The Real Sqrt

Travelling through the jungle of mathematics

Thursday, August 16, 2007

These are a few of my favourite maths

(On the melody of "My Favourite Things" from "Sound Of Music".)

Functors, espec'ly of higher dimensions
Infinite spectra of loops and suspensions
A topological theory on strings
These are a few of my favourite things.

Finally grokking the bar resolution
Solving equations by guessing solutions
zero-dimensional Gorenstein rings
These are a few of my favourite things

When a conference,
a real good one, ends
Then I'm feeling sad
But then I remember my favourite things
And then I don't feel so bad

Posing a question without hesitation
Knowing the proof and in four variations
Dreams of the future, what ever it brings
These are a few of my favourite things

When a complex
is too complex
driving me insane
I take a good look at my favourite things
and know this work's not in vain

Saturday, November 11, 2006

The lecturer.

Feeling like an almost full-grown mathematician, I now have performed my first mathematics lecture.

Actually, I 'performed' quite some weeks ago, but I came to remember that I hadn't told you guys.

The lecture was a part of a course on representation theory, where we had just gone through complex representations, character tables, and stuff like that.

The topic was perfect for me: The correspondence between complex group representations and modules over the complex group ring.

Some of the younger maths students who were gatherd there (younger as in number of theorems understood, not as in age), and thus I was to introduce them to the concept of 'modules' for the fist time of their lives. I am positive, that they or at least some of them came to love modules just a little bit. Or at least like them. Just a little.

As usual, the lecture notes are available to you, by simply clicking the blog-topic.
Note that I use "R-module map" as a word for an R-morphism of R-modules in the exercise.

And by the way: if this made you love modules over the group ring just a little bit more: tell me! ;)

See you.

Tuesday, August 01, 2006

Bachelor's Thesis

Hi, and welcome to my new maths blog.

I recently finished my "thesis" on Group Cohomology.

In this thesis we describe the basic properties of homology and cohomology with coefficients. Induction and coinduction is shown to coincide when the subgroup has finite index. We then proceed with a proof of Shapiro's Lemma.

We study the group algebra over a field and prove Maschke's theorem. The group algebra over a field is a self-injective ring, and we notice that the group algebra of a p-group over a field of characteristic p is a local Gorenstein ring.

Click the title of this entry to download the pdf.