff. What’s your advice on how to make a compromise between amount of information absorbed and memory?

    Also, how do you break your day apart? I mean how many hours do you spend for math and how do you use that time efficiently?

    Thanks again Akhil!

    1. Name Says:

      I second this. A blog post with advice for learning math would be both helpful and interesting.

      1. Tim Says:

        Akhil is not a genius. He works very hard. Look at how much notes he takes. Its all about practice.

        1. timothy Says:

          There isn’t such thing as genius anyway. I’m pretty sure most people could become leading mathematicians if they put enough hard work into it (and did everything right).

          1. enaslund Says:

            I don’t believe this. Up to a point hard work matters, but people who are fields medalists for example, not only work extremely hard, but also have incomparable innate talent for mathematics.

  19. timothy Says:

    Hi Akhil. Please take a look at . It’s the princeton general qual. exams for terry tao. Could you answer all those questions? Just wondering how much harmonic analysis you’ve done.

  20. Justin Says:

    Akhil, it’s always intriguing, but how did you get interested in math? I mean you must have started calculus, but I, for one, never liked calculus. Math mainly starts getting interesting beyond calculus, so how did you manage to overcome the obstacle and get interested in math? Did you just pick up a book one day and liked it?

    Finally, how do you come up with these blog posts. Do you think about the proofs of the results and write them in your own words and give ways of thinking about them? Or do you just copy the results out of the book?

  21. 喻伟 Says:

    You math blog update is too fast..thank share.Do you give some related problems in the blog?left to the reader thinking.

  22. 1ly1 Says:

    Akhil: In video I saw of you, I think you mentioned “complex dimensions” and deligne category? Is there a reference for this? Also is a notion of “complex cardinality” possible for cardinality of a set?

    1. Akhil Mathew Says:

      Most of these ideas come from Deligne’s paper (there’s a post on the Secret Blogging Seminar about it, as well as a couple here). I don’t know of any notion of “complex cardinality.”

  23. Efaim Says:

    Hi Akhil,

    Your blog is really great. Sadly it’s kind of hard to really browse the past posts and thus to get a quick overview of everything you have written. I think a really cool thing would be to create a page where there would be a kind of table of contents (ordered by topics for example) of your favorite posts. This way it would make it much more efficient for your readers to retrieve informations and I believe that many, like me, would be grateful.

    Anyway congratulations for the amazing job you are doing here.


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