Michi’s blog » archive for March, 2007

Sudden responsibilities

  • March 28th, 2007

I just met up with the workgroup in the Deutsche Mathematikervereinigung (German Association of Mathematicians) with interest spanning “Information and Communication” - which turns out to mean that they care about libraries, about communicative tools for mathematicians, and spend their time thinking about these things, and meeting at conferences.

Met up with is to say participated in their workshop.

Stunned them all with the relevation that I blog. “Wow, a real, live blogger! Here!”

And promptly got elected to their executive committee.

Kinda unexpected result of going to a conference.

(91 words)

4th Carnival of Mathematics posted

  • March 23rd, 2007

The fourth Carnival of Mathematics is up at EvolutionBlog.

Featured this time around are homological algebra, representation theory, Rubik’s cube solutions, Bernoulli processes, topology, number theory, and much much more.

(31 words)

Representation theory - basics

  • March 21st, 2007

Many interesting groups have a very geometrical definition: transformations that fix certain symmetries is one of the historical origins of group theory.

Thus, one of the most interesting classes of finite groups are the rotation and reflection symmetries of a regular polygon. These are called D_n, for a n-sided polygon. Thus, for a triangle, we can label the corners a,b,c, reading clockwise, and enumerate the possible transformations by the positions the corners end up in. Thus we get the elements:
identity: abc -> abc
τ: abc -> bca (rotation by 120 degrees)
τ2: abc -> cab
σ: abc -> acb (reflection fixing a)
στ: abc -> bac
στ2: abc -> cba

Now, if we fix one equilateral triangle - say the one spanned by the points a=(1,0), b=(-\frac12,\frac{\sqrt3}2) and c=(-\frac12,-\frac{\sqrt3}2), then these transformations of the triangle can be extended to rotations and reflections of the entire 2-dimensional plane \mathbb R^2. As such, we can write down matrices for the group elements, starting with
\sigma=\begin{pmatrix}0&1\\1&0\end{pmatrix} and
\tau=\begin{pmatrix}\frac{\sqrt3}2 & -\frac12 \\ -\frac12 & -\frac{\sqrt3}2\end{pmatrix}
The rest of the group elements we can realize as matrices by just multiplying these with each other.

This is a preview of Representation theory - basics. Read the full post (1199 words, 12 images)

Carnival of Mathematics #3

  • March 9th, 2007

And it is with pride that I welcome you all to my first issue, and the third issue all in all, of the Carnival of Mathematics. I probably should apologize as well - my announcement stated March 8th, but that was before I really looked at the dates involved, so we did, alas, miss the international women’s day. We haven’t had quite the rush that Mark CC enjoyed, but we’ll make a good one even so.

First out, from the first half of our submissions, we have a grand tour of didactic topics, starting out with Michael Tang, who shows us why negative times negative is positive, with a touch of ring theory into the mix. Following that, Rebecca Newburn discusses equation solving strategies and Laurie Bluedorn takes a historical view on the age of introduction of formal arithmetic. To finish it up, jd2718 tells us about teaching complex numbers and your humble host has a manifesto of sorts about stimulating strong students.

This is a preview of Carnival of Mathematics #3. Read the full post (460 words)

Bright students and topology

  • March 2nd, 2007

Today, I started an experiment together with the local specialised secondary school. I’ll be taking care of two of their brightest students, meeting them roughly once a week, and taking them on a charge through algebraic topology. At the far end shimmers knot theory and other funky applications; and on the way there, I hope for many interesting spinoffs and avenues.

They got, today, Armstrong’s Basic Topology, and an extract from the German topology book by Jänich, and on monday, we shall go over the formal definition of topological spaces, and of continuous functions together.

I plan to keep updates on our progress here on the blog - with the questions I send them off with each meeting as well as some sort of discussion about how this setup is working out, if at all.

For the first trip, the questions I dumped in their laps were:

  • What topologies are possible on the set {0,1}?
  • What topologies are possible on the set {0,1,2}?
  • Which are the continuous functions between the topologies above?
  • Give an example of a continuous and a discontinuous function each for the following cases
    1. f\colon\mathbb R\to\mathbb R with the standard topology on \mathbb R
    2. f\colon\mathbb C\to\mathbb C with the standard topology on \mathbb C
    3. f\colon\mathbb Z\to\mathbb Z with the discrete topology
    4. f\colon\mathbb Z\to\mathbb Z with the finite-complement topology
(206 words, 6 images)

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Michi is a PhD student working in homological algebra. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
Not all here is mathematics. All here, though, are my personal thoughts and opinions. Please read the about page (linked above) for more details.
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