Goemetric Algebra Vs Algebraic Geometry.

[MathematicalPhysicist]

is there any difference between the two discplines, and what is it?

p.s
what do you think of emil artin's classical book on geometric algebra, is it still appropiate for learning the subject, and also go into the advanced stuff of this subject?

thanks in advance.

 

[Feldoh]

http://en.wikipedia.org/wiki/Geometric_algebra vs. http://en.wikipedia.org/wiki/Algebraic_geometry

 

[mathwonk]

yes artins book is wonderful for learning the subject of geometric algebra, but i have no idea what connection it has to algebraic geometry.

by the way, anytime anyone asks whether it s worthwhile to read a book by a great mathematician, the answer is always YES, YES, YES!

get a clue.

 

[MathematicalPhysicist]

and what about a book for algebraic geometry, any recommendations?

 

[MathematicalPhysicist]

but does artin's book cover also advanced topics in goemetric algebra?
I gave a look at the contents, and it seems that it doesn't cover a lot of material, perhaps the subject doesn't have a lot of material, don't know.

 

[mathman]

and what about a book for algebraic geometry, any recommendations?

There is a classic, entitled "Algebraic Geometry", by Andre Weil. Since this is not my field, I couldn't say if there are good more recent books.

 

[mathwonk]

read mumfords red book. or shafarevich's book BAG. or reid's undergrad alg geom.

 

[MathematicalPhysicist]

waht are the prequisites before reading the books, what one needs to know beforehand.
i gues undergrad calculus 1,2 and linear algebra 1,2 are mandatory (which i have), but will it suffice?

 

[Chris Hillman]

Classical groups in algebraic geometry

is there any difference between the two discplines, and what is it?

Since Artin's day, "geometric algebra" has been employed to refer to a rather complicated notation related to Clifford algebras which has been much touted by a small group of physicists in the UK. It seems fair to say that this notation hasn't really caught on with physicists elsewhere. This is (at time of writing this post) the principle subject of the Wikipedia article someone cited above, http://en.wikipedia.org/w/index.php?title=Geometric_algebra&oldid=147444736

However, Artin used this term in the title of his book to mean something pretty close to what Weyl called (in the title of his own book) "the classical groups" (orthogonal, symplectic, unitary, general and special linear (and indeed, Artin's book does touch on Clifford algebras in his discussion of the spin groups). I would characterize the overall thrust of Artin's book as a modest foray into Kleinian geometry--- he is concerned with the interplay of algebra and geometry, in the important special case of those geometries (projective, euclidean, symplectic) for which the classical groups serve as "geometrical symmetry groups" a la Klein. Unfortunately the natural sequel (the "dream team" coauthors would have been Artin, Noether, Hilbert) explicitly expositing Klein's ideas in the context of invariant theory never appeared.

"Algebraic geometry" originates in the study of curves and surfaces (originally only studied in two and three dimensional settings) defined by systems of simultaneous polynomial equations. In this sense, algebraic geometry is a natural generalization of linear algebra from first order polynomials to higher degree polynomials. Also in this sense, algebraic geometry concerns two types of closely related Kleinian geometries, affine and projective, especially over the the complex field. Once we get bored with spaces of points and planes (Grassmannians), we can form spaces of curves of a given degree; one of the characteristic features of algebraic geometry is that in a sense this gives nothing new! The urge to generalize to finite fields is irresistable, and this does result in something new, finite affine and projective geometries. By adding extra structure (in either algebraic or geometric guise, for example, a quadratic form) we obtain the other geometries considered by Artin.

As recently as 1965, the field has been reinvigorated with the discovery by Buchberger of Groebner bases, for which see the modern classic by Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer, 1992, which is IMO one of the best books yet published. The introduction of Groebner bases made it practical to compute with ideals, and fostered the reinvigoration of the study of such classical topics as resultants. The methods discussed in IVA (and more besides) are implemented in many symbolic computational systems, and every serious student of mathematics must learn and use these techniques, since such mundane tasks as elimination of variables can be difficult or at least tedious without sophisticated algorithmic aids. Not to mention counting (e.g. configurations in finite geometries), computing homological quantities, etc., all of which are facilitated by these methods.

To better understand the relation between "geometric algebra" in the sense of Artin and "algebraic geometry", I think the best textbook is Joe Harris, Algebraic Geometry, Springer, 1982. This book makes a concerted effort to explain the role played by some of the classical groups in algebraic geometry, particularly as regards complex projective geometry, the setting for "classical" algebraic geometry.

I also really like the little book by C. G. Gibson, Elementary Geometry of Algebraic Curves, Cambridge University Press, 1998, as a further antidote to the highly abstract ways in which algebraic geometry developed in the last century. (Not that there are not good reasons for that, but I think it's essential to obtain a solid grounding in the concrete inspiration for these abstract fancies.) This book should be helpful in understanding the relation between the affine and projective planes. You asked about prerequisites--- this book at least has only modest prerequisites, and I think the textbook by Harris cited above is one of the most readable algebraic geometry books aimed at beginning graduate students; it should be quite accessible to ambitious undergraduates.

what do you think of emil artin's classical book on geometric algebra, is it still appropiate for learning the subject, and also go into the advanced stuff of this subject?

I'm with mathwonk--- Artin's book is worth reading. I'll go out on a limb and guess that you might sometimes read John Baez's "This Weeks Finds", if so, he has stated that in future weeks he intends to exposit Kleinian geometry, at the intersection of the classical groups (c.f. Artin's book) with his "groupoidification" and "n-categorification" programs, which might turn out to be important for mathematics, and (thus?) perhaps for physics.

I have urged JB to set up a "closed edit" wiki for his study group (to be called, of course, Weekipedia), since at present there is, amazingly enough, no book which makes a credible attempt even to explain the Kleinian geometry of Klein's time, much less of our own, so his group will have to write one. Or so I argue In the mean time, see http://golem.ph.utexas.edu/category/ but only if you have MathML. The above books should however assist serious students in figuring this out on their own--- this is more challenging, but then one learns more that way!

 

[mathwonk]

one should also read les miserables, by victor hugo.

 

[Emmanuel114]

one should also read les miserables, by victor hugo.

Seconded!