## Monthly Archives: December 2007

### 3704 summary

Although it’s still quite incomplete, I’ve made available on the course website a preliminary summary of the material you should review for the exams. I’m likely to modify and expand it, especially if I receive comments, so you should check back every now and then.

Happy Christmas and New Year!

MK

### 2201 Summary

Although it’s still quite incomplete, I’ve made available on the course website a preliminary summary of the material you should review for the exams. I’m likely to modify and expand it, especially if I receive comments, so you should check back every now and then.

Happy Christmas and New Year!

MK

### Non-commutative constructions

In June of next year, there will be a small workshop at UCL with the title

non-commutative constructions in arithmetic and geometry.’

I will write later in more detail about the goals of the meeting, but here is a link to a preliminary web page.

MK

### Mathematical form

During a lecture on linear algebra, I was giving a loose description of orthonormal bases in the space of vibrational modes of a metal string. It seemed noteworthy that the underlying vector space was infinite-dimensional, but was approximated by finite-dimensional spaces in practice, corresponding to ignoring high frequencies. That is to say, it is a prototype example where the thing being approximated is not a number or a function, but a *space*. At this point, I received the question Can you prove that?’ from a student at the very rear of the lecture hall. I was a bit taken aback, and did not manage to deal with the question in a suitable manner. Immediately afterwards, of course, I regretted not having looked more deeply into the perspective of the student. I do hope to discuss the issues surrounding that question at some point in a separate essay. It was somewhat on my mind when I wrote the short note on education. I was thinking of it also during the last day of classes when, in the course of a brief conversation with some students, I asserted that mathematics is not primarily about proofs.’ Needless to say, this was also an unsatisfactory statement to make on the fly, and ended up inviting the awkward question of what then it might be about. I didn’t answer, and will not attempt any real answer for quite a few more years. I still fear the sterility that might result from excessive rumination on the nature of mathematics, at least for a mediocre practitioner like myself. But I will pass along another Grothendieck quote:

If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither “number” nor “size”, but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.

Incidentally, when I direct your attention to a paragraph of this sort, the intention (to the extent that it matters) is not necessarily to express full sympathy with the sentiment therein.

MK

### Fundamental groups

Yesterday, at the last algebraic number theory lecture of the term, I defined the fundamental group \pi_1(O_K) of the ring of algebraic integers O_K in an algebraic number field K as the Galois group Gal(F/K), where F is the compositum of all the unramified extensions of K. And then, it was stated that the ideal class group Cl_K that we’ve been studying is just the abelianization of \pi_1(O_K). The point is that it seems reasonable now to start making computations of the whole \pi_1 rather than just the class group, if we want to capture finer invariants reflecting the complexity of a field. Of course, I’m not sure yet how to begin! But that’s what the best problems are like. Incidentally, the definition I gave isn’t quite right. One can give a better definition of \pi_1(O_k), closer to topological loops, after which the isomorphism with the Galois group becomes a theorem.

One of these days, I really will write some expository introduction at the undergraduate level. In the meanwhile the connection to topological fundamental groups is alluded to in the London-Paris lecture mentioned in an earlier post.

The basic reference in the field explaining this view is the great work Séminaire de Géométrie Algébrique, Vol. I by Alexandre Grothendieck and his students. Of course it’s hard to read if you’ve only now taken my course. But it’s good to get a sense of the classics even at this stage.

MK

### Isometries in dimension 4?

It occurs to me that I’ve never thought about higher-dimensional generalizations of the two theorems contained in the previous post, say even in dimension 4. I’m sure it’s well-known in some form, but by thinking about it, you might even find a nice new description! To be precise, we are asking for a natural description of isometries of determinant 1 on R^4. One can think immediately of some families, like the ones that leave a line fixed and then rotate inside the three-space orthogonal to it. But then, I guess such a transformation is actually leaving a whole *plane* fixed and rotates inside the orthogonal complement plane. One could then compose two such rotations along two different planes. Hmm. I have a feeling the answer is a version of some theorem in standard Lie theory, but I can’t quite see it….Even so, you shouldn’t hesitate to think it through using your own intuition.

MK

### Isometries in two and three dimensions

The 2201 course website for linear algebra now has a write-up of the last lecture. The note on self-adjoint maps has been corrected and somewhat revised.

MK

### Spec(Z) and three-manifolds

The following question comes from an old friend, John Baez. John is a renowned mathematical physicist and internet guru. You should look at his website and his blog for an enormously informative and entertaining survey of mathematics and physics.

MK

Hi! Why is Spec(Z) 3-dimensional?

I think I understand what the number theorist James wrote in response to “week257” of This Week’s Finds:

“Instead of considering Z, let’s consider F[x], where F is a finite field. They are both principal ideal domains with finite residue fields, and this makes them behave very similarly, even on a deep level. I’ll explain why F[x] is three-dimensional, and then by analogy we can hope Z is, too. Now F[x] is an F-algebra. In other words, X = Spec(F[x]) is a space mapping to S = Spec(F). I already explained why S is a circle from the point of view of the etale topology. So, if X is supposed to,be three-dimensional, the fibers of this map better be two-dimensional. What are the fibers of this map? Well, what are the points of S? A point in the etale topology is Spec of some field with a trivial absolute Galois group, or in other words, an algebraically closed field (even better, a separably closed one). Therefore a etale point of S is the same thing as Spec of an algebraic closure Fbar of F. What then is the fiber of X over this point? It’s Spec of the ring Fbar[x]. Now, *this* is just the affine line over an algebraically closed field, so we can figure out its cohomological dimension. The affine line over the complex numbers, another algebraically closed field, is a plane and therefore has cohomological dimension 2. Since etale cohomology is kind of the same as usual singular cohomology, the etale cohomological dimension of Spec(Fbar[x]) ought to be 2.

Therefore X looks like a 3-manifold fibered in 2-manifolds over Spec(F), which looks like a circle. Back to Spec(Z), we analogously expect it to look like a 3-manifold, but absent a (non-formal) theory of the field with one element, Z is not an algebra over anything. Therefore we expect Spec(Z) to be a 3-manifold, but not fibered over anything.”

But, it would be nice to go a bit deeper than mere “reasoning by analogy with Fbar[x]”, without yet diving into a full-blown calculation of the etale cohomology of Spec(Z). Can you say anything that’s helpful and fun? related question is this. Nowadays various people are making serious attempts to define the field with one element and do algebraic geometry over this field. With these, Z *is* an algebra over the field with one element. How much does this help?