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Michi’s blog » archive for May, 2007

 Going to T’bilisi

  • May 25th, 2007
  • 1:37 pm

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.

 8th Carnival of Mathematics

  • May 19th, 2007
  • 9:40 am

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.

 7th Carnival of Mathematics

  • May 4th, 2007
  • 7:15 pm

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.

 Young Topology: The fundamental groupoid

  • May 4th, 2007
  • 3:26 pm

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.