Tim Gowers just announced that the Tricki – a wikipedia of tricks and tips for proving mathematical results – is up and running (it has been visible but un-editable for a few days). It features articles from various prominent people, including Terry Tao. After a quick glance it looks pretty awesome and has the potential to become even awesome-er.

Over at Ars Mathematics Walt points out two articles that blaim mathematicians and physicists for the fall of Wall Street. His conclusion is that mathematics is dangerous when it falls into the wrong hands. I think the problems is more that mathematics is dangerous when it falls into stupid and greedy hands!

I’m reading the book Prime Obsession, by John Derbyshire, which tries to explain the Riemann Hypothesis to non-mathematicians. I got the book early in my undergraduate but never got around to reading it and then came across it in my book shelf the other day. I’m about half-way through and it has been a delightful read. Derbyshire manages to explain a lot of difficult mathematics without getting too technical and I would recommend the book to anyone starting an undergraduate in mathematics.

When I turned page 268 I was a bit surprised to see Derbyshire’s notation for the quotient rings $\mathbb{Z}/n \mathbb{Z}$; he uses $\mathbb{CLOCK}_n$. The reason for his choice is that arithmetic modulo $n$ is just like adding numbers (or positions) on a (analog) clock. Let’s work this out for $n = 12$. We begin by replacing the $12$ on our clock by ${0}$. Then we can start adding things. Say it’s $11$ and we’re starting to drive home from work which will take $2$ hours (damn commute). What will the time be when we arrive at our doorstep (late for dinner)? Well it’s $1$ of course, $11 + 2 = 1$ on the clock face. This is exactly the same as addition modulo $12$.

In my last post I talked about using MatLab to help with teaching multivariable calculus at Reykjavik University. Well, what happened was that Mathworks, the makers of MatLab, were dragging their feet in getting us the Symbolic Toolbox for MatLab (needed for defining and working with variables); so we decided to switch to some other computer algebra system. The open-source SAGE came out on top from the choices we had. I’ve been working with it for two weeks now and I can honestly say that it is friggin’ awesome! Here are a few good reasons to try it out:

1. It’s free.
2. It can be run locally on a Mac, PC or Linux; or through the SAGE website: They give you a free account and you can access your computations from anywhere. Has anyone tried it on their iPhone?
3. It combines the power of many open-source math packages like Maxima, R, Singular, etc. and can interface with programs such as MatLab, Mathematica, etc.
4. It is built on Python which makes coding really nice and simple.
5. It is actively developed and growing fast.

Give it a spin!

I’m teaching a course on multivariable calculus at Reykjavik University, in Iceland, with a slight twist. The students turn in four problem sets over the semester, each one containing four problems. Most of the problems utilize MatLab for either vizualization or for the solution. Here’s an example.

Problem. Part A. Given a function of two variables $x,y$ write a MatLab function that can make a rudimentary test for local maxima, minima and saddle points in the following way: The input should be a function $f$, a point $P_0 = (x_0,y_0)$, a real number $\epsilon > 0$, and a natural number $n \geq 1$. What the MatLab function should do is randomly select $n$ points $P_1, P_2, \dots, P_n$ on a circle (in the $xy$-plane) of radius $\epsilon$ and center $P_0$; calculate the directional derivatives along the vectors $\vec{P_0P_n}$. The output should be:

1. “The function has a local maximum at $P_0$.” if all the directional derivatives are negative.
2. “The function has a local minimum at $P_0$.” if all the directional derivatives are positive.
3. “The function has a saddle point at $P_0$.” if the directional derivatives are both positve and negative.

Part B. Give an example of a function with the property that: It accurately detects a local maximum or minimum at a point $P_0$ for $\epsilon < 1$ but fails to detect it if $\epsilon > 1$.

I haven’t assigned this problem yet but am very excited to see the results. I think this problem should give the students a good feeling for what local maxima, minima and saddle points “look like”.

You know you’ve been asked “what’s math research good for?”  I’m talking about at that party that wasn’t full of other math grad students.  You know, right after, “you do new mathematics?  I thought it was all finished!”

I’ve made a stab at writing an answer to this for the general public, and I’d love input from other mathematicians.  Do you think I’m right?  So what is it good for?