Math Tricki now live

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.

 

A new grad student math blog

The newest AMS Headlines and Deadlines announces a new AMS graduate student math blog, managed by Frank Morgan, AMS vice-president. They already have posts ranging from finding an advisor to advice on organizing a reading seminar.

 

Mathematics is dangerous when …

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!

 

The Clock Notation

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 ; he uses . The reason for his choice is that arithmetic modulo is just like adding numbers (or positions) on a (analog) clock. Let’s work this out for . We begin by replacing the on our clock by . Then we can start adding things. Say it’s and we’re starting to drive home from work which will take hours (damn commute). What will the time be when we arrive at our doorstep (late for dinner)? Well it’s of course, on the clock face. This is exactly the same as addition modulo .

 

NOT Teaching Multivariable Calculus with MatLab

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!

 

Teaching Multivariabe Calculus with MatLab

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 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 , a point , a real number , and a natural number . What the MatLab function should do is randomly select points on a circle (in the -plane) of radius and center ; calculate the directional derivatives along the vectors . The output should be:

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

 

So what is it good for?

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?

 

Teaching without Permission!

Many of the math blogs that I read (and you can find in the blogroll) are writing about the plight of Ali Nesin and his Summer School. Apparently, Prof. Nesin was arrested for running his yearly Summer School in Turkey since he was “teaching without permission”! You can even see pictures of black boards behind police tape on Mathematics under the Microscope. A petition was set up to protest and I urge you to sign it!

 

New Paper on Arxiv: Nonisomorphic Verdier Octahedra on the same Base

The paper Nonisomorphic Verdier Octahedra on the same Base, by Matthias Künzer, popped up on the Arxiv today. It answers the following question: Given a morphism in a triangulated category the cone on is determined up to a (non-unique) isomorphism; is the same true for a Verdier octahedron constructed from a commutative triangle? The answer is: No, there exist non-isomorphic Verdier octahedra constructed from the same base! The non-uniqueness mentioned above has been a thorn in the side of the standard axioms for triangulated categories and this result doesn’t exactly make their life easier!

To prove this result Künzer uses something called Heller triangulated categories which he describes in detail in the paper Heller Triangulated Categories. These two papers are definitely being added to my list of things that I want to understand!

On a related note, J.P. May has proved that the standard axioms for triangulated categories are over-determined in the sense that one of the axioms can be proved using the other three. This is explained in his paper The Axioms for Triangulated Categories.

 

Arguing with Other Mathematicians

I use an RSS thread to follow what is new on the arXiv and recently I came across a paper that claims that the theory of mathematics isn’t consistent. That sounded intriguing so I downloaded the paper and read it. I discovered, what I think is, a serious error and notified the authors. Every night since then I have been in email contact with the authors. We are in complete disagreement about whether there is an error or not but our discussions are (still) very polite and it doesn’t look like either one of us will convince the other(s).

I would appreciate any comments or advice from anyone reading this post that has similar experiences of conversing/arguing with the author(s) of a paper about errors and how such conversations/arguments have turned out.

I don’t want to post a link to the paper, since I feel like that would be like trying to make fun of other people’s work. I can email a link to anyone who is interested in taking a look at the paper, just drop a comment.