User gonçalo marques - MathOverflow

Books you would like to read (if somebody would just write them...)

2011-01-24T10:23:39Z

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).

Let me start with some books I would like to read (again with self-explanatory titles)

1) The Weil conjectures for dummies

2) 2-categories for the working mathematician

3) Representations of groups: Linear and permutation representations made side by side

4) The Burnside ring

5) A functor of points approach to algebraic geometry

6) Profinite groups: An approach through examples

Any other suggestions ?

Answer by Gonçalo Marques for reference request: John Baez on (-1)- and (-2)-categories and properties+structure+stuff

2011-11-08T10:49:30Z

I think you mean the paper "Lectures on n-Categories and Cohomology". You can also look at the appropriate pages at the nLab.

Answer by Gonçalo Marques for What are some examples of colorful language in serious mathematics papers?

2010-09-17T13:32:45Z

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:

Satellite- Voir Cartan-Eilenberg et non Paris-Match.

Answer by Gonçalo Marques for Books you would like to see translated into English.

2010-03-19T00:52:22Z

Chebotarev's "Grundzüge der Galois'schen Theorie"

Answer by Gonçalo Marques for Equivalent singular chains and differential forms, as functionals on forms, on compact Riemannian manifolds

2010-03-15T06:20:59Z

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"

Answer by Gonçalo Marques for Linearity of the inner product using the parallelogram law

2009-12-04T21:27:28Z

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).

Answer by Gonçalo Marques for Differences between reflexives and projectives modules

2009-12-01T20:09:44Z

According to Lam's "Lectures on Modules and Rings" every f.g. projective module is reflexive. See page 55, exercise 7.

Answer by Gonçalo Marques for What is a projective space?

2009-12-01T19:36:52Z

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 "Continuous geometry". (see here). Another book to look might be Baer's classic "Linear algebra and projective geometry".

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 this. (But, as far as I know, these are all finite dimensional results so they might not be that interesting to Andrew...)

Answer by Gonçalo Marques for References/literature for pushouts in category of commutative monoids? [ed. - amalgams]

2009-12-01T15:12:38Z

Have you looked at the book Monoids, Acts and Categories by Mati Kilp, Ulrich Knauer and Alexander V. Mikhalev? Pushouts are discussed in chapter II.

Answer by Gonçalo Marques for Which math paper maximizes the ratio (importance)/(length)?

2009-12-01T12:49:16Z

The so called "Weil conjectures" are in the last pages of André Weil's short paper 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.

Answer by Gonçalo Marques for de Rham Cohomology of surfaces

2009-12-01T02:49:58Z

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.

Answer by Gonçalo Marques for determinant of a perfect complex

2009-11-29T10:53:37Z

You should take a look at the appendix A of "Discriminants, Resultants, and Multidimensional Determinants" by Gelfand, Kapranov and Zelevinsky

Comment by Gonçalo Marques

2011-08-25T23:39:55Z

That's a really hard book to find...

Comment by Gonçalo Marques

2011-03-06T15:02:42Z

There is also the book Emerton reffered in a comment above. It's called "Mathematical Developments Arising from Hilbert Problems" and it is published by the AMS

Comment by Gonçalo Marques

2011-01-24T15:07:39Z

Michael Spivak has recently written a book called "Physics for Mathematicians: Mechanics I". 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 "only" about classical mechanics...)

Comment by Gonçalo Marques

2011-01-24T14:42:24Z

I think Steve Awodey's "Category theory" might be just right for you.

Comment by Gonçalo Marques

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 "leitmotiv" 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...).

Comment by Gonçalo Marques

2011-01-24T12:28:47Z

Thanks Dylan. Tim: it would be great indeed if JB would publish his "higher algebra" 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.

Comment by Gonçalo Marques

2011-01-23T23:10:44Z

Another reference is "Algèbre et Théories galoisiennes" by Régine et Adrien Douady namely pag. 63-64

Comment by Gonçalo Marques

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.

Comment by Gonçalo Marques

2010-07-14T13:17:57Z

You should check Keith Conrad's handouts on the tensor product too. They are really good. math.uconn.edu/~kconrad/blurbs

Comment by Gonçalo Marques

2010-03-19T00:42:34Z

It would be really good to see all 3 volumes of Weber's "Lehrbuch der Algebra" translated.

Comment by Gonçalo Marques

2010-03-17T08:55:58Z

"a species is a functor from the category of finite sets into the category of finite sets". A small detail: it's the category of finite sets and bijections (so it's actually a groupoid).

Comment by Gonçalo Marques

2009-12-06T18:08:39Z

The book "Measure theory and probability" by Guillemin and Adams is also very good.

Comment by Gonçalo Marques

2009-12-06T17:22:53Z

You're right. Nice notes and written with great sense of humour.

Comment by Gonçalo Marques

2009-12-04T19:44:37Z

This is on page 31 of Peter Freyd's "Abelian Categories" book.