# Author Archives: Jason Polak

## 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: Liebeck, Martin W.; O'Brien, E. A.; Shalev, Aner; Tiep, Pham Huu. The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008. It's quite a […]

## 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 […]

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

Sir Michael Atiyah's preprints are now on the internet: The Riemann Hypothesis The Fine Structure Constant 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 […]

## 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 […]

## 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 […]

## 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 […]

## 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 […]

## 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 […]