From the Noether boys to Bourbaki

Next year I’ll be teaching a master course on the “History of Mathematics” for the first time, so I’m brainstorming a bit on how to approach such a course and I would really appreciate your input.

Rather than giving a chronological historic account of some period, I’d like this course to be practice oriented and focus on questions such as

  • what are relevant questions for historians of mathematics to ask?
  • how do they go about answering these questions?
  • having answers, how do they communicate their finds to the general public?

To make this as concrete as possible I think it is best to concentrate on a specific period which is interesting both from a mathematical as well as an historic perspective. Such as the 1930’s with the decline of the Noether boys (pictures below) and the emergence of the Bourbaki group, illustrating the shift in mathematical influence from Germany to France.

(btw. the picture above is taken from a talk by Peter Roquette on Emmy Noether, available here)

There is plenty of excellent material available online, for students to explore in search for answers to their pet project-questions :

There’s a wealth of riddles left to solve about this period, ranging from the genuine over the anecdotic to the speculative. For example

  • Many of the first generation Bourbakis spend some time studying in Germany in the late 20ties early 30ties. To what extend did these experiences influence the creation and working of the Bourbaki group?
  • Now really, did Witt discover the Leech lattice or not?
  • What if fascism would not have broken up the Noether group, would this have led to a proof of the Riemann hypothesis by the Noether-Bourbakis (Witt, Teichmuller, Chevalley, Weil) in the early 40ties?

I hope students will come up with other interesting questions, do some excellent detective work and report on their results (for example in a blogpost or a YouTube clip).

Farey symbols in SAGE 5.0

The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$.

This Janko group has a 100-dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such that the fundamental domain $\mathbb{H}/G$ for the action of $G$ on the upper-half plane by Moebius transformations consists of 100 triangles in the Dedekind tessellation.

Four years ago i tried to depict this fundamental domain in the Farey symbols of sporadic groups-post using Chris Kurth’s kfarey package in Sage, but the result was rather disappointing.

Now, the kfarey-package has been greatly extended by Hartmut Monien of Bonn University and is integrated in the latest version of Sage, SAGE 5.0, released a few weeks ago.

Using the Farey symbol sage-documentation it is easy to repeat the calculations from four years ago and, this time, we do obtain this rather nice picture of the fundamental domain

But, there’s a lot more one can do with the new package. By combining the .fractions() with the .pairings() info it is now possible to get the corresponding Farey code which consists of 34 edges, starting off with

Perhaps surprisingly (?) $G$ turns out to be a genus zero modular subgroup. Naturally, i couldn’t resist drawing the fundamental domain for the 12-dimensional permutations representation of the Mathieu group $M_{12}$ and compare it with that of last time.

a hamlet with avg. 0.2 kB/sec internet-connection

Aaron Siegel on transfinite number hacking

One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication.

Some time ago we did run a couple of posts on this. In transfinite number hacking we recalled Cantor’s ordinal arithmetic and in Conway’s nim arithmetics we showed that Conway’s simplicity rules for addition and multiplication turns the set of all ordinal numbers into a field of characteristic zero : $\mathbb{On}_2$ (pronounced ‘Onto’).

In the post extending Lenstra’s list we gave Hendrik Lenstra’s effective construction of the mystery elements $\alpha_p$ (for prime numbers $p$) needed to do actual calculations in $\mathbb{On}_2$. We used SAGE to check the values for $p \leq 41$ and solved the conjecture left in Lenstra’s paper Nim multiplication that $(\omega^{\omega^{13}})^{43} = \omega^{\omega^7} + 1$ and determined $\alpha_p$ for $p \leq 67$.

Aaron Siegel has now dramatically extended this and calculated the $\alpha_p$ for all primes $p \leq 181$. He mails :

“thinking about the problem I figured it shouldn’t be too hard to write a dedicated program for it. So I threw together some Java code and… pushed the table up to p = 181! You can see the results below. Q(f(p)), excess, and alpha_p are all as defined by Lenstra. The “t(sec)” column is the number of seconds the calculation took, on my 3.4GHz iMac. The most difficult case, by far, was p = 167, which took about five days.

I’m including results for all p < 300, except for p = 191, 229, 263, and 283. p = 263 and 283 are omitted because they involve computations in truly enormous finite fields (exponent 102180 for p = 263, and 237820 for p = 283). I'm confident that if I let my computer grind away at them for long enough, we'd get an answer... but it would take several months of CPU time at least. p = 191 and 229 are more troubling cases. Consider p = 191: it's the first prime p such that p-1 has a factor with excess > 1. (190 = 2 x 5 x 19, and alpha_19 has excess 4.) This seems to have a significant effect on the excess of alpha_191. I’ve tried it for every excess up to m = 274, and for all powers of 2 up to m = 2^32. No luck.”

Aaron is writing a book on combinatorial game theory (to be published in the AMS GSM series, hopefully later this year) and will include details of these computations. For the impatient, here’s his list

Quiver Grassmannians and $\mathbb{F}_1$-geometry

Reineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects over the elusive field with one element $\mathbb{F}_1$.

In a comment to the previous post Markus Reineke recalls motivating discussions with Javier Lopez Pena and Oliver Lorscheid (the guys responsable for the map of $\mathbb{F}_1$-land above) and asks about potential connections with $\mathbb{F}_1$-geometry. In this post I will ellaborate on Javier’s response.

The Kapranov-Smirnov $\mathbb{F}_1$-floklore tells us that an $n$-dimensional vectorspace over $\mathbb{F}_1$ is a pointed set $V^{\bullet}$ consisting of $n+1$ points, the distinguished point playing the role of the zero-vector. Linear maps $V^{\bullet} \rightarrow W^{\bullet}$ between $\mathbb{F}_1$-spaces are then just maps of pointed sets (sending the distinguished element of $V^{\bullet}$ to that of $W^{\bullet}$). As an example, the base-change group $GL_n(\mathbb{F}_1)$ of an $n$-dimensional $\mathbb{F}_1$-space $V^{\bullet}$ is isomorphic to the symmetric group $S_n$.

This allows us to make sense of quiver-representations over $\mathbb{F}_1$. To each vertex we associate a pointed set and to each arrow a map of pointed sets between the vertex-pointed sets. The dimension-vector $\alpha$ of quiver-representation is defined as before and two representations with the same dimension-vector are isomorphic is they lie in the same orbit under the action of the product of the symmetric groups determined by the components of $\alpha$. All this (and a bit more) has been worked out by Matt Szczesny in the paper Representations of quivers over $\mathbb{F}_1$.

Oliver Lorscheid developed his own approach to $\mathbb{F}_1$ based on the notion of blueprints (see also part 2 and a paper with Javier).

Roughly speaking a blueprint $B = A // \mathcal{R}$ is a commutative monoid $A$ together with an equivalence relation $\mathcal{R}$ on the monoid semiring $\mathbb{N}[A]$ compatible with addition and multiplication. Any commutative ring $R$ is a blueprint by taking $A$ the multiplicative monoid of $R$ and $\mathcal{R}(\sum_i a_i,\sum_j b_j)$ if and only if the elements $\sum_i a_i$ and $\sum_j b_j$ in $R$ are equal.

One can extend the usual notions of prime ideals, Zariski topology and structure sheaf from commutative rings to blueprints and hence define a notion of “blue schemes” which are then taken to be the schemes over $\mathbb{F}_1$.

What’s the connection with Reineke’s result? Well, for quiver-representations $V$ defined over $\mathbb{F}_1$ they can show that the corresponding quiver Grassmannians $Gr(V,\alpha)$ are blue projective varieties and hence are geometric objects defined over $\mathbb{F}_1$.

For us, old-fashioned representation theorists, a complex quiver-representation $V$ is defined over $\mathbb{F}_1$ if and only if there is an isomorphic representation $V’$ with the property that all its arrow-matrices have at most one $1$ in every column, and zeroes elsewhere.

Remember from last time that Reineke’s representation consisted of two parts : the Veronese-part encoding the $d$-uple embedding $\mathbb{P}^n \rightarrow \mathbb{P}^M$ and a linear part describing the subvariety $X \rightarrow \mathbb{P}^n$ as the intersection of the image of $\mathbb{P}^n$ in $\mathbb{P}^M$ with a finite number of hyper-planes in $\mathbb{P}^M$.

We have seen that the Veronese-part is always defined over $\mathbb{F}_1$, compatible with the fact that all approaches to $\mathbb{F}_1$-geometry allow for projective spaces and $d$-uple embeddings. The linear part does not have to be defined over $\mathbb{F}_1$ in general, but we can look at the varieties we get when we force the linear-part matrices to be of the correct form.

For example, by modifying the map $h$ of last time to $h=x_0+x_7+x_9$ we get that the quiver-representation

is defined over $\mathbb{F}_1$ and hence that Reineke’s associated quiver Grassmannian, which is the smooth plane elliptic curve $\mathbb{V}(x^3+y^2z+z^3)$, is a blue variety. This in sharp contrast with other approaches to $\mathbb{F}_1$-geometry which do not allow elliptic curves!

Oliver will give a talk at the 6th European Congress of Mathematics in the mini-symposium Absolute Arithmetic and $\mathbb{F}_1$-Geometry. Judging from his abstract,he will also mention quiver Grassmannians.