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Michi’s blog » archive for May, 2007

Going to T’bilisi

May 25th, 2007

In about 23 hours, I’ll step on to the train in Jena, heading for T’bilisi, Georgia.

On Monday, I’ll give a talk on my research into $A_\infty$ -structures in group cohomology. If you’re curious, I already put the slides up on the web.

I’ll try to blog from T’bilisi, but I don’t know what connectivity I’ll have at all.

(59 words, 1 image)

8th Carnival of Mathematics

May 19th, 2007

Is up at the GeomBlog.

This fortnight has a lot of goodies, among those a call for reading Grothendieck and a blogpost by Ian Stewart.

Go read.

(28 words)

7th Carnival of Mathematics

May 4th, 2007

I have been somewhat remiss in announcing these lately - but over at nOnoscience, the 7th Carnival of Mathematics just got posted.

I’m featured again - as are many other very readable bloggers. Go. Read.

(36 words)

Young Topology: The fundamental groupoid

May 4th, 2007

Today, with my bright topology 9th-graders, we discussed homotopy equivalence of spaces and the fundamental groupoid. In order to get the arguments sorted out, and also in order to give my esteemed readership a chance to see what I’m doing with them, I’ll write out some of the arguments here.

I will straight off assume that continuity is something everyone’s comfortable with, and build on top of that.

Homotopies and homotopy equivalences

We say that two continuous maps, f,g:X→Y between topological spaces are homotopical, and write $f\simeq g$ , if there is a continuous map $H\colon X\times[0,1]\to Y$ such that H(x,0)=f(x) and H(x,1)=g(x). This captures the intuitive idea of step by step nudging one map into the other in formal terms.

Two spaces X,Y are homeomorphic if there are maps $f\colon X\to Y$ , $f^{-1}\colon Y\to X$ such that $ff^{-1}=\operator{Id}_Y$ and $f^{-1}f=\operator{Id}_X$ .

Two spaces X,Y are homotopy equivalent if there are maps $f\colon X\to Y$ , $f^{-1}\colon Y\to X$ such that $ff^{-1}\simeq\operator{Id}_Y$ and $f^{-1}f\simeq\operator{Id}_X$ .

This is a preview of Young Topology: The fundamental groupoid. Read the full post (840 words, 46 images)

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about: 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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