Here is a (periodically updated) list of books and sources that I have referred to, or plan to in the future, sorted by field:

Algebra

  • Representation Theory: A First Course, by William Fulton and Joseph Harris
  • Algebra, by Serge Lang
  • Commutative Algebra: With a View Towards Algebraic Geometry, by David Eisenbud.
  • Commutative Algebra, by Nicolas Bourbaki
  • An introduction to Homological Algebra, by Charles Weibel
  • Introduction to Commutative Algebra, by Michael Atiyah and Ian Macdonald
  • Linear Representations of Finite Groups, by Jean-Pierre Serre
  • Lie Groups and Lie Algebras, by Jean-Pierre Serre
  • Introduction to Lie Algebras and Representation Theory, by James Humphreys
  • Complex Semisimple Lie Algebras, by Jean-Pierre Serre

Algebraic geometry

  • Algebraic Geometry, by Robin Hartshorne
  • Elements de Géometrie Algébrique, by Alexandre Grothendieck and Jean Dieudonné
  • FGA Explained, by Barbara Fantechi
  • The Geometry of Schemes, by David Eisenbud and Joe Harris
  • Basic Algebraic Geometry, by Igor Shafarevich
  • Topologie algébrique et théorie des faisceaux, by Roger Godement
  • Algebraic curves and Riemann surfaces, by Rick Miranda
  • Introduction to Algebraic Geometry and Algebraic Groups, by Michel Demazure and Peter Gabriel

Differential geometry

  • Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason
  • Morse Theory, by John Milnor
  • A Comprehensive Introduction to Differential Geometry, by Michael Spivak
  • Foundations of Differential Geometry, by Shoshichi Kobayashi and Katsumi Nomizu
  • Riemannian Geometry, by Manfredo do Carmo

Analysis

  • Functional Analysis, by Peter Lax
  • Real and Complex Analysis, by Walter Rudin
  • Singular Integrals and Differentiability Properties of Functions, by Elias Stein
  • Riemann Surfaces, by Hershel Farkas and Irwin Kra
  • Partial Differential Equations, by Michael Taylor
  • Introduction to Partial Differential Equations, by Gerald Folland

Number theory

  • Algebraic Number Theory, by Serge Lang
  • Local Fields, by Jean-Pierre Serre
  • Algebraic Number Theory, by John Cassels and Albrecht Frohlich, eds.
  • The Arithmetic of Elliptic Curves, by Joseph Silverman
  • A Course in Arithmetic, by Jean-Pierre Serre
  • Introduction to Cyclotomic Fields, by Lawrence Washington

Logic and computer science

  • Introduction to the Theory of Computation, by Michael Sipser
  • Introduction to the Metamathematics of Algebra, by Abraham Robinson
  • Lectures on the Hyperreals, by Robert Goldblatt
  • Mathematical Logic, by H.D. Ebbinghaus, J. Flum, and W. Thomas
  • Model Theory, an Introduction, by David Marker
  • Computational Complexity, by Christos Papadimitriou

Dynamical systems and ergodic theory

  • Introduction to Ergodic Theory, by Peter Walters
  • Ergodic Theory, by Paul Halmos
  • Introduction to the Modern Theory of Dynamical Systems, by Anatole Katok and Boris Hasselblatt

Topology

  • Elements of Algebraic Topology, by James Munkres
  • Algebraic Topology, by Allen Hatcher
  • Algebraic Topology, by Edwin Spanier
  • Algebraic Topology: Homotopy and Homology, by Robert Switzer
  • Topology and Geometry, by Glen Bredon
  • Topology, by James Dugundji
  • General Topology, by Nicolas Bourbaki
  • A Concise Course in Algebraic Topology, by Peter May
  • Spectral sequences in Algebraic Topology, by Allen Hatcher
  • Homotopy theory, by Sze-Tsen Hu
  • Stable homotopy theory and generalized homology, by J. Frank Adams
  • Characteristic Classes, by John Milnor

Online sources

Here are some online sources:

Here is a more current reading list:

  • Goerss-Jardine, Simplicial Homotopy Theory
  • SGA 1, 4, 4 1/2; EGA
  • Kashiwara and Schapira, Categories and Sheaves
  • Lurie, Higher Topos Theory
  • Tamme, Introduction to Etale Cohomology

44 Responses to “Bibliography”

  1. Arnav Says:

    Very impressive! I wish you the best at Harvard; please feel free to contact me (contact information generally available through the harvard facebook) if you have any questions about Harvard, or math more generally.

    1. Akhil Mathew Says:

      Thanks! I suppose I’ll see you around campus.

  2. Tom Says:

    Akhil, you wrote that you read real and complex analysis. How did you find the exercises? I’ve had a crack at that book and the exercises in the first 2 chapters were pretty OK as was the 4th chapter but the third chapter was ridiculously difficult. If you’ve done it could you offer me some hints on how to do the exercises in chapter 3 specifically exercise 5? Also what whas your insight on the Riesz Fischer theorem?

    Thanks!

    1. Akhil Mathew Says:

      Sorry that your comment took so long to appear—it got caught in the spam filter, and I only just saw it.

      I think that, in general, Rudin’s book has nice exercises. For the one you asked about, geometrically the condition \phi((x+y)/2) \leq (\phi(x)+\phi(y))/2 says that on the graph of \phi, given two points (x, \phi(x)), (y, \phi(y)), the point corresponding to the midpoint (s+y)/2 lies below the line drawn through the two initial points. Repeating this inductively, one can get the desired inequality on \phi(tx+(1-t)y) for t a dyadic fraction, and continuity then implies it for all t in the unit interval. If I am not mistaken, this is the idea of the argument (which can also be done formally).

      The Riesz-Fischer theorem essentially says that each Hilbert space is isomorphic to some \ell^2(A) for some $A$ (which is determined up to cardinality, and can be taken e.g. as an orthonormal basis). Some people call different things by the same name, though.

      1. Tom Says:

        Hi Akhil,

        Sorry I think I wasn’t clear. Exercise 5 is the one about a measure mu with mu(X) = 1 and the question asks to show that the L^r norm is <= L^s norm when r<s. Do you have any ideas on how to do it?

        BTW, how far have you gotten into the book? Would you recommend the complex analysis portion or should I go to another book for that? Also would you recommend Rudin's functional analysis as the next step in analysis or are there better functional analysis books on the market?

        Thanks!

        1. Akhil Mathew Says:

          My mistake. For this one, I believe you can apply the Holder inequality with one of the functions being f and the other being 1. The condition of measure one is necessary for one of the factors to be one (in general, this argument shows that on a space of *finite* measure, L^r \subset L^s when r>s and the inclusion is continuous).

          I went through all of Rudin’s book at some point. I actually think the complex analysis part of the book is extremely cleanly done. It gets into some interesting material (e.g. nontangential estimates on Poisson integrals, Hardy spaces), but also does the basic material (e.g. Cauchy’s theorem) very thoroughly and rigorously (in that case, by giving an argument which is actually rather recent).

          Rudin’s functional analysis book seemed comparatively dry to me, but it is thorough. I have enjoyed the book by Lax. Anyway, I would recommend talking to someone who actually knows analysis for such advice, though.

          1. Tom Says:

            Hi Akhil,

            Firstly, sorry. I use a shared computer and had your website open and I think someone posted some kind of vulgarity on your website using the same computer (and I think the vulgarity came under my name too). Hopefully the spam filter caught it or something but in case it didn’t, please ignore it.

            Anyway, may I ask how long you took to do Rudin’s real and complex analysis? It’s taking me a while. I don’t want to be spending too much time on it since it might not be worth the investment but would something like 1 year be about par? Or should I be finishing the book quicker? What was your experiences?

            When you did the book, could you do all the exercises and were they do-able? I mean, did it take you “not so long” to do most or all of them? Just interested. I’m kind of hoping I’m not alone in my attacks at the problems that seem to come to no end! I know some very important results are there in the exercises so I want to do ‘em all so to speak but I don’t want to be spending too much time on them. What’s your advice on how to approach them and how hard they are? I know they’re interesting but are there any exercises whose proofs go for a page or longer?

            Thanks Akhil!

            Tom

  3. Tom Says:

    BTW, you wrote “I’d recommend talking to someone who actually knows analysis for such advice, though”. Surely this is modesty! I mean I’ve heard professors in mathematics say that completing Rudin’s book is like more than what an average math student learns in his entire PhD in analysis. And I’ve also heard that once you’ve done Rudin’s real and complex analysis, you could well begin reading books that would get you to a point of research in the area. Is this talk of Rudin’s book being “so great” just over-emphasising the point or are you just being modest? ;)

    1. Akhil Mathew Says:

      I’m pretty sure a PhD student (especially in analysis) needs to and should know much more than what’s in Rudin, which is very much an introductory textbook. I’m afraid I can’t evaluate the last claim, because I haven’t read too many research papers in analysis.

      I usually read and re-read books, so I can’t answer your question on length. I also work on the exercises in a rather piecemeal fashion. I never took a formal course on this material, but a more systematic approach would probably have been better. Some of the exercises are quite difficult.

      1. Tom Says:

        At Harvard, the requirement for PhD students seems to be only a knowledge of Rudin’s text for the analysis portion of the qual. exams. I get that PhD students in analysis need to know more but would it be fair to say Rudin is only an introductory book in analysis? I mean just by way of comparison and nothing more, no university in the world requires a knowledge of analysis for the general qual. exams more than what’s covered in Rudin.

        You say Rudin’s functional analysis is a good book. Would that be research level in functional analysis or would you need to know more to do research in functional analysis?

        I’m just wondering, surely there’s some standard of research. I mean reading Rudin takes a while and you don’t have 10 years to do a PhD! So there must be some finite value of textbooks one should be reading in analysis to get to a point of research and considering that not all undergraduates are as nearly well-prepared as yourself.

        I understand that you’re still a student but you seem to have done a bit of mathematics in several directions. Which direction are you most experienced in and can you tell how much one should know before doing research in that direction for example? There’s this dilemma nearly all PhD students have: It’s near impossible to do research in topic X unless you have the “right” background but what is the right background? What’s your experiences with this and how much someone should know to do research? Thanks Akhil …

        1. Akhil Mathew Says:

          I believe Rudin is considered an introductory book in analysis in the same sense that Hartshorne is an introductory algebraic geometry book, or Spanier an intro algebraic topology book. I.e., the material is foundational, but the prior acquaintance assumed is at the level of a intro-level undergraduate course (e.g. point-set topology, elementary real analysis); “introductory” does not necessarily mean “easy reading.”

          I’m afraid that I simply don’t know enough to answer your question about the background in functional analysis to do research. My current interests tend to be reflected in the choice of topics (i.e., algebraic topology as I write this), but I have not done research in it, and even had I done a little I would not be qualified to comment. Perhaps you might find this MO thread interesting, though.

          1. Tom Says:

            OK, I understand. That seems to make sense. But can you please clarify about Hartshorne? I mean R&C and algebraic topology don’t assume much background but Hartshorne does. I’ve heard you need to have read the tome commutative algebra by Eisendud. And surely that’s grad. level background.

            Actually that brings me to an interesting question which I’d like to ask you. I know you read Hartshorne earlier (or started reading it). Can you tell me based on what you’ve read so far how much background is necessary to read Hartshorne? I mean is Atiyah and Macdonald enough for the comm. algebra. Or do you need more like Eisenbud? Basically I’m interested in the comm. algebra background necessary but it’d also be nice if you could tell me the necessary background in the other areas.

            Thanks!

            1. Akhil Mathew Says:

              There are definitely points in Hartshorne that require more than what’s covered in Atiyah-Macdonald (e.g. the section on Kahler differentials). I’m pretty sure that Eisenbud includes everything you need, though. On the other hand, if you’re willing to accept the properties of Kahler differentials without proof, you can just read Hartshorne straight away I suppose. You also need to know basic general topology and homological algebra (e.g. some familiarity with derived functors). (Note that Hartshorne is generally *not* used as a first course on algebraic geometry, despite the fact that it opens with a chapter on varieties over alg. closed fields.)

  4. Wei Yu Says:

    I am a Chinese student, would like to ask you a few questions, English is not very good, do not know can not express what I mean. On differential geometry, that is,1. how to finding Griffiths-positive metric on an ample vector bundle.2.On moduli space,for example,moduli space on Calabi-Yau manifold.How to Finding the absruct new structure on it.I do not know if you have interest in this area, that area you are interested in mathematics? Physics are not interested in? For example, string theory, quantum field theory, mirror symmetry. Want to introduce you to learn way. My English is not good, and very worried about not properly express what I mean. Hope that it will communicate with you more mathematics or physics. I strive to learn English now.Hope to be friends with you.

    1. Akhil Mathew Says:

      Dear Wei, my background is insufficient for me to answer your questions. Perhaps you should try asking on MathOverflow.

  5. Wei Yu Says:

    Why the every pure sheaf has a unique Harder-Narasimhan filtration,but any semi-stable sheaf need not unique Jordan-Holder filtration.The geometry of moduli space of sheaf is very difficult.Whether proof the existence of symplectic structure of moduli space of quasi-coherent sheaf?Ask you for hyperbolic geometric flow of interested?For example,hypobolic Ricci flow,hypobolic Kahler-Ricci flow,hypobolic mean curvature flow,hypobolic Calabi flow and so on.

  6. Tom Says:

    (previous thread was getting skinny!)

    Thanks Akhil. I appreciate your advice. But can you tell me what book you used for a
    first course in alg. geo. if you search up “alg. geo. first course” you end up with this book
    with that title by some guy called “Joe Harris”. How did you find this book by Joe Harris?
    Did you read it and if so how much background do you need to tackle it?

    Also is maclane a good book for homological algebra. I see you’ve used Weibel but how’s
    saunders maclane’s book? Both look good but I’d appreciate your views on which is good
    in which respects.

    Finally, did you read A&M and then Hartshorne or did you read A&M, Eisendud and then
    Hartshorne? I’m sometimes uncomfortable with accepting facts as true without knowing
    their proofs. Did you do this with Hartshorne or did you know all the necessary comm. algebra before reading it? If not how did you find it accepting facts with proofs in Hartshorne?

    Thanks Akhil.

    1. Akhil Mathew Says:

      I think I have used Shafarevich and James Milne’s notes (cf. the link on the page) as intro sources. I haven’t read Harris’s book, I’m afraid, but it’s probably good.

      Maclane’s book is on general category theory, not homological algebra (I don’t think it covers derived functors, for instance). I didn’t really read this books in any kind of order, but it probably makes sense to start at least with A/M before trying Hartshorne (or any other book on schemes). I don’t think reading all of Eisenbud (which is a long book) is necessary before one starts Hartshorne; you can always refer back as necessary.

      1. Tom Says:

        Sorry, I think I wasn’t clear. I was referring to “Maclane’s Homology” book
        in the grundlehren math series. I’m pretty sure this book covers hom.
        algebra. But I think you’re right about Maclane’s “Categories for the working
        mathematician” which is about category theory.

        Algebraic geometry is such a interconnecting discipline. Do you need to
        know things like algebraic topology or complex analysis (or even number theory) to read Hartshorne? Or are the prerequisites just restricted to comm.
        algebra and a graduate algebra course?

        I heard of this excellent book called “Principles of Algebraic Geometry” by Harris and Griffiths. Have you attacked this book yet? ;) It looks like a real killer in terms of background math necessary but it also looks like the holy bible of alg. geo. I don’t think a knowledge of schemes is necessary for this book but it looks pretty good on the classical side of things. You might want to have a look at it if you haven’t already. I think it might supplement Hartshorne well since you’ve already done diff. geometry complex analysis and the like which are the only prereqs. for the book.

        Thanks for being patient with me and sorry for asking so many questions! I’m
        very much a beginner in alg. geo. I’m kind of scratching my head. The subject is like this vast theory of theories and it seems like there’s no clear path to take. How do you plan to approach alg. geo.?

        1. Akhil Mathew Says:

          Ah. Right, I haven’t read _Homology._ Ditto for Griffiths and Harris. Hartshorne doesn’t invoke any algebraic topology, but Griffiths and Harris do, as well as some several complex variables (which is sketched at the beginning). Lack of prerequisites prevented me from reading G+H in the past, but I should take another look sometime soon. I understand it’s standard.

          For algebraic geometry, I’m currently not actively studying the subject outside of class, because I am distracted with algebraic topology. If I were, though, (and I probably will be soon) I would probably be trying to a) read Hartshorne and EGA and b) looking at other scheme-theoretic books, e.g. Ueno’s and Liu’s.

          For how to learn algebraic geometry, cf. my question on MO: http://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry

          You will find answers from people immensely more qualified than I.

  7. jujahju Says:

    Akhil, how do you work efficiently? I hear harvard have tonnes of work to do so how do you manage your time? Do you get any sleep?

    1. Akhil Mathew Says:

      I wish I knew a good answer to that question!

  8. gffsjr Says:

    Hi Akhil, I’m a undergraduate student at UFPI Brazil,and I want to study complex algebraic geometry, but first I know that I should learn the basic of algebraic geometry, what good references could you give to me for a first read?I’m in the second year and have a background in elementary algebra(I studied Dummit’s book until module theory) , topology and Real analysis.By the way I’m building a blog too,I hope see your comments there! Thanks !

    1. Akhil Mathew Says:

      Shafarevich’s book is what we are using for algebraic geometry in my (introductory) course, though I have not read it all that carefully. I can also vouch for Fulton’s _Algebraic curves_ (available at people.reed.edu/~davidp/332/CurveBook.pdf), which I found useful in preparing for a reading course on elliptic curves some time back. Also, I recommend Milne’s online notes on algebraic geometry.

      1. gffsjr Says:

        Thanks for the references , by the way could you give me good introductory text in complexity theory?Thanks again.

        1. Akhil Mathew Says:

          I know very little complexity theory. But my understanding was that Papadimitriou was the standard introduction.

  9. David Says:

    Akhil, have you heard of the book “Partial differential equations” by Lawrence Evans? It’s a really good book on PDE’s and I noticed you listed some books on the topic above, so if you’re interested in PDE’s, you might wish to have a look at this book.

    Also might be worthwhile taking a look at Gunning and Rossi’s “Several Complex Variables” at some point because of its connections with algebraic geometry. The book is very good too.

    1. Akhil Mathew Says:

      Dear David, thanks! I do intend to read those books at some point in my undergraduate education, perhaps this summer: the problem is, during the academic year, I seem to get very little time for sustained reading on a given topic because of coursework. (I tried to read Gunning-Rossi a while back but was unsuccessful; my current plan is to start with “Coherent analytic sheaves,” which seemed more accessible when I flipped through it.)

  10. David Says:

    I got the wrong email address in my previous comment.

  11. Max Muller Says:

    Hi Akhil, that’s one impressive list…

    I was wondering how you obtained all of these books. Did you actually buy them all? It seems to me that’s quite expensive to do so. Or did you borrow them at a library?

    Thanks,
    Max

    1. Akhil Mathew Says:

      Hi Max,
      I was rather fortunate to be living, when in high school, close to several university libraries.

  12. David Says:

    Another suggestion regarding analysis. Have you looked at Loukas Grafakos’ Classical Fourier Analysis and Modern Fourier Analysis? These are comprehensive and the materal is presented in a more “textbook” style unlike Stein’s monographs (which are also excellent, of course). You might already be familiar with the first book of Grafakos in which case the second book is ideal. (I’m not sure if you’ve seen the proof of the Carleson-Hunt theorem but if you haven’t, Grafakos’ Modern Fourier Analysis is great.)

    But as you say, these are good books to read you’re not busy with your courses …

    1. Akhil Mathew Says:

      Thanks for the recommendation! I’ve never read Grafakos (or, for that matter, thought about analysis anytime recently). I’ve never actually worked through the proof of Carleson-Hunt, but would like to someday — probably I’ll have the time to catch up on things like this once I don’t have problem sets due.

  13. Marcus Says:

    Dear Akhil,

    I really liked reading your algebraic geometry notes. But I unforunately didn’t have the background in category theory so I ddn’t appreciate the categorical language. (I pretended everything was for abelian groups, rings etc.) Can you please recommend some books on category theory that you think provide sufficient background to understand your AG notes and maybe EGA later on?

    Thanks,

    Marcus

    1. Akhil Mathew Says:

      Dear Marcus, EGA 0 provides a fair bit of introductory material on categories (starting with Yoneda’s lemma, fibered products, etc.). I found it very helpful. Other sources on categories that might be helpful are MacLane’s “Categories for the Working Mathematician” and Kashiwara-Schapira’s “Categories and Sheaves.” (I recommend the last one in particular.) At least for me, though, it was easier to learn basic category theory by seeing lots of examples, and thinking about it in the context of algebraic geometry (and algebraic topology).

      For EGA, you’ll also need basic sheaf theory. (There are even points in EGA III where Grothendieck uses the Cech-to-derived-functor spectral sequence.) You can find this in Godement’s “Theorie des faisceaux,” or Chapter II of Hartshorne. Though I should note that you really have to do things differently for general sites (in which case you don’t have stalks to reason with, and you have to define e.g. sheafification in a more general way). Cf. for instance Tamme’s book “Introduction to Etale Cohomology.”

      1. Marcus Says:

        Thank you Akhil! I wanted to ask you another question about Hartshorne’s book: I hear from a lot of people that Hartshorne’s book is very difficult to read and that the exercises are also very difficult. How much truth is there in this? I know you can only speak for yourself but I found the exercises in Chapter 2 of Hartshorne very routine (at least for the first few sections). Is it that these exercises are much easier than the ones in the other chapters? For example, how are the Chapter 1, 4 and 5 exercises? It seems plausible that these could be harder since they are on geometry whereas exercises on sheafs for example are routine but tedious. What did you find of Hartshorne and his exercises?

        Marcus

        1. Marcus Says:

          Also thanks for recommending “Categories and Sheaves”. This looks like a very good book!

        2. Akhil Mathew Says:

          I’m not sure if I’d describe most of Hartshorne’s exercises as routine! I think they get a lot harder after the first few sections of chapter II and general sheaf theory. I should mention that sheaf theory on general sites feels a bit different (because you don’t have stalks, for instance) and the proofs (e.g. that the category of abelian sheaves has enough injectives) have to appeal to general principles. I’ve never really looked at chapters 4 or 5 (except for Riemann-Roch). Also, a lot of them turn out to be special cases of (often nicer) results in EGA. (One example: in II.7, he asks you to prove using divisors that for a regular noetherian scheme X and a vector bundle E on X, the Picard group of the projectivization is that of X times \mathbb{Z}; but it’s possible to do this very cleanly using the formal function theorem as Grothendieck points out (without a full proof) in III.4, which I might blog about sometime.) Anyway, regardless of this, I’ve certainly learned a lot from the exercises, which strike me as one of the best features of the book.

          On the other hand, I still think that Hartshorne often seems to leave out much (possibly because it’s such a short book) far too often: he omits the functorial characterization of smoothness/etaleness/unramifiedness (the latter two of which figure in only briefly as exercises), the quasi-finite form of Zariski’s Main Theorem, the Leray spectral sequence, etc. I think this is actually one of the reasons it is such a hard book; EGA, where things are developed both more leisurely and in a much more general (and, at least to me, natural) setting, gives a very different picture, and is certainly easier reading (at least if you don’t try to read it linearly!).

          I would also note that there are plenty of successful algebraic geometers who have never read EGA. I suppose it’s a matter of taste, and as I am a beginning student, what I say should not be taken too seriously.

  14. nehsb Says:

    Hi,

    What are your thoughts on reading etale cohomology? I’m not sure as to whether it’s worth reading it thoroughly, as I’ve heard the advice that it’s better to just black-box all the etale cohomology theorems at the start. Do you have any thoughts on this, and what are the standard references? I’ve heard SGA 4/4.5, which is workable, but I’m at least an hour away (by car) from any library which has them, and I find it somewhat hard to read the scanned copies on a computer screen.

    Thanks.

    1. Akhil Mathew Says:

      Hi,

      Sorry for the slow response — it’s been a busy past week. I’m certainly no expert on etale cohomology. I did a reading course about it once, which was a lot of fun, but I don’t know a lot about number theory. I do think that it definitely makes sense to black box a lot of it at the start. Tamme’s “Introduction to Etale Cohomology” is by far the friendliest introduction I’ve seen. SGA 4.5 is very helpful though terse, while SGA 4 seems to go on to no end (but which also has a lot of really important and useful foundational stuff on topoi which I’ve never properly learned). Freitag-Kiehl’s book on etale cohomology and Kiehl-Weissauer “Weil conjectures, perverse sheaves, and the l-adic Fourier transform” are pretty inspirational (and hard) stuff. You might look at Emerton’s advice at

      http://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/

      Something I’m trying to learn more about recently — and which I didn’t study at all when I took the reading course, because I didn’t know homotopy theory — is the refinement of etale cohomology to etale homotopy theory. Instead of getting cohomology groups (with torsion or l-adic coefficients), you get a pro-homotopy type, which is a lot more information. I’ve heard things to the effect that you can get Poincar\’e duality in etale cohomology (which is very difficult to prove directly) using homotopy-theoretic methods, although I do not know the details. (I could suggest people who know a lot about this that you could contact if you’re interested.)

      Anyway, one reason it may make sense to black-box things is that modern technology not available in the days of SGA 4 enables simpler proofs!

      Incidentally, I saw your blog — very impressive! You should, of course, feel free to contact me if you have further questions about math or other things (e.g., college applications).

      1. nehsb Says:

        Thanks for the suggestions!

        I unfortunately don’t really know normal topological homotopy theory. It seems to motivate a lot of the recent work on algebraic geometry, so I should probably get to reading it at some point.

  15. Toussaint Says:

    Hello Mr Akheel

    I’m seeking advice and looking at your knowledge, your advice will really be useful.
    I am new to abstract mathematics. By abstract I mean proving theorems and understanding what they mean in contrast to the computational side of mathematics.
    I have read the book “How to prove it, a structured approach” and the book of proof and now I want to pick one branch of mathematics and practice theorem proving. Which one would you advice? Analysis, algebra, topology? I have tried with analysis with the Rudin book but it seems hard to me. So what would you advice and which book(s)?
    I equally hope your answer will be useful to someone else in the future.
    Regards

    1. Akhil Mathew Says:

      Dear Toussaint,

      Usually when starting out with math the goal is not so much to prove theorems, but to understand how mathematical proofs work and to practice with exercises, which you’ll find in any textbook, and in any of the above three fields. I would only mention that learning general topology without knowing a little about metric spaces and real analysis is likely risky.

      You might try math.stackexchange.com to look for lists of suggested textbooks (I myself found Rudin’s book very helpful, and I think Herstein’s algebra textbook; I’ve also heard that Munkres’s book is a good introductory topology textbook but have never read it). The book “Proofs from the book” is a lot of fun but is written more for enjoyment than serious study. Terence Tao’s blog (under “career advice”) is another helpful resource.

      1. Toussaint Says:

        Dear Mr Akheel!

        Thank you for the helpful advice! I was right about to give up Rudin’s book in desperation because I could not prove the theorems. But now, it is clear one must first emphasize understanding of the proofs and doing the exercises (for the sake of repetition).

        Indeed, I am on math.stackexchange.com and it is a wonderful resource for exercises, alternative proofs and references. And I did read Terrenece Tao advice(s) as well!

        Regards

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