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Introduction to Algebraic Geometry (3 in a series)

March 4th, 2008

I’m going to move on with the identification of geometric objects with functions from these objects down to a field soon enough, but I’d like to spend a little time nailing down the categorical language of this association. Basically, we have two functors I and V going back and forth between two categories. And the essential statement of the last post is that these two functors form an equivalence of categories.

Now, first off in this categorical language, I want to nail down exactly what the objects are. In the category $\mathcal{AV}ar_k$ the objects are solution sets of systems of polynomial equations. And in the category $\mathcal{RA}lg_k$ , the objects are finitely presented Noetherian reduced k-algebras.

The functor $V:\mathcal{RA}lg_k\to \mathcal{AV}ar_k$ acts on objects by sending an algebra R to the solution set of the polynomial equations generating the ideal in a presentation of the algebra.

This is a preview of Introduction to Algebraic Geometry (3 in a series). Read the full post (723 words, 49 images)

Introduction to Algebraic Geometry (2 in a series)

February 21st, 2008

I want to lead this sequence to the point where I am having trouble understanding algebraic geometry. Hence, I won’t take the usual course such an introduction would take, but rather set the stage reasonably quickly to make the transit to the more abstract themes clear.

But that’s all a few posts away. For now, recall that we recognized already that any variety is defined by an ideal, and that intersections and unions of varieties are given by sums and intersections or products of ideals.

This is the first page of what is known as the Algebra-Geometry dictionary. The dictionary is made complete by a pair of reasonably famous theorems. I won’t bother proving them - the proofs are a good chunk of any decent commutative algebra course - but I’ll quote the theorems and discuss why they matter.

This is a preview of Introduction to Algebraic Geometry (2 in a series). Read the full post (565 words, 11 images)

Introduction to Algebraic Geometry (1 in a series)

February 21st, 2008

I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.

So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.

Varieties

The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set $f_1,\dots,f_r$ of polynomials in some polynomial ring $k[x_1,\dots,x_n]$ over some field . And we write $V(f_1,\dots,f_r)$ for the set of all simultaneous roots to all these polynomials:
$V(f_1,\dots,f_r)=\{p\in k^n:f_1(p)=0, \dots, f_r(p)=0\}$

This is a preview of Introduction to Algebraic Geometry (1 in a series). Read the full post (1162 words, 58 images)

Algebraic surface toys!

January 25th, 2008

At the start of the German Year of Mathematics, the Oberwolfach research institute has released an exhibition and the software they used to produce it. The software, surfer, is a really nice GUI that sits on top of surf and lets you rotate and zoom your algebraic surfaces as well as pick colours very comfortably.

They have a whole bunch of Really Pretty Images at the exhibition website, and I warmly recommend a visit. If you can get hold of the exhibition, they also have produced real models - with a 3d-printer - of some of the snazzier surfaces, so that one could have a REALLY close encounter with them.

But also, I’d really like to show you some of my own minor experiments with the program.

Tuba Mirum - the innards of a Klein Bottle
This is the interior of a Klein Bottle, using the “standard” realization as an algebraic surface given by Mathworld. In other words, I’m using
(x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1)=0
for the defining equation. It kinda looks a bit like a Sousaphone in my opinion.

This is a preview of Algebraic surface toys!. Read the full post (300 words, 3 images)

The why and the what of homological algebra

July 12th, 2007

I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.

Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.

And I received a letter from the Haskellite crowd - more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.

I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess - especially with the carnival dry-out I’ve been having - that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.

So here goes.

This is a preview of The why and the what of homological algebra. Read the full post (2290 words, 2 images)

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about: Michi is a PhD student working in homological algebra. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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