Best algebraic geometry books according to redditors
We found 96 Reddit comments discussing the best algebraic geometry books. We ranked the 54 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 96 Reddit comments discussing the best algebraic geometry books. We ranked the 54 resulting products by number of redditors who mentioned them. Here are the top 20.
Preach it, brother!
Let me highly recommend Conceptual Mathematics: A First Introduction to Categories and Topoi: The Categorial Analysis of Logic as introductions to the topic requiring no more than a completely rudimentary grasp of set theory to get started--and really, they even motivate the rudimentary set theory. These are basically "Category Theory for Dummies," or at least the closest things that I've found so far.
Algebraic Number Theory - Cassels & Frolich and The Arithmetic of Elliptic Curves - Silverman
Pick up a copy of Algebraic Geometry: A Problem Solving Approach and work through the first chapter.
It shouldn't require much more than high school algebra, with just a smidgen of understanding of partial derivatives.
The first chapter defines algebraic sets of a polynomial, which is a subset of the plane defined by a polynomial: {(x, y) | P(x, y) = 0}.
The degree of the polynomial determines the degree of the curve. Degree 1 polynomials give straight lines, as you might expect. Degree 2 polynomials give the conic sections. You might remember conic sections from your high school algebra II class, but chances are it was mostly an exercise in memorizing equations.
It goes on to classify the conics up to affine change of coordinates. In R^2, there are ellipses (including the circle), hyperbolas, parabolas, and the degenerate conics, a double-line and a pair of crossing lines.
The chapters are fairly short and filled with super easy exercises that get you thinking about the material you're reading.
The chapter builds up some of the basic notions studied in algebraic geometry. While working over R^2 is great, it is harder to study because not every polynomial will have roots. So you upgrade to C^2 instead. In C^2, though, ellipses and hyperbolas become equivalent, thanks to allowing complex numbers in our affine change of coordinates.
Lastly, it builds up to projective geometry in CP^2. Even in C^2, there are cases where two intersecting lines may fail to meet if they are parallel to each other. By moving to CP^2, we force all lines to eventually greet each other (at some point of infinity if at no finite point).
This final upgrade is a bit technical, but it is a key ingredient to world-famous Bezout's Theorem, studied in chapter 3. But one immediately awesome result is that all nondegenerate conics become equivalent: ellipses, hyperbolas, and parabolas are just three ways of looking at the same geometrical object.
Algebraic geometry is an amazing field whose roots go back to at least Desargues in the 17th century. It has intimate ties with complex analysis (Chow's Theorem says that curves in the projective plane are actually compact Riemann surfaces) and number theory (where we work over the rationals, rather than the reals or complex numbers). In the 1930s, the field was put on a rigorous algebraic basis by Hilbert and Noether (this is essentially what Commutative Algebra is). And in the 1960s, Alexandre Grothendieck went totally ham and rephrased the entire subject in terms of categories and schemes.
Adding on to this, we need like a workbook for working with sheaves. They are difficult for me to get a feel for
On the other hand, I know there are very concrete problems in https://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963
Particularly the last chapter when sheaves and cech cohomology are introduced.
However when I think of sheaves, I cannot see the trees in the forest, I just see the forest
Perhaps we can get the special flair users in /r/math to setup some of this (the ones with the red background in their flair)?
I know nothing about any of these topics but we could use course notes from a school's Open Courseware.
Here are the relevant ones I've found. If a cell says "none" that just means I've left a placeholder for if people find something I can put in that spot. The ones with all nones means I either wasn't sure what to look for, or if what I found was the right thing (Lie Theory = Lie Groups? for example)
Subject | Source1 | Source2 | Source3| Source4
---|---|----|----|----
Algebraic Topology | MIT Seems to have all relevant readings as PDFs | Introductory Algebraic Topology I don not know the source for this one| Algebraic Topology by Hatcher is free | A Basic Course in Algebraic Topology by Massey - Not free
Algebraic Geometry | MIT Fall 2003 Has lecture notes| MIT Spring 2009 Also has lecture notes | Vakil's course notes| Eyal Goren Syllabus and course notes
Functional Analysis | MIT Lecture notes and assignments with solutions | Nottingham 2010 | Nottingham 2008 These ones not only have lecture notes, but audio of the lecture. | none
Lie Theory | MIT - Intro to Lie Groups | MIT - Topics in Lie Theory: Tensor Categories | none | none
General Relativity | Sean Carroll's Lecture Notes | Stanford video lectures on general relativity, Leonard Susskind | Lecture notes from Nobel Laureate Gerard Hooft on GR | Semi-Riemannian geometry with Applications to Relativity - Not free
Dynamical Systems | Very applied (Strogatz style) course notes for dynamical systems | More theoretical (Perko style) course notes for dynamical systems by the same author | none | none
Numerical Analysis | MIT Spring 2012 | MIT Spring 2004 | none | none
This is obviously not an exhaustive list. I thought Stanford and their own open courseware thing but it seems to just be a list of courses they have on Coursera.
Categories for the Working Mathematician. Of course, that's geared more towards people who need some category theory in their own work. Category theory by itself is like a bookshelf. A nice way to organize your stuff, but nothing substantive is actually there until you fill it with books.
Some great classics:
I really enjoy Reid's book, but for a first introduction I would look into Cox, Little and O'Shea's book Ideals, Varieties, and Algorithms. It has very good extended examples, and ties in computation in a very interesting way.
As far as algebraic topology goes, while Hatcher is available for free (and legally at that), I've found him quite difficult to use for independent study. He tries a bit too hard, I think, to illustrate his geometric intuition, and ends up with extremely verbose, confusing explanations. His proofs are difficult to follow if you're new to the subject as well (it took me hours to understand his proof that
[; \pi_1(S^1) = \mathbb{Z} ;]).An alternative text that I have greatly enjoyed reading is A Basic Course in Algebraic Topology by Massey. His text is much more appropriate for a student's first introduction to the subject because he explains every relevant detail, rather than assuming some indefinite body of prerequisite knowledge. The text is also split up into many short chapters, rather than the four long chapters of Hatcher, and each chapter includes a generous selection of exercises. It's very readable and very rewarding to work through.
If we do decide to use Massey, make sure to get the text labeled v. 127, as opposed to this one labeled v.56. The latter only includes the material on fundamental groups and covering spaces, without any mention of homology or cohomology.
Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.
I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.
At some point these "Pop" reading books get wholly unsatisfying and you need textbooks, but I think that's a story for a different semester. Theres a good set of books written by Avner Ash and Robert Gross (Boston College) that anyone with calculus 1 can easily get into:
Elliptic Tales:
https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199
Fearless Symmetry:
https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710/ref=pd_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=JG1NQ2F2XS0WJJ5PBKVV
Well worth the read, entertaining, and great introductions to their respective subjects!
Interesting. Did you ever think about self-publishing? Lulu.com or amazon self-publish or something?
My PhD advisor asked me and his other students for some advice on how he should try to get his book priced so that graduate students would be able to buy it. We all suggested between like $25 and $35, largely because we all have read this excellent book which is $36 to buy and free to download. I wonder how Hatcher negotiated that deal, although B&W probably drives the cost down a lot.
I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.
Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.
Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.
Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.
Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.
Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.
The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.
An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.
Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)
If you feel like you have the time, I could recommend http://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963 which is a very gentle introduction to the subject using classical curves. Only in the last chapter does it introduce sheaves and cohomology. I suspect something like this might be helpful to place everything in a concrete context, and also build up motivation for all the modern machinery that you'll find in Hartshorne.
Theres definitely some relations (curves in projective spaces are Riemann surfaces).
Here is an AG book that is taught from a CG perspective.
https://www.amazon.com/Principles-Algebraic-Geometry-Phillip-Griffiths/dp/0471050598
You might want to check out Chapter 5, Robotics and Automated Geometric Theorem Proving, of Cox, Little and O'Shea's Ideals, Varieties and Algorithms for applications of algebraic geometry to motion planning in robotics.
https://www.amazon.com/Dirichlet-Branes-Symmetry-Mathematics-Monographs/dp/0821838482 is a good book. Treats both the SYZ conjecture and Homological conjecture.
I'd suggest reading it and figuring out what background you are missing. Then fill in the details as necessary.
an introduction to manifolds by loring tu has very reasonable exercises that give you a good working feel for the material. many of them have hints or solutions at the back of the book. plus, it’s an excellent book. the same goes for his differential geometry book.
there is also analysis and algebra on differentiable maniflds: a workbook for students and teachers. it has lots of fully worked problems.
For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.
I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.
If you're interested in a sequence of books...keep reading.
If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.
Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.
Would Conway's Zip proof count?
Slightly off-topic: could you post a source for the picture-hanging problem? I'd be really interested to see how the proof goes.
[EDIT: Oh and in Conceptual Mathematics they prove the Brouwer Fixed-Point Theorem using only very elementary category theory... but I forget all the details and how rigorous the proof is, and my copy's on loan to a friend...]
You could easily spend a semester on chapter 1! That book is dense.
For those who have never heard of Hartshorne, the comment is referring to the standard graduate level Algebraic Geometry text by Robin Hartshorne.
There's a lot of category theory, but this only uses the basics. Galois theory is deeper than anything used so far. Category, functor, natural transformation (co)limit, and maybe adjoint should be plenty (it looks like gibbon's tries to explain everything about adjoints that he uses). That's all in the first four or five (short) chapters of MacLane.
>I was reading Frenkel's book "Love and Math" about the Langlands program.
I'm not sure what level you're coming at it from, but for an intro to Langlands I'd recommend Gelbart which is where I first read about it and still serves a good intro, in more recent times there is Bump et al's An Introduction to the Langlands Program also excellent. Frenkel's book may well whet your appetite but it's pretty light on details.
As I see it there are four kinds of books that fall into the sub $30 zone:
You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
So that's a lot of suggestions, but two of them are free so you can't go wrong with those.
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
Categories for the Working Mathematician
The basic idea for identifying faces of a polytope P in R^n is to identify linear inequalities of the form c⋅x ≤ a (here c and x are vectors in R^(n), and a is a constant) that are valid for P, i.e., that are satisfied by every vector x contained in P. Then the set P ∩ { x ∈ R^(n) : c⋅x = a } is a face of P.
The trick is to identify interesting valid inequalities, so that the corresponding face is not the empty set. To do this you need to investigate the (affine) dimension of the set P ∩ { x ∈ R^(n) : c⋅x = a }. Often the most interesting kinds of faces are facets, which are faces of dimension n − 1.
This is going to be difficult if your polytope is the convex hull of just some random set of points. It is easier if your polytope has some kind of structure; for example, if it is the space of all feasible solutions to a linear program. In that case you can use this structure to prove that inequalities are valid and that the corresponding faces are nonempty, or have a certain dimension, or whatever it is you want to show. That's not to say it is easy, but at least you have some structure to work with.
There is a lot of theory behind this. A standard textbook on the subject is Lectures on Polytopes by Günter M. Ziegler.
Neal KoblitzIntroduction to Elliptic Curves and Modular Forms is fairly short as far as math books go, though not as short as the others here.
Gouvea's Intro to the p-adics is also not quite as short as what others have suggested, but it felt short to me when I was going through it.
That is the intro. The Bible is the following:
For non-liars
I am using the word liar in the sense of the people who interrupt talks to ask can you prove that is well defined.
According to the back cover (available in the Amazon preview), only linear algebra is required; could you be more specific about what you have trouble understanding?
Also, I realize that this book is much older and doesn't quite cover the same sort of material, but you might want to look at Coxeter too.
Loring Tu has a new book which discusses the basics of principal bundles and their characteristic classes towards the end. I have already read a significant chunk of it to get the forms perspective of characteristic classes on regular ol' vector bundles, and I can say that the book is excellent. To get a more detailed or advanced perspective though, Kobayashi & Nomizu seems to be the best place.
I unfortunately don't know of a good generally accessible source on this. It's really one specific (and particularly nice looking) example of a more general theory. As such it's the sort of thing you usually learn as an example while you're learning about the more arithmetic side of modular forms, and I don't know of any good self contained sources for it.
What's your current background in math? If you haven't already taken algebraic number theory, you'd definitely want to start there (although you'd need abstract algebra, and especially Galois theory, as a prerequisite). The most important concept to pick up there would be the notion of a "Frobenius element".
If you're already familiar with algebraic number theory, then (as other people have said here), primes of the form x^(2)+ny^(2) by Cox is a good place to start, although if I remember correctly they don't actually do the n = 23 example there. That combined with a book on modular forms, such as Diamond and Shurman should give you a pretty good understanding of this.
I haven't read the following books, but they're supposed to be ultra simple (in this case, easy).
Algebraic Geometry for Scientists and Engineers by Abhyankar
Algebraic Geometry: A Problem Solving Approach by Garrity et al
I am not sure there are AG books more elementary than those listed.
I hope others will chime in here, but I'll answer as well as I can.
Euclidean and Non-Euclidean Geometry
Euclidean and non-Euclidean geometries are interesting and important for various reasons, so I certainly wouldn't say it's a bad idea to study them in depth.
If you want to study these subjects first because you find them interesting and you have plenty of years to spend, then go for it! However, it's not necessary (more on this below).
Multivariable Calculus and Linear Algebra
Before attempting even an elementary treatment of differential geometry, you'll want to have a working knowledge of calculus (single and multivariable) and linear algebra.
Elementary Differential Geometry
You could potentially skip the elementary treatments of differential geometry, but these might be useful for tackling more advanced treatments. Studying elementary differential geometry first is perhaps similar to taking a calculus class (with an emphasis on computation and hopefully on intuition) before taking a class in real analysis (with an emphasis on abstraction and rigorous proofs).
If you do want to work through an elementary treatment, then you have options. One well reviewed book, and the one I learned from as an undergraduate, is Elementary Differential Geometry by Barrett O'Neill.
Note that O'Neill lists calculus and linear algebra as prerequisites, but not Euclidean and Non-Euclidean geometry. Experience with Euclidean geometry is definitely relevant, but if you understand calculus and linear algebra, then you already know enough geometry to get started.
Abstract Algebra, Real Analysis, and Topology
The next step would probably be to study a semester's worth of abstract algebra, a year's worth of real analysis, and optionally, a semester's worth of point-set topology. These are the prerequisites for the introduction to manifolds listed below.
Manifolds
An Introduction to Manifolds by Loring W. Tu will give you the prerequisites to take on graduate-level differential geometry.
Note: the point-set topology is optional, since Tu doesn't assume it; he expects readers to learn it from his appendix, but a course in topology certainly wouldn't hurt.
Differential Geometry
After working through the book by Tu listed above, you'd be ready to tackle Differential Geometry: Connections, Curvature, and Characteristic Classes, also by Loring W. Tu. There may be more you want to learn, but after this second book by Tu, it should be easier to start picking up other books as needed.
Caveat
I myself have a lot left to learn. In case you want to ask me about other subjects, I've studied all the prerequisites (multivariable calculus, linear algebra, abstract algebra, real analysis, and point-set topology) and I've tutored most of that material. I've completed an elementary differential geometry course using O'Neill, another course using Calculus on Manifolds by Spivak, and I've studied some more advanced differential geometry and related topics. However, I haven't worked through Tu's books yet (not much). The plan I've outlined is basically the plan I've set for myself. I hope it helps you too!
Why are you doing the exercises? Is it for a class? Are you self studying?
I've done all the exercises in Hungerford. When we had a section assigned for homework, I would just do everything in that section. Maybe I was doing 1 or 2 sections a week. I can't really remember. After the class ended, I finished the book in the name of completionism (and because I enjoyed the material). It was a really fun project because my professor had done the same thing years before. She still had her hand-made answer key, and we'd compare solutions during office hours.
I've also done all the exercises in an old version of Vakil's notes and all the exercises in the first few chapters of Hartshorne. However, in both of those books the exercises contain key material so you have to do them.
If I could do it again, I'd force myself to type up everything. I did all this before I learned TeX. Now I have pages and pages of exercises wasting away in a box somewhere...
The link http://www.amazon.co.uk/Advanced-Euclidean-Geometry-Dover-Mathematics/dp/0486462374/ref=sr_1_1?s=books&ie=UTF8&qid=1332359201&sr=1-1
Griffiths & Harris ;)
As an academic, I don't think it is, at least in my field. The authors of textbooks make 0 money on their books. They do it for the professional exposure. There is almost no need for publishers, and yet it's the publishers who are making a lot of money basically acting as a middle-man for digital content.
You don't need expensive middle men for digital content.
You can buy the best Algebraic Topology book for $40, or you can download it for free on the author's webpage. This is not uncommon. Instead of amazon selling that book, you can imagine Hatcher (the author) just selling the rights to some reasonable online distribution company, and Hatcher would probably see more money that way.
Related: the Elsevier Boycott
Milnor