signs of life

eight categories

preprojective component

some tubes

representation space of the four subspace quiver

imaginary Schur roots

Earlier I watched a video of (fields medallist) Cédric Villani [ http://bit.ly/kYpJ2p ]. He spoke about the feeling

When you arrive in a new problem, the first thing is total obscurity. You don’t understand anything, What’s going on? I can’t understand! Like darkness everywhere. Like Bilbo the hobbit in Gollum’s cave for those who know. Everywhere so dark. And then at some point you feel some tiny breeze of wind, something that shows you that there is an opening: you will see the light. That’s the moment I prefer, excitement. And third stage is you understand it all, light comes and it comes all of a sudden and it’s so bright and we understand how the various mathematical concepts can be put together to solve your problem.

I’m in this dark cave at the moment. Grothendieck describes a similar place

A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet it finally surrounds the resistant substance.

[ http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty1.pdf ]

To me it feels like a dense thicket of underbrush. Directly in front of your face you can see quite clearly the leaves, the individual theorems and lemmas of the literature, and you can understand them, with work. You’re pushing through the vegetation. Or rising up, very slowly through the forest. Today I went through a proof of this lemma

Let R be a Henselian Cohen-Macaulay local ring. The category of Cohen-Macaulay modules over R admits Auslander-Reiten sequences if and only if R is an isolated singularity.

And sure, I understand it now (though it took all day). I believe it. But what does it mean? What is a Cohen-Macaulay module, really? Why do I care that I have AR sequences?

Then gradually, gradually as you keep examining the underbrush you start to float upwards. Your focus broadens, and you begin to see parts of how it all fits together.

blowing up [from Hartshorne]

Yukawa coupling

Hodge decomposition