## Commutators and the Ore Conjecture

In a talk yesterday by Boris Kunyavski at the University of Ottawa, I learned a little about the Ore conjecture, which in 2010 was proved a theorem in:

It's quite a fascinating result that arises by considering commutators in groups. If $G$ is a group, its commutator subgroup $[G,G]$ is the subgroup of $G$ generated by all the commutators $[g,h] = ghg^{-1}h^{-1}$ of $G$. It's easy to see that the commutator subgroup is normal. A group $G$ is said to be **perfect** if $G = [G,G]$.

So let's assume $G$ is perfect. This implies that every element of $G$ can be written as a product of commutators. But can every element of $G$ be written as a single commutator? That's really far from obvious. For example, take your favourite perfect group and an element in it: can you prove that this single element is a commutator? Not so easy, right?

In fact, we can define the commutator length of any $g\in G$ to be the minimum number of commutators in all products of commutators equal to $g$. If $g$ can't be written as the product of commutators, then its commutator length is infinite.

The **commutator width** of a group is defined to be the supremum over commutator lengths of all the elements of $G$. (Note: I think this should just be called the commutator length of $G$ as well, but that's how the terminology ended up!)

It turns out that finding a perfect group $G$ with commutator width greater than one is quite tricky. In fact, the theorem proved in *loc. cit.* is:

**Former Ore Conjecture/Now Theorem**. If $G$ is a finite nonabelian simple group, then every element of $G$ is a commutator.

That's pretty cool, though the proof is very long. That's not surprising since it is a theorem about all finite nonabelian simple groups. What's perhaps even more surprising is that there are examples of finitely-generated *infinite* simple groups containing elements that are *not* commutators. In fact, examples exist of such $G$ with infinite commutator length, as given in Alexey Muranov's paper:

This result makes use of small cancellation theory, which is a geometric group theory machinery that studies presented groups whose relations don't have too much in common.

## What is a residually finite group?

We say that a group $G$ is **residually finite** if for each $g\in G$ that is not equal to the identity of $G$, there exists a finite group $F$ and a group homomorphism

$$\varphi:G\to F$$ such that $\varphi(g)$ is not the identity of $F$.

The definition does not change if we require that $\varphi$ be surjective. Therefore, a group $G$ is residually finite if and only if for each $g\in G$ that is not the identity, there exists a finite index normal subgroup $N$ of $G$ such that $g\not\in N$.

Hence, if $G$ is residually finite, then the intersection of all finite-index normal subgroups is trivial. The converse holds, too (why?).

…read the rest of this post!

## Links to Atiyah's preprints on the Riemann hypothesis

Sir Michael Atiyah's preprints are now on the internet:

The meat of the claimed proof of the Riemann hypothesis is in Atiyah's construction of the Todd map $T:\C\to \C$. It supposedly comes from the composition of two different isomorphisms

$$\C\xrightarrow{t_+} C(A)\xrightarrow{t^{-1}_{-}} \C$$ of the complex field $\C$ with the $C(A)$, the center of a von Neumann hyperfinite factor $A$ of type II-1. Understanding Atiyah's work boils down to understanding this Todd map, and therefore in understanding what is in the paper "The Fine Structure Constant".

Assuming there is a zero $b$ of the Riemann zeta function $\zeta(s)$ off the critical line, Atiyah defines a function

$$F(s) = T(1 +\zeta(s + b)) – 1.$$ The function $F$ satisfies $F(0) = 0$ because one of the properties of the Todd map $T$ is that $T(1) = 1$, and $F$ is also supposedly analytic. According to some of the basic properties of the Todd function which is a polynomial on a closed rectangle containing this zero, this would imply that $F(s) = 0$ and therefore that $\zeta$ is identically zero, which is the contradiction.

Now, I have very little understanding of von Neumann algebras so I won't comment at all on the Todd map. I have no doubt that the experts will dissect this because there's so much attention on it. Even assuming all the properties of the Todd function, I find the proof difficult to follow. For example, the assumed zero $b$ off the critical strip: I can't find where "$b$ is off the critical strip" is even being used. In fact, it's hard to see where *any* of the basic properties of the zeta function are being used.

## For real? Atiyah's proof of the Riemann hypothesis

Well this is strange indeed: according to this New Scientist article published today, the famous Sir Michael Atiyah is supposed to talk this Monday at the Heidelberg Laureate Forum. The topic: a proof of the Riemann hypothesis. The Riemann hypothesis states that the Riemann Zeta function defined by the analytic continuation of $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ has nontrivial zeros only on the critical line whose numbers have real part $1/2$. Check out this MathWorld article for more details.

The Riemann hypothesis is considered by many to be *the* outstanding problem in mathematics. Many people have tried to prove it and failed.

Is this for real?

## Some 2018 Springer Math Texts

When I was a student at McGill I loved looking at the latest Springer texts in the now-nonexistant Rosenthall library. So, I thought that I'd list some of the cool looking titles that have come out in 2018:

- Walter Dittrick, Reassessing Riemann's Paper: This book is an analysis of Riemann's paper "On the Number of Primes Less Than a Given Magnitude", and could be a great historical starting point into the subject
- Andreas Hinz, Sandi Klavžar, and Ciril Petr, The Tower of Hanoi – Myths and Maths: Looks like a fun recreational math book about the Tower of Hanoi game
- Wojciech Chachólski, Tobias Dyckerhoff, John Greenlees, Greg Stevenson, Building Bridges Between Algebra and Topology: It contains notes from four different mini-lectures on Hall algebras, triangulated categories, homotopy invariant commutative algebra, and idempotent symmetries!
- Karin Erdmann, Thorsten Holm, Algebras and Representation Theory: Nice-looking text on representation theory. Has classical things like the Artin-Wedderburn theorem and more modern topics like quivers
- V. Lakshmibai, Justin Brown, Flag Varieties: An introduction to flag varieties, assuming some background in commutative algebra and algebraic geometry
- Nicolas Privault, Understanding Markov Chains
- Masao Jinzenji, Classical Mirror Symmetry
- Berthé, Michel Rigo (editors), Sequences, Groups, and Number Theory: Now this looks interesting! It is a curious volume of lectures on the interactions between words (as in formal languages and presented groups), number theory, and dynamical systems
- Ibrahim Assem, Sonia Trepode (editors), Homological Methods, Representation Theory, and Cluster Algebras: A series of lectures from CRM minicourses

*N.B. I do not work for Springer and I do not get compensated for posting these links. I just think they look cool.

## For (most) PIDs: Trace zero matrices are commutators

Let $R$ be a commutative ring and $M_n(R)$ denote the ring of $n\times n$ matrices with coefficients in $R$. For $X,Y\in M_n(R)$, their commutator $[X,Y]$ is defined by

$$[X,Y] := XY – YX.$$ The trace of any matrix is defined as the sum of its diagonal entries.

If $X$ and $Y$ are any matrices, what is the trace of $[X,Y]$? It's zero! That's because the trace of $XY$ is the same as the trace of $YX$. Therefore:

What about the converse? Is any trace zero matrix also a commutator? In other words, given a trace zero matrix $Z\in M_n(R)$, can we find matrices $X$ and $Y$ such that $Z = [X,Y]$? Albert and Muckenhoupt proved that you can, assuming that $R$ is a field.

What happens if you also want $X$ and $Y$ to have trace zero?

Good question. In general, this is not possible. For example, let's consider the simplest field of all, the field with two elements denoted by $\F_2$. Okay, it's actually debatable whether $\F_2$ really is the simplest field, because so many problems happen in characteristic two. For example, this problem we've been considering: in $\F_2$, if $X$ and $Y$ are $2\times 2$ matrices of trace zero, then $[X,Y]$ will have zero off-diagonal entries. So for example, the matrix

$$\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix}\in M_2(\F_2)$$ cannot be written as a commutator of two matrices with trace zero.

It seems that characteristic two is the only obstruction, in the case of $2\times 2$ matrices. In fact, Alexander Stasinski proved in his paper [1] the following:

**Theorem.**Let $R$ be a principal ideal domain. If $n\geq 3$ then any matrix in $M_n(R)$ of trace zero can be written as the commutator of two matrices in $M_n(R)$, each having trace zero. The same holds for $n=2$ if two is invertible in $R$.

Notice how the characteristic two problem only happens in the $2\times 2$ case.

[1] Stasinski, A. Isr. J. Math. (2018). https://doi.org/10.1007/s11856-018-1762-5

## Calculator Review: Casio FX-991MS

Some believe that if you're main profession is pure math research, you don't need a scientific calculator. That's simply not true. Although I don't use one nearly as much as when I was an undergrad, I still need a calculator and the only one I'm willing to use is the Casio FX-991MS.

## My top nine favourite math texts

Here they are:

## Keith Devlin, *The Joy of Sets*

If you're not a set theorist but want to understand set theory, this book is awesome and one of a kind. I have not read it all, but what I have read I can actually understand!

## Frank de Meyer and Edward Igraham, *Separable Algebras over Commutative Rings*

This classic book explains Galois theory but for commutative rings. Even though there are many more technicalities in the general commutative ring case compared to fields, I actually found the approach in this book more natural than the Galois theory for fields that I learned in undergrad algebra. There are some exercises and this book is easy to read.

…read the rest of this post!

## On reasonably sure proofs

I happened to come across a 1993 opinion piece, Theorems for a price: Tomorrow's semi-rigorous mathematical culture by Doron Zeilberger. I think it's a rather fascinating document as it questions the future of mathematical proof. Its basic thesis is that some time in the future of mathematics, the expectation of proof will move to a "semi-rigorous" state where mathematical statements will be given probabilities of being true.

It helps to clarify this with an example even more simple than in Zeilberger's paper. Take the arithmetic-geometric mean inequality for two variables $a,b\geq 0$. It says that

$$\frac{a + b}{2} \geq \sqrt{ab}.$$ This simple identity is just a rearrangement of the inequality $(a – b)^2 \geq 0$. For simplicity, let's say that $a,b\in [0,1]$. Instead of actually proving this inequality, we could generate uniform random numbers in $[0,1]$ and see if this inequality actually works for them. So if I test this inequality 1000 times, of course I will get that it works 1000 times.

…read the rest of this post!