Timeline for Books you would like to read (if somebody would just write them…)
Current License: CC BY-SA 4.0
52 events
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| Dec 9, 2022 at 19:02 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| May 30, 2017 at 2:24 | history | edited | Alex Shpilkin | CC BY-SA 3.0 |
fix arXiv link
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 9, 2012 at 0:46 | comment | added | Jay Kangel | The book amazon.com/Frames-Locales-Topology-Frontiers-Mathematics/dp/… by Picado and Pultr is a general topology book written in the language of locales. Against your wishes this book does mention topologies. Much of this material is to point out differences between topologies and locales. Since the likely audience for this book will likely have some knowledge of topology I don't think one can expect the authors to do otherwise. | |
| Feb 6, 2011 at 15:44 | comment | added | Johannes Ebert | @Harry: I taught linear algebra last year. Once bases (as sets of vectors) were understood, the passage to bases as arbitrary families did not cause many problems. Moreover, after several weeks of practice with matrices, writing linear maps into the entries of a matrix wasn't a big problem, either. Writing polynomials as entries (to define the characteristic polynomial), however, is a much more subtle thing to grasp, at least if you wish to do it correctly. | |
| Feb 4, 2011 at 6:36 | comment | added | Harry Gindi | @Johannes: It extends one's abilities to use matrices immensely! The "basis" is really a choice of decomposition, not just a sequence of elements. | |
| Feb 1, 2011 at 11:44 | comment | added | Johannes Ebert | @Harry: sure, but what is the point? Once you understand both, matrices, bases, linear map and categories, coproducts, products, this is a triviality. If you think there is a pedagogical or mathematical advantage of defining matrices in this manner, what is it? | |
| Feb 1, 2011 at 10:31 | comment | added | Harry Gindi | @Pete: There's a very nice way to think of matrices category theoretically (as I'm sure you know), where an $I\times J$ matrix is given by a family of morphisms $f_ij:A_i\to B_j$, and the induced map is given $f:\coprod_i A_i \to \prod_j B_j$. | |
| Jan 31, 2011 at 15:46 | comment | added | Pete L. Clark | Having thought things through a bit more, I wish to affirm the principle that the author of a math book ought not to be required to include any material beyond that which is of firm personal interest to herself. (Diligent application of this principle could lead to better books.) So I don't want to discourage anyone from writing this particular take on linear algebra. Rather what I mean to say is that such a book should be used for good rather than ill: raising a generation of mathematicians for whom bases and matrices are no more than an afterthought would be nothing to be proud of. | |
| Jan 31, 2011 at 15:32 | comment | added | Pete L. Clark | Also: anyway, as a general rule I would regard suggestions for textbooks to be written after a complete overhaul of American high school mathematics to be somewhat hypothetical, if not actually counterfactual. | |
| Jan 31, 2011 at 15:31 | comment | added | Pete L. Clark | @Harry: "your comment speaks to the inadequacy of high school mathematics in the United states to teach the skills necessary in higher mathematics." It does? I was educated entirely in the US, and I think I learned all the necessary skills for higher mathematics. (At any rate, right after high school I learned higher mathematics. Isn't that the point? Are you truly arguing that my learning about matrix multiplication at age 16 was an impediment to learning about abstract vector spaces at age 17 and transfinite constructions in linear algebra at 18? I was there, and I'm not buying it.) | |
| Jan 31, 2011 at 15:24 | comment | added | Johannes Ebert | @Dmitri: if your students struggle with the notion of an abstract vector space, do you really think they'd struggle less with the notion of a fully dualizable object in a symmetric monoidal category? I somehow doubt it. | |
| Jan 31, 2011 at 14:09 | comment | added | Harry Gindi | The reference is New Thinking in School Mathematics, and it is the proceedings from a conference. Dieudonné's paper is included in full on page 32. Anyway, the reason why I bring this up is that this sort of thinking should be taught in highschool, which makes Dmitri's approach much easier to swallow. | |
| Jan 31, 2011 at 14:07 | comment | added | Harry Gindi | @Pete, your comment speaks to the inadequacy of highschool mathematics in the United states to teach the skills necessary in higher mathematics. I actually took out the book from the famous New Math conference in 1959 where Dieudonné set out his original ideas, and they're not at all what you would expect. In particular, the major setpiece of his programme is removing Euclidean geometry and replacing it with more depth in algebra. Before you jump to the critics' side, I suggest you read it for yourself to see if it sounds so crazy. | |
| Jan 31, 2011 at 4:30 | comment | added | Pete L. Clark | Let me put it another way: at what age and point in one's development does a person realize that she is going to be the kind of mathematician that wants to think in an especially categorical / abstract / basisfree way? Many if not most American students are not even exposed to these concepts until grad school. But my (American) high school had a course in which we multiplied matrices, solved linear systems and so forth. I presume that most of the people in this course are not now pure mathematicians. So how could I have avoided learning about matrices first? (And why? What was the harm?) | |
| Jan 31, 2011 at 4:24 | comment | added | Pete L. Clark | @Dmitri: I think I for one have not learned anything "literally in a few minutes". Do you have experimental evidence for this claim? Anyway, I agree with you that most people take a lot longer to learn the abstract approach. But moreover, for many people understanding concrete examples is a necessary route to abstraction. If you're going to teach people about dualizable objects in categories, you can go ahead and teach them about bases and matrices first, I think, without wasting anyone's time. | |
| Jan 30, 2011 at 23:55 | comment | added | Dmitri Pavlov | @Pete: I guess my point here is that once you know coordinate-free linear algebra, you can learn bases, coordinates, and matrices literally in a few minutes. The reverse transition is much more difficult to make. I have seen lots of students in my linear algebra discussion section struggling with “abstract” vector spaces. The same applies to manifolds: Charts and atlases are very easy to learn once you have mastered the coordinate-free approach. Also, if you have a coordinate-free proof it is very easy to turn it into a coordinate proof. The reverse process is highly non-trivial. | |
| Jan 30, 2011 at 21:09 | comment | added | Pete L. Clark | ...arguments and computations with matrices come up in very sophisticated branches of modern number theory, and not of course because these number theorists do not know how to think in a coordinate-free way. But even if someone absolutely never finds it helpful to think about matrices, nevertheless he needs to know about them in order to communicate with other people. Thus I think a linear algebra text entirely without matrices creates the wrong impression, and could tempt zealous youngsters to literally never learn about the basis/matrix approach. That would be very bad. | |
| Jan 30, 2011 at 21:06 | comment | added | Pete L. Clark | @Dmitri: I heartily agree with you that for an important subject in mathematics, there will not be one approach which is most satisfactory to all clients. In fact, I would take it a step further: for sufficiently basic and important subjects -- like linear algebra! -- even one person needs to learn multiple approaches, so here both the basis/matrix viewpoint and the coordinate-free viewpoint. IMO a mathematician who really does not know about matrices will be at a serious professional disadvantage:... | |
| Jan 28, 2011 at 21:08 | comment | added | Dmitri Pavlov | @Johannes: I think we already went through this story in the case of integration on smooth manifolds. I don't think that the situation here is different. As for positive characteristinc, the definition with exterior powers works perfectly in all cases. | |
| Jan 27, 2011 at 21:27 | comment | added | Johannes Ebert | I think once you try to write down all the details of that approach, you will quickly start to appreciate bases as a convenient theoretical tool for setting up the coordinate-free theory. | |
| Jan 27, 2011 at 21:23 | comment | added | Johannes Ebert | @Dmitri: checking the details of (1) and (2) (i.e., that the maximal objects do what they should do) amounts to the same (simple) arguments that are used in the traditional approach. Ad (3): the trace of the identity is not a good definition of dimension, since it is wrong in positive characteristic. In characteristic $0$: how do I know that the trace is a nonnegative integer? And how do I check that a finitely generated vector space or a subspace of a finite-dimensional space is dualizable? | |
| Jan 27, 2011 at 19:57 | comment | added | Yemon Choi | @Dmitri: nothing could be further from the truth, and I apologize if that is the impression you got. I misread you as arguing for the "rightness" of this approach over others, rather than (as you clarify) arguing for its "rightness" alongside others. Sometimes I want to take a basis or an approximating net and get my hands dirty; sometimes I want to think of the proj. tensor product of Banach or operator spaces as left adjoint to the internal Hom. I think that functional analysis, or at least the corner I happen to have landed in, benefits from both perspectives | |
| Jan 27, 2011 at 19:05 | comment | added | Dmitri Pavlov | @Yemon: One size does not fit all. You and darij seem to subtly imply (or at least this is my feeling when I read your comments) that for any mathematical theory there is the best way to expose it, whereas I am more inclined towards diversity of expositions. Some people (like me) like coordinate-free expositions, while others prefer bases and matrices. There are plenty of linear algebra textbooks written using bases and matrices, but very few or none are written in a coordinate-free way. That's why I included linear algebra in my list. | |
| Jan 27, 2011 at 18:42 | comment | added | Yemon Choi | This discussion betwen Johannes and Dmitri touches on one of my misgivings. If one wants to train as an operator theorist - not an operator algebraist - and hence do messy things with particualr operators, does Dmitri's presentation still possess conceptual or pedagogical advantages over the "conventional" one? I don't claim it would be worse, because I haven't though about this, but would it really be better? In my experience functional analysis can be learned quite well by going from the particular to the general and back again, although this requires more time from the teachers | |
| Jan 27, 2011 at 17:48 | comment | added | Dmitri Pavlov | @Johannes: I fail to see how (1) and (2) use bases in disguise. The definition via traces is not circuitous and I don't need to show that the map V*⊗V→End(V) is an isomorphism. By definition, a finite-dimensional vector space is a dualizable object in the category of vector spaces, hence its endomorphisms possess traces, which are defined here: ncatlab.org/nlab/show/trace | |
| Jan 27, 2011 at 13:29 | comment | added | Johannes Ebert | @Dmitri: (1) and (2) are elegant arguments (but they use bases in disguise). I am not satisfied with (3). Your definition of dimension is really the same as the standard one (there exists a basis of length $n$), and the work to be done is then that each finitely generated vector space has a dimension (or, equivalently, that any subspace of $k^n$ is isomorphic to some $k^m$). The definition of the dimension via the trace is circuitous since you have to know that the natural map $V^* \otimes V \to End(V)$ is an isomorphism in order to define the trace without matrices. | |
| Jan 27, 2011 at 5:04 | comment | added | Dr Shello | @Dmitri's answers to Johannes' questions: that's really great! That would be an interesting book... | |
| Jan 27, 2011 at 5:00 | comment | added | Dmitri Pavlov | @darij: I prefer to characterize finitely generated projective modules as dualizable objects in the category of modules and finitely generated modules as factorobjects of dualizable objects. I don't think that maps between direct sums of modules are that important to justify the introduction of matrices. | |
| Jan 27, 2011 at 4:38 | comment | added | Dmitri Pavlov | @Johannes: (3) is trivial for infinite dimensional spaces if you use the usual definition of dimension for infinite-dimensional spaces (dim V=n if V is isomorphic to k^n). For finite-dimensional spaces you have much better definitions (with traces or exterior powers). In this case choose a maximal pair of isomorphic subspaces (no need to use Zorn's lemma here). By maximality the subspaces must coincide with the entire spaces. You also don't need Zorn's lemma in (1) and (2) if all spaces are finite-dimensional. | |
| Jan 27, 2011 at 4:34 | comment | added | Dmitri Pavlov | @Johannes: For (1) use Zorn's lemma to find the maximal subspace that is disjoint with the image of the first map in the exact sequence. This maximal subspace is the image of the splitting map. For (2) use Zorn's lemma to find the maximal extension of your linear functional. This maximal extension is defined on the whole subspace. Note that the use of some form of the axiom of choice is inevitable because these statements are equivalent to some weak forms of the axiom of choice. | |
| Jan 26, 2011 at 21:42 | comment | added | darij grinberg | "Linear algebra without bases" really means "module theory over commutative rings". Which is why it is probably a bad idea to write a full book without even once using a basis: Many properties of modules require finitely generated modules, i. e. generating sets, and the step from generating set to basis isn't that huge. Also, matrices are extremely important pretty much everywhere including the most abstract algebra (maps between direct powers of modules are written as matrices of maps, for example). But I agree that it would be better to START a linear algebra course with generic modules. | |
| Jan 26, 2011 at 20:43 | comment | added | Johannes Ebert | @Dmitri: here are a couple of theorems that I need whenever I find a vector space on my desk: 1. any short exact sequence of $K$-vector spaces is split; 2. any linear functional on a subspace can be extended to the whole space; 3. Two vector spaces of the same dimension are isomorphic. How can these results be shown without bases (or an equivalent notion)? | |
| Jan 26, 2011 at 2:42 | comment | added | Dmitri Pavlov | @Michael: Could you please be more precise? What kind of theorem or definition do you have in mind? | |
| Jan 25, 2011 at 23:11 | comment | added | Not Mike | @Dmitri -1 for bullet point number one. Locales are not sufficient for any thoroughly general treatment of topology. No mater what ncat may have you believe (sorry ncat fans, just know I do respect the site). Just because you can squint your eyes and make it look like a topology does not mean it can actually handle all of the constructions from topology. | |
| Jan 25, 2011 at 19:14 | comment | added | Dmitri Pavlov | @Johannes: Take a look at Bourbaki's book to see how linear algebra can be done without bases, coordinates, and matrices. If you have a concrete theorem in mind, feel free to ask. | |
| Jan 25, 2011 at 18:58 | comment | added | Johannes Ebert | @Dmitri: How do you plan to do linear algebra, beyond the first definitions, without the notion of a basis? | |
| Jan 25, 2011 at 16:42 | comment | added | Dmitri Pavlov | @Pete: Sorry, I got it all wrong. Replace “maximal” by “prime” and “prime” by “radical”. I am not sure what I was thinking about when I wrote my previous comment, but points in spectra correspond to prime ideals and open sets correspond to radical ideals. Nullstellensatz can be reformulated in such a way that it does not refer to maximal or prime ideals. A proof of such a localic version can be found in this paper: ams.org/journals/proc/1980-079-01/S0002-9939-1980-0560591-4/… | |
| Jan 25, 2011 at 11:20 | comment | added | Pete L. Clark | @Dmitri: thanks for responding. Part of my point is that maximal ideals are part of the core subject matter of commutative algebra in the sense that many of the key results refer to them (e.g. the Nullstellensatz). So I don't understand how it could still be commutative algebra without them. But about prime ideals...why are they so much better from a constructive point of view? The existence of a prime ideal in any ring still requires at least the Boolean Prime Ideal Lemma, right? | |
| Jan 25, 2011 at 6:36 | comment | added | Dmitri Pavlov | @Yemon: My confusion extends to this example also. For a set A the dual of l^1(A) is l^∞(A), l^∞(A)=C^0(B), where B is the Stone-Čech compactification of A, and the dual of C^0(B) is the space of Radon measures on B. In other words, passing to the framework of locales again does not change anything. Note that the Stone-Čech compactification of locales does not require any form of the axiom of choice. | |
| Jan 25, 2011 at 6:12 | comment | added | Yemon Choi | @Dmitri: fair point about just wanting some coordinate-free texts, given that "working in coordinates" approaches are indeed well-covered. Also, yes, I rather stupidly wrote "bidual of c_0" when I meant "bidual of l^1". | |
| Jan 25, 2011 at 5:27 | comment | added | Dmitri Pavlov | @Pete: You only need to consider maximal ideals as long as you insist that the spectrum of a commutative ring consists of actual points. Once you pass to the localic framework you only need to keep track of open sets, which correspond to prime ideals. There are plenty of papers in this area (for example, Thierry Coquand has lots of papers on his homepage: cse.chalmers.se/~coquand/algebra.html), but I am not aware of any books written in this language. | |
| Jan 25, 2011 at 4:57 | comment | added | Dmitri Pavlov | @Yemon: Only 2 out of 8 proposals mention coordinates, not 7. My point is that since there are so many books that teach you how to do all kinds of things using coordinates, it would be nice to have at least one book that would teach you how to do things in a coordinate-free way. I find that coordinate-free proofs often enhance my intuition. | |
| Jan 25, 2011 at 4:46 | comment | added | Dmitri Pavlov | @Yemon: Could you please be more specific about “getting things done in functional analysis”? What kind of theorem do you have in mind? I can guess that RKHS means reproducing kernel Hilbert space, but I have no idea what kind of result you are referring to. | |
| Jan 25, 2011 at 4:42 | comment | added | Dmitri Pavlov | @Yemon: I am somewhat confused by your question about the bidual of c_0. The dual of c_0 is l^1 and the dual of l^1 is l^∞. I don't think that passing to locales affects the validity of any of these statements. | |
| Jan 24, 2011 at 22:47 | comment | added | Pete L. Clark | Does a localic approach to commutative algebra actually exist (albeit not written down in textbook form)? I'm puzzled at the suggestion that such an approach could avoid maximal ideals: aren't these part of the subject? (Note that I barely even know what a locale is...) | |
| Jan 24, 2011 at 19:30 | comment | added | Yemon Choi | (Last question honest, not rhetorical. Maybe you can do it in a chart-free way, but it's not clear to me what the benefits would be.) | |
| Jan 24, 2011 at 19:29 | comment | added | Yemon Choi | Finally: having taken some time to read the exposition in Connes NCDG part II of how to work out the cyclic cohomology of $C^\infty(M)$ ... I am puzzled as to how this would be made clearer by refusing to use charts to work in local coordinates. You'd presumably have to use a double-complex argument and resolve by acyclics, but how does one check these are acyclic without doing some kind of local computation? | |
| Jan 24, 2011 at 19:26 | comment | added | Yemon Choi | Might I also suggest separating off your suggestions about higher categories and writing them as a separate answer? They seem to have a different flavour from your other "down with coordinates! define everything in terms of the coordinate ring!" suggestions | |
| Jan 24, 2011 at 19:23 | comment | added | Yemon Choi | Also: I thought it was acknowledged that while you can (and to some extent, should) set up linear algebra without coordinates and bases and matrices, getting things done in functional analysis rather often needs you to choose bases, etc. (Cf. the difference between categories of Hilbert spaces and categories of RKHS) | |
| Jan 24, 2011 at 18:50 | history | edited | Dmitri Pavlov | CC BY-SA 2.5 |
added 161 characters in body
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| Jan 24, 2011 at 17:21 | history | answered | Dmitri Pavlov | CC BY-SA 2.5 |