The latest issue of the Notices of the AMS contains the first part of a long biographical article about Grothendieck written by Allyn Jackson. Evidently Winfried Scharlau is writing a biography of Grothendieck, and Jackson’s article is partially based on materials he has gathered. Much of this material is brought together at a website maintained by the “Grothendieck Circle”.

This issue of the Notices also contains a short expository piece on one of the most abstract ideas due to Grothendieck, that of a “topos”. Illusie was a student of Grothendieck’s, and Jackson’s article has some of his reminiscences about what that experience was like. Illusie’s piece is not very accessible; a better place to try and get some feeling for these ideas is Pierre Cartier’s Bulletin article.

Regarding books on algebraic geometry, I’d like to point out two sources not mentioned above.

a) EGA – This is Grothendieck’s monumental work and probably not what most people are looking for since it is extremely detailed, formal and dry. But if you prefer abstract nonsense, who knows? This might just be your thing since it is pretty easy to read.

b) Algebraic Geometry and Arithmetic Curves – This relatively new Oxford University Press title does a fine job of explaining schemes in the first 7 chapters before delving into details about arithmetic curves. Complexity wise, this comes somwhere between Hartshorne and EGA (EGA being way too detailed, Hartshorne being way too dense). This will also be more useful if you’re a number theorist since Liu, unlike Hartshorne, does not shy away from char. p and, in fact, devotes many sections/examples to it.

BTW, I asked about G. last time I was in Paris and I was told he left France. The last published notice was about him being hiddeng at some place in the Pyrinees. Any gossip about?

Levi, Dan

Years ago I went through the 2nd volume of Shaferevich’s “basic algebraic geometry” books, which covered algebraic geometry from the complex manifold perspective like Griffiths & Harris’ book. At the time I ended up glossing over the chapters on schemes, largely out of laziness.

For the entire time I was in college (both undergrad and grad school), I never really appreciated the notion of abstract math which didn’t involve doing heavy computations. When I took math courses like abstract algebra and real analysis, I was always thinking to myself “what’s the point of this junk?!?!?!” Even doing particle and/or string theory research, most of it was largely doing computations and not really much hardcore abstract math.

It was after grad school when I started to look at abstract math again like algebraic geometry, commutative algebra, number theory, category theory, etc … and started to appreciate it a lot more. I guess it came with some mathematical maturity and a change in perspective, that I started to appreciate abstract math more. It took me awhile to become adjusted from the heavy computation mentality to a mindset of proving general theorems and seeing the “beauty” in it. I even went back to some old books on general topology and functional analysis and started to see them in a new light, in comparison to when I previously thought it was all a “waste of time” in the past.

I guess the only way I can describe “beauty” in abstract math, would be similar how an artist or musician sees “beauty” in a particlar piece of art or music. To a person who has no interest in math, art, or music, they would not see much “beauty” in it and wonder “what’s the point” of it all.

I find the biggest barrier to *any* math learning in a modern context is that *all* of it seems dry and lifeless as presented – of course it is anything but, only the formalist methods of presentation just make it seem so IMO. I wish there were a modern Klein or Weyl running around. This seems to be a niche that someone should fill.

I was told by a former Princeton math PhD that oral tradition is now essential in math, and that learning from textbooks without some personal mentor is very hard. In Klein’s day, his “Vorlesungen” were taken directly from his personal style, so the presentation and the results were mixed in a way that made for exciting reading. We need something like that again IMO. Shlomo Sternberg comes close to filling the bill. Harris tries, but assumes too much of readers with my level of patience and skill.

I always found the biggest barrier to understanding scheme theory and/or the more abstract formalism of algebraic geometry, was a solid understanding of commutative algebra. Years ago I worked through Atiyah’s book on commutative algebra, which I thought was kind of dry and boring at the time.

Awhile ago I went through Diers’ book “categories of commutative algebras” which seems to make the subject more interesting and less dry, though I don’t know how much more useful it is for understanding algebraic geometry.

There is a smaller book called “Algebraic Geometry – A First Course” by Harris alone that is good, and is accessible to physics types (i.e. me). At one time I had the preposterous idea of mastering scheme theory.

-drl

JC–Some friends and I are currently reading “Basic Algebraic Geometry” by Shaferevich. It may not be what you are looking for, though, since it makes a concerted effort to be no more abstract than is absolutely necessary.

First off, I appreciate the information in your article on Grothendieck. I am deeply interested in how different minds came to discover the foundational values of mathematics, albeit from the sidelines, of your perspectives and collegues, Peter.

As a general reader of Smolin’s, Three Roads to Quantum Gravity I saw this same road taken, and assumed the essence of his distilliation, was topos theory.

http://math.ucr.edu/home/baez/topos.html

For more information I added this to my main page because of this set relationship I found being demonstrated in the developement of LQG perspectives in terms of quantum gravity.

At a fundamental level of this association was quantified in terms of quantum grvaity, then I saw computerization deeply valued in relation to discriptors of that early uiverse.

Like gamma ray detection, we saw the deeper fundamental realites emerge on the Windows of the universe. We might have to wait, to go deeper then this?

I found it difficult to find some comparison on how such a view could have been exemplified, so by studing the Glast perspective I found a general relevance, that would have supported LQG.

But did this go far enough? I see the limitations now, in relation to early universe perspective?

The universe had to be smooth at its most earliest time?

The “easiest” books on algebraic geometry I’ve come across over the years would be “ideals, varieties, and algorithms” by Cox, Little, & O’Shea, and perhaps the two undergradate books by Reid on algebraic geometry and commutative algebra. None of them seem to really go into the more abstract formalism of schemes, advocated by Grothendieck.

I’ll take a look at Eisenbud & Harris’ book on the geometry of schemes.

JC: Eisenbud, Harris: “The Geometry of Schemes” seems to be a more accessible introduction to the language, with examples and motivation behind the ideas. Relatively cheap, too…

Peter,

How common is it for physics folks to change into math? What’s the most popular area of math that physics folks like to change into?

In recent years, the cases of physics folks changing into math that I’ve heard of were frequently some string theory folks who changed into some algebraic geometry areas related to Calabi-Yau manifolds (ie. the sort of stuff in the language of Griffiths and Harris’s “principles of algebraic geometry” book). I haven’t personally come across many physics folks who changed into the more abstract algebraic geometry stuff (ie. the more abstract Grothendieck style, such as in Hartshorne’s book). Though on the surface, I can understand why string folks would prefer the “Griffiths & Harris” language of algebraic geometry. Last time I tried to tackle Hartshorne’s book, I still found it a difficult read.

Anybody know of an easier book than Hartshorne, which covers the more abstract Grothendieck style of algebraic geometry?

I just fixed that link, should work now.

The link to the expository piece on topoi is empty.