Thanks for clicking through… I guess.

If nothing else, it shows that just as much as the stock market is fueled by greed, mathematical reasearch is driven by frustration (or the pleasure gained from knowing others to be frustrated).

I did spend the better part of the day doing a lengthy, if not laborious, calculation, I’ve been postponing for several years now. Partly, because I didn’t know how to start performing it (though the basic strategy was clear), partly, because I knew beforehand the final answer would probably offer me no further insight.

Still, it gives the final answer to a problem that may be of interest to anyone vaguely interested in Moonshine :

What does the Monster see of the modular group?

I know at least two of you, occasionally reading this blog, understand what I was trying to do and may now wonder how to repeat the straightforward calculation. Well the simple answer is : Google for the number 97239461142009186000 and, no doubt, you will be able to do the computation overnight.

One word of advice : don’t! Get some sleep instead, or make love to your partner, because all you’ll get is a quiver on nine vertices (which is pretty good for the Monster) but having an horrible amount of loops and arrows…

If someone wants the details on all of this, just ask. But, if you really want to get me exited : find a moonshine reason for one of the following two numbers :

(the dimension of the monster-singularity upto smooth equivalence), or,

(the dimension of the moduli space).

Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.

(more…)

The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least one eigenvector. Take a vector , then as the collection of vectors must be linearly dependent, there are complex numbers such that But then as is algebraically closed the polynomial on the left factors into linear factors and therefore as from which it follows that at least one of the linear transformations has a non-trivial kernel, whence A has an eigenvector with eigenvalue . Okay, fine, nice even, but does this simple minded observation warrant the extreme conclusion of his paper (on page 18) ?

As mathematicians, we often read a nice new proof of a known theorem, enjoy the different approach, but continue to derive our internal understanding from the method we originally learned. This paper aims to change drastically the way mathematicians think about and teach crucial aspects of linear algebra.
The simple proof of the existence of eigenvalues given in Theorem 2.1 should be the one imprinted in our minds, written on our blackboards, and published in our textbooks. Generalized eigenvectors should become a central tool for the understanding of linear operators. As we have seen, their use leads to natural definitions of multiplicity and the characteristic polynomial. Every mathematician and every linear algebra student should at least remember that the generalized eigenvectors of an operator always span the domain (Proposition 3.4)—this crucial result leads to easy proofs of upper-triangular form (Theorem 6.2) and the Spectral Theorem (Theorems 7.5 and 8.3).
Determinants appear in many proofs not discussed here. If you scrutinize such proofs, you’ll often discover better alternatives without determinants. Down with Determinants!

I welcome all new proofs of known results as they allow instructors to choose the one best suited to their students (and preferable giving more than one proof showing that there is no such thing as ‘the best way’ to prove a mathematical result). What worries me is Axler’s attitude shared by extremists and dogmatics world-wide : they are so blinded by their own right that they impoverish their own lifes (and if they had their way, also that of others) by not willing to consider other alternatives. A few other comments : 1. I would be far more impressed if he had given a short argument for the one line he skates over in his proof, that of being algebraically closed. Does anyone give a proof of this fact anymore or is this one of the few facts we expect first year students to accept on faith? 2. I dont understand this aversity to the determinant (probably because of its nonlinear character) but at the same time not having any problems with successive powers of matrices. Surely he knows that the determinant is a fixed -polynomial in the traces (which are linear!) of powers of the matrix. 3. The essense of linear algebra is that by choosing a basis cleverly one can express a linear operator in a extremely nice matrix form (a canonical form) so that all computations become much more easy. This crucial idea of considering different bases and their basechange seems to be missing from Axler’s approach. Moreover, I would have thought that everyone would know these days that ‘linear algebra done right’ is a well developed topic called ‘representation theory of quivers’ but I realize this might be viewed as a dogmatic statement. Fortunately someone else is giving the basic linear algebra courses here in Antwerp so students are spared my private obsessions (at least the first few years…). In [his post](http://golem.ph.utexas.edu/category/2007/05/ linearalgebradoneright.html) Leistner askes “What are determinants good for?” I cannot resist mentioning a trivial observation I made last week when thinking once again about THE rationality problem and which may be well known to others. Recall from the previous post that rationality of the quotient variety of matrix-couples under simultaneous conjugation is a very hard problem. On the other hand, the ‘near miss’ problem of the quotient variety of matrix-couples is completely trivial. It is rational for all n. Here is a one-line proof. Consider the quiver then the dimension vector (n-1,n) is a Schur root and the first fundamental theorem of (see for example Hanspeter Krafts excellent book on invariant theory) asserts that the corresponding quotient variety is the one above. The result then follows from Aidan Schofield’s paper Birational classification of moduli spaces of representations of quivers. Btw. in this special case one does not have to use the full force of Aidan’s result. Zinovy Reichstein, who keeps me updated on events in Atlanta, emailed the following elegant short proof Here is an outline of a geometric proof. Let and . Applying the no-name lemma to the -equivariant dominant rational map given by (which makes X into a vector bundle over a dense open -invariant subset of Y), we see that is rational over On the other hand, is an affine space. Thus is rational. The moment I read this I knew how to do this quiver-wise and that it is just another Brauer-Severi type argument so completely inadequate to help settling the genuine matrix-problem. Update on the paper by Esther Beneish : Esther did submit the paper in february.

This morning, Esther Beneish arxived the paper The center of the generic algebra of degree p that may contain the most significant advance in my favourite problem for over 15 years! In it she claims to prove that the center of the generic division algebra of degree p is stably rational for all prime values p. Let me begin by briefly explaining what the problem is all about. Consider one n by n matrix A which is sufficiently general, then it will have all its eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely determined upto conjugation (that is, base change) by its characteristic polynomial. In other words, the conjugacy class of a sufficiently general n by n matrix depends freely on the coefficients of the characteristic polynomial (which are the n elementary symmetric functions in the eigenvalues of the matrix). Now what about couples of n by n matrices (A,B) under simultaneous conjugation (that is all couples of the form for some invertible n by n matrix g) ??? So, does there exist a sort of Jordan normal form for couples of n by n matrices which are sufficiently general? That is, are there a set of invariants for such couples which determine it is freely upto simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old result due to James Sylvester saying that this is indeed the case and that the set of invariants consists of the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio did a lot more in his paper. He showed that if you could prove this for couples of matrices, you can also do it for triples, quadruples even any k-tuples of n by n matrices under simultaneous conjugation. He also related this problem to the center of the generic division algebra of degree n (which was introduced earlier by Shimshon Amitsur in a rather cryptic manner and for a while he simply refused to believe Claudio’s description of this division algebra as the one generated by two generic n by n matrices, that is matrices filled with independent variables). Claudio also gave the description of the center of this algebra as a field of lattice-invariants (over the symmetric group S(n) ) which was crucial in subsequent investigations. If you are interested in the history of this problem, its connections with Brauer group problems and invariant theory and a short description of the tricks used in proving the results I’ll mention below, you might have a look at the talk Centers of Generic Division Algebras, the rationality problem 1965-1990 I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally settled in 1979 by Ed Formanek and a year later he was able to solve also the case of couples of 4 by 4 matrices in a fabulous paper. In it, he used solvability of S(4) in an essential way thereby hinting at the possibility that the problem might no longer have an affirmative answer for larger values of n. When I read his 4×4 paper I believed that someone able to prove such a result must have an awesome insight in the inner workings of matrices and decided to dedicate myself to this problem the moment I would get a permanent job… . But even then it is a reckless thing to do. Spending all of your time to such a difficult problem can be frustrating as there is no guarantee you’ll ever write a paper. Sure, you can find translations of the problem and as all good problems it will have connections with other subjects such as moduli spaces of vectorbundles and of quiver representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with Christine Bessenrodt we were able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of 7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if you could do it for couples of k by k and l by l matrices and if k and l were coprime then you could also do it for couples of kl by kl matrices, so the n=6 case was already done). Or did we? Well not quite, our methods only allowed us to prove that the center is stably rational that is, it becomes rational by freely adjoining extra variables. There are examples known of stably rational fields which are NOT rational, but I guess most experts believe that in the case of matrix-invariants stable rationality will imply rationality. After this paper both Christine and myself decided to do other things as we believed we had reached the limits of what the lattice-method could do and we thought a new idea was required to go further. If today’s paper by Esther turns out to be correct, we were wrong. The next couple of days/weeks I’ll have a go at her paper but as my lattice-tricks are pretty rusty this may take longer than expected. Still, I see that in a couple of weeks there will be a meeting in Atlanta were Esther and all experts in the field will be present (among them David Saltman and Jean-Louis Colliot-Thelene) so we will know one way or the other pretty soon. I sincerely hope Esther’s proof will stand the test as she was the only one courageous enough to devote herself entirely to the problem, regardless of slow progress.

Here are my nominees for the 2006 paper of the year award in mathematics & mathematical physics : in math.RA : math.RA/0606241 : Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I by Maxim Kontsevich and Yan Soibelman. Here is the abstract :

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an “abstract” Calabi-Yau manifold).

why ? : Because this paper probably gives the correct geometric object associated to a non-commutative algebra (a huge coalgebra) and consequently the right definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve tried to explain how this links up with my own interpretation and since then I’ve thought more about this, but that will have to wait for another time. in hep-th : hep-th/0611082 : Children’s Drawings From Seiberg-Witten Curves by Sujay K. Ashok, Freddy Cachazo, Eleonora Dell’Aquila. Here is the abstract :

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called “dessins d’enfants” or “children’s drawings” on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck’s programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.

why ? : Because this paper gives the best introduction I’ve seen to Grothendieck’s dessins d’enfants (slightly overdoing it by giving a crash course on elementary Galois theory in appendix A) and kept me thinking about dessins and their Galois invariants ever since (again, I’ll come back to this later).