Believe it or not yesterday night it rained in Trento. I was dreaming I was about to have an icecream and the wind banging the open windows woke me up. Went I went back to sleep my icecream was gone.

This morning was wasted on applications and getting a couple of things I forgot at home as well as finding out how to get to the classroom, which is way outside of town. Tomorrow afternoon I will have to replace a couple of things more I forgot, a pair of trousers (I actually took Sanja’s) and a belt.

The lectures started promptly and we had another round of lovely Italian coffee and pear juices at the break, just like in Cortona, although we couldn’t serve ourselves as much as I wanted. The first lecture was (another) introduction to affine toric geometry. Sandra di Rocco had promised she would discuss open problems from lecture 2, but after seeing the outline I consider it unlikely. We’ll see. Grisha Mikhalkin gave a motivational lecture mixing tropicalisation (I will not write tropicalization, sorry) of Algebraic Varieties as inspired from Thermodynamics. To be honest I understood little, but in times where mathematicians have to fight to show the ‘impact’ of their work, it ocurred to me that it wouldn’t be crazy to see these things in post-doc applications soon. I mentioned it to him at the end of the lecture and he said that if anything, in this case it was Mathematics been ‘impacted’ by Thermodynamics. But I do not think he is aware of how desperate the situation is turning in the UK. Other fields are already being less honest. Is it OK to justify your research with some dodgy hypothetical application as long as it pleases your sponsor and you do a work that you think is worth doing? Or should you actually work towards an application even if it is a dodgy one, like the Manhattan Project? Even those like Hardy who thought their mathematics had no real application, ended up with applications (not directly) to war. Anyways… I disgress…

The idea Grisha suggested, if I understood it right, is that the integral points of a given polytope would represent the possible minimal energy states of a quantised system. The dimension of the lattice where these live in would represent the dimension of the space of energy states. For instance, a polytope in 2 dimensions, with 7 lattice points would have 2 dimensions for all the possible energy states, the minimal one corresponding to the integral points given by those 7 points. These 7 points form ‘abstractly’ a simplex of dimension 6 and any other state would live in within. To me this sounds like science fiction to me but apparently Okounkov has done some work in it. The tropical picture comes when these states have something to do with some amoebas (pretty much like a real amoeba from which you extract the nucleus) that when contracted become tropical curves (if we started in dimension 2).

Together with Pau and two students of Sandra di Rocco I decided to come back to town walking. We got a bit lost and the expected 30 minutes turned out to be an hour and a half. Then we went for dinner and they served us a monster pasta-dish. I usually eat automatically until there is no food in the plate but I reckon there were 200g of pasta, so at some point I started feeling a bit sick and I stopped. Too bad. I want

This picture is taken coming down from Povo. This area is beautiful, like a warm Asturias.

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