User gon&#231;alo marques - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:19:06Z http://mathoverflow.net/feeds/user/2162 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them Books you would like to read (if somebody would just write them...) Gonçalo Marques 2011-01-24T10:23:39Z 2012-12-13T15:42:48Z <p>I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different approach. (I do hope this is not inappropriate for MO).</p> <p>Let me start with some books I would like to read (again with self-explanatory titles)</p> <p>1) The Weil conjectures for dummies</p> <p>2) 2-categories for the working mathematician</p> <p>3) Representations of groups: Linear and permutation representations made side by side</p> <p>4) The Burnside ring</p> <p>5) A functor of points approach to algebraic geometry</p> <p>6) Profinite groups: An approach through examples</p> <p>Any other suggestions ?</p> http://mathoverflow.net/questions/80372/reference-request-john-baez-on-1-and-2-categories-and-propertiesstructur/80375#80375 Answer by Gonçalo Marques for reference request: John Baez on (-1)- and (-2)-categories and properties+structure+stuff Gonçalo Marques 2011-11-08T10:49:30Z 2011-11-08T10:49:30Z <p>I think you mean the paper <a href="http://arxiv.org/abs/math/0608420" rel="nofollow">"Lectures on n-Categories and Cohomology"</a>. You can also look at the appropriate pages at the <a href="http://ncatlab.org/nlab/show/HomePage" rel="nofollow">nLab</a>. </p> http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/39099#39099 Answer by Gonçalo Marques for What are some examples of colorful language in serious mathematics papers? Gonçalo Marques 2010-09-17T13:32:45Z 2010-09-17T17:10:11Z <p>In the huge and austere book "Groupes algébriques" by M. Demazure and P. Gabriel we find in the last pages a "Dictionaire "Fonctoriel"", a dictionary of terms related to category theory where they have:</p> <blockquote> <p>Satellite- Voir Cartan-Eilenberg et non Paris-Match.</p> </blockquote> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/18695#18695 Answer by Gonçalo Marques for Books you would like to see translated into English. Gonçalo Marques 2010-03-19T00:52:22Z 2010-03-19T00:52:22Z <p>Chebotarev's "Grundzüge der Galois'schen Theorie"</p> http://mathoverflow.net/questions/18238/equivalent-singular-chains-and-differential-forms-as-functionals-on-forms-on-co/18247#18247 Answer by Gonçalo Marques for Equivalent singular chains and differential forms, as functionals on forms, on compact Riemannian manifolds Gonçalo Marques 2010-03-15T06:20:59Z 2010-03-16T18:19:04Z <p>I think that what you are getting at is called a "Current". See any book on geometric measure theory like the recent one by Krantz called "Geometric Integration Theory" or the older one by Federer called "Geometric Measure Theory"</p> http://mathoverflow.net/questions/123/linearity-of-the-inner-product-using-the-parallelogram-law/7814#7814 Answer by Gonçalo Marques for Linearity of the inner product using the parallelogram law Gonçalo Marques 2009-12-04T21:27:28Z 2009-12-04T21:27:28Z <p>There is a book by A. C. Thompson called "Minkowski Geometry". In this context "Minkowski Geometry" means the geometry of a vector space with a norm (not the geometry of special relativity as one might think...). Chapter 3.4 is called "Characterizations of the Euclidean Space" and it has many theorems stating that a norm comes from a inner product iff such and such (mostly geometric) conditions is satisfied. They might be helpful (as far as I can tell... I don't know the book in depth). In Thompson's book there's a reference to Dan Amir's book "Characterizations of inner product spaces" that might also be useful (but I haven't even seen this book).</p> http://mathoverflow.net/questions/7490/differences-between-reflexives-and-projectives-modules/7496#7496 Answer by Gonçalo Marques for Differences between reflexives and projectives modules Gonçalo Marques 2009-12-01T20:09:44Z 2009-12-01T20:09:44Z <p>According to Lam's "Lectures on Modules and Rings" every f.g. projective module is reflexive. See page 55, exercise 7. </p> http://mathoverflow.net/questions/7407/what-is-a-projective-space/7489#7489 Answer by Gonçalo Marques for What is a projective space? Gonçalo Marques 2009-12-01T19:36:52Z 2009-12-01T19:49:59Z <p>The book "Modern Projective Geometry" has a system of axioms for projective spaces in a set $X$ using a function $f\colon X\times X\to P(X)$ (see page 30). Also there's a lot of projective geometry that can be done in the context of lattices. I thing this is related to Greg's answer through Von Neumann's "<a href="http://books.google.pt/books?id=onE5HncE-HgC&amp;dq=Continuous+geometry&amp;printsec=frontcover&amp;source=bl&amp;ots=gbfVKzAAbj&amp;sig=qYzkgkbW6ZJu7v4subUL2jrOieI&amp;hl=pt-PT&amp;ei=S2QVS8jBLIWy4QbX3aHiBg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=3&amp;ved=0CBEQ6AEwAg#v=onepage&amp;q=&amp;f=false" rel="nofollow">Continuous geometry</a>". (see <a href="http://planetmath.org/encyclopedia/VonNeumannLattice.html" rel="nofollow">here</a>). Another book to look might be Baer's classic "<a href="http://books.google.pt/books?id=Ol8YFnZRJfIC&amp;dq=baer+linear+algebra&amp;printsec=frontcover&amp;source=bn&amp;hl=pt-PT&amp;ei=5mQVS4qvJoSl4QauoN3cBg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=4&amp;ved=0CBgQ6AEwAw#v=onepage&amp;q=&amp;f=false" rel="nofollow">Linear algebra and projective geometry</a>". </p> <p>On Greg's comment to Qiaochu's comment: let's not forget that for every projective space $X$ we can always find a division ring $D$ such that every $X=P^n(X)$ for some n (in the the plane case you need that $X$ be a Desargue's plane).See <a href="http://www.cs.elte.hu/geometry/csikos/proj/pr5.dvi" rel="nofollow">this</a>. (But, as far as I know, these are all finite dimensional results so they might not be that interesting to Andrew...)</p> http://mathoverflow.net/questions/6442/references-literature-for-pushouts-in-category-of-commutative-monoids-ed-ama/7444#7444 Answer by Gonçalo Marques for References/literature for pushouts in category of commutative monoids? [ed. - amalgams] Gonçalo Marques 2009-12-01T15:12:38Z 2009-12-01T15:18:59Z <p>Have you looked at the book <a href="http://books.google.pt/books?id=4gPhmmW-EGcC&amp;dq=Monoids,+Acts+and+Categories+with+Applications+to+Wreath+Products+and+Graphs&amp;printsec=frontcover&amp;source=bl&amp;ots=1lTFC5WTuc&amp;sig=ZrU_auTCCfLSQZsxa4xfSvW534Y&amp;hl=pt-PT&amp;ei=ezAVS6LDO4uH4QbQqsTKBg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=4&amp;ved=0CBwQ6AEwAw#v=onepage&amp;q=&amp;f=false" rel="nofollow">Monoids, Acts and Categories</a> by Mati Kilp, Ulrich Knauer and Alexander V. Mikhalev? Pushouts are discussed in chapter II. </p> http://mathoverflow.net/questions/7330/which-math-paper-maximizes-the-ratio-importance-length/7423#7423 Answer by Gonçalo Marques for Which math paper maximizes the ratio (importance)/(length)? Gonçalo Marques 2009-12-01T12:49:16Z 2009-12-01T12:49:16Z <p>The so called "Weil conjectures" are in the last pages of André Weil's short <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.bams/1183513798&amp;page=record" rel="nofollow">paper</a> in 1949, "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society 55: 497–508. They probably were around before though.</p> http://mathoverflow.net/questions/2479/de-rham-cohomology-of-surfaces/7347#7347 Answer by Gonçalo Marques for de Rham Cohomology of surfaces Gonçalo Marques 2009-12-01T02:49:58Z 2009-12-01T02:49:58Z <p>For surfaces there's very nice and down-to-earth approach in Fulton's "Algebraic Topology" specially chapter 18 "Cohomology of Surfaces". For higher dimensional classical manifolds including the projective spaces see Karoubi's "Algebraic Topology via Differential Geometry" specially chapter V "Computing Cohomology". All of these use de Rham cohomology.</p> http://mathoverflow.net/questions/7124/determinant-of-a-perfect-complex/7125#7125 Answer by Gonçalo Marques for determinant of a perfect complex Gonçalo Marques 2009-11-29T10:53:37Z 2009-11-29T10:53:37Z <p>You should take a look at the appendix A of "Discriminants, Resultants, and Multidimensional Determinants" by Gelfand, Kapranov and Zelevinsky</p> http://mathoverflow.net/questions/73640/discrete-version-of-nullstellensatz/73643#73643 Comment by Gonçalo Marques Gonçalo Marques 2011-08-25T23:39:55Z 2011-08-25T23:39:55Z That's a really hard book to find... http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/57545#57545 Comment by Gonçalo Marques Gonçalo Marques 2011-03-06T15:02:42Z 2011-03-06T15:02:42Z There is also the book Emerton reffered in a comment above. It's called &quot;Mathematical Developments Arising from Hilbert Problems&quot; and it is published by the AMS http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/53068#53068 Comment by Gonçalo Marques Gonçalo Marques 2011-01-24T15:07:39Z 2011-01-24T15:07:39Z Michael Spivak has recently written a book called &quot;Physics for Mathematicians: Mechanics I&quot;. I haven't seen it and it's a bit expensive on Amazon, but it might be just what you want (but as far as I can tell it's &quot;only&quot; about classical mechanics...) http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/53063#53063 Comment by Gonçalo Marques Gonçalo Marques 2011-01-24T14:42:24Z 2011-01-24T14:42:24Z I think Steve Awodey's &quot;Category theory&quot; might be just right for you. http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them Comment by Gonçalo Marques Gonçalo Marques 2011-01-24T14:16:08Z 2011-01-24T14:16:08Z Thank you for your suggestions I will look for Katz and Deligne's articles. I am aware of Freitag and Kiehl's textbook, unfortunately it's a hard to find. I was thinking of a textbook that would use the Weil conjectures as a &quot;leitmotiv&quot; while introducing some of the more modern characters in algebraic geometry. But maybe it can't be done (at least at level I would understand...). http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them Comment by Gonçalo Marques Gonçalo Marques 2011-01-24T12:28:47Z 2011-01-24T12:28:47Z Thanks Dylan. Tim: it would be great indeed if JB would publish his &quot;higher algebra&quot; but his past expository work is already amazing. Tom: I am aware of that paper (and you also have some sections about it on your wonderful book) but I was thinking about a full textbook presentation that might have more examples and applications. http://mathoverflow.net/questions/16917/what-is-a-reference-for-profinite-sets/16929#16929 Comment by Gonçalo Marques Gonçalo Marques 2011-01-23T23:10:44Z 2011-01-23T23:10:44Z Another reference is &quot;Alg&#232;bre et Th&#233;ories galoisiennes&quot; by R&#233;gine et Adrien Douady namely pag. 63-64 http://mathoverflow.net/questions/5786/how-do-i-check-if-a-functor-has-a-left-right-adjoint/5911#5911 Comment by Gonçalo Marques Gonçalo Marques 2010-08-20T13:52:40Z 2010-08-20T13:52:40Z Is there a good place to learn about ends and coends with good (and not too sophisticated) examples? The page on the n-lab didn't help me that much because it goes very quickly into the enriched context and I think that your wonderful notes on category theory don't cover it. http://mathoverflow.net/questions/31842/tensor-product-and-category-theory/31844#31844 Comment by Gonçalo Marques Gonçalo Marques 2010-07-14T13:17:57Z 2010-07-14T13:17:57Z You should check Keith Conrad's handouts on the tensor product too. They are really good. <a href="http://www.math.uconn.edu/~kconrad/blurbs/" rel="nofollow">math.uconn.edu/~kconrad/blurbs</a> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/17784#17784 Comment by Gonçalo Marques Gonçalo Marques 2010-03-19T00:42:34Z 2010-03-19T00:42:34Z It would be really good to see all 3 volumes of Weber's &quot;Lehrbuch der Algebra&quot; translated. http://mathoverflow.net/questions/18365/what-is-the-relationship-between-the-bell-numbers-the-bell-polynomials-and-the/18373#18373 Comment by Gonçalo Marques Gonçalo Marques 2010-03-17T08:55:58Z 2010-03-17T08:55:58Z &quot;a species is a functor from the category of finite sets into the category of finite sets&quot;. A small detail: it's the category of finite sets and bijections (so it's actually a groupoid). http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7967#7967 Comment by Gonçalo Marques Gonçalo Marques 2009-12-06T18:08:39Z 2009-12-06T18:08:39Z The book &quot;Measure theory and probability&quot; by Guillemin and Adams is also very good. http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7963#7963 Comment by Gonçalo Marques Gonçalo Marques 2009-12-06T17:22:53Z 2009-12-06T17:22:53Z You're right. Nice notes and written with great sense of humour. http://mathoverflow.net/questions/7793/what-are-the-auto-equivalences-of-the-category-of-groups/7805#7805 Comment by Gonçalo Marques Gonçalo Marques 2009-12-04T19:44:37Z 2009-12-04T19:44:37Z This is on page 31 of Peter Freyd's &quot;Abelian Categories&quot; book.