Author Archives: Jason Polak

A finitely generated flat module that is not projective

Let's see an example of a finitely-generated flat module that is not projective! What does this provide a counterexample to? If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module […]

Kourkovka Notebook: Open problems in group theory

Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved […]

Britton's lemma and a non-Hopfian fp group

In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free […]

How to make your own WordPress theme

This is a meta post on blogging, not mathematics. Recently, I got it into my head that I should design my own WordPress theme from scratch. As a consequence, you may have noticed that the theme of this blog has changed a little. I don't know if many other math bloggers will want to try […]

Book Review: Riot at the Calc Exam by Colin Adams

When it comes to math humour, there's not much out there. There is a good list of jokes on MathOverflow. There's also Mathematical Apocrypha by Krantz, many of whose folklore stories are also amusing. The other day at the library I found another one: Riot at the Calc Exam by Colin Adams. Adams' book is […]

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 […]