It is March 2012 and I got an e-mail announcing EPSRC fellowships. Of course Pure Mathematics is not among them. In a few months it will be a year without fellowships. If Newton resurrected he would revolt. But again, there are no real mathematicians in Westminster, anymore. Only bureaucrats.

I start a new trip in a week: Barcelona on holidays and then Cambridge and Paris.

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Some people thought enough is enough and created a sort of public commitment declaration to boycott Elsevier including not publishing, peer-reviewing or joining Editorial Boards of Journals own by Elsevier. In principle it seems like a good idea. Let alone the moral stand: Our university is facing cuts in research budget (for seminars, travelling, even teaching) every year for the last three. Giving money to a company whose only effort is some secretary work, printing books and running a website making over 20% of benefit a year seems wrong. Especially if we take in account that there are lots of academics behind, funded by public bodies and usually Governments, doing most of the work for free. What’s more, paying for it, since these universities and research centres are the ones which pay for the journals.

So I was all ready to sign and commit and then I stopped and thought: What if when I have a paper ready (and I almost do) it turns out that the appropriate place for publishing it is an Elsevier journal? Or if I write something with my supervisor and he says ‘we will try to publish it in this journal’ that is Elsevier? How do I tell him no?

It is hard to commit to this when one is not established enough. And that is why Elsevier is still there. It uses the status quo. We need his bananas. We need to buy the bananas directly from the producers. A boycott is good, but it needs strategy.

Last week I’ve been in a Winter School and one of the students I talked to said he did not care at all about this. If he was aware that the money saved could go into his benefit maybe his point of view would change.

For a start we need alternatives for those who are young. And the community can get organised easily. Possible steps are:

- Contacting the members of Elsevier Journal editorial boards to lobby Elsevier to change its policy or resign, with a deadline for change to happen, including at the very least the end of bundling and reducing the prices of the journals. This should be done by senior, well-established mathematicians. It is enough to convince one member to discuss it with the others to get the snowball moving.
- A database (a wiki in wikia would be enough) with alternatives to each of the journals of Elsevier (similar subject, similar impact factor). This would help young researchers to find alternatives. It is easy to set it up, but to fill it in is harder.
- An easy way of creating alternative journals for each of those that Elsevier is willing to lose before they agree to change their policy. I think the Journal of Topology is the way to go. In case they have doubts they may not the impact factor was doubled after the change. It is necessary to contact alternative publishers who are reasonable is a good idea.
- Some incentive for those who behave like mushrooms to take action (I’m not sure what this incentive would be).

Any more ideas?

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To be honest in the school there has been lots of free time, but I somehow managed to fill it in one way of another one. Walking, for instance, took easily an hour or an hour an a half away. Our accommodation was in some bunker-style houses near the former building of the Mathematics Institute which happens to be 25 minutes walking from the current building. The supermarket was even further.

I had very fruitful discussions with my supervisor, and in fact for the first time since I started I also had a fruitful one with my second supervisor (it is rare in Britain to interact with second supervisors). Actually it seems both of us are interested in deformations of singularities of surfaces which is quite cool. We saw how given a deformation over a base of dimension n one can get one over a base of dimension 2. If the properties that the deformation satisfies are preserved by base change, then it is enough to consider deformations over the germ of a curve to classify all varieties which accept that kind of deformation.

Moreover I luckily happened to learn certain things in the school (like quick and exhaustive calculation of data regarding cyclic quotient singularities) that I am using at the moment.

I also met really nice and interesting people: Kuzma Khrabov, Jesus Tapia, Nathan (Glasgow), Alberto and Emmanuele (Oxford), and the usual russian troupe, of course!

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I am aware I left last trip half done but I rather not post about it now. The reason was mainly the lack of Internet in Poland. Then, when I went to Germany and later to Croatia I had better things to do than write here, I guess.

Since then I had not travelled much: to Moscow in HSE 10 days, with their generous support and the LMS. It was december, really cold but it is nice to see how they would not reduce their activity due to this. I met Kuzma Khrabov, an undergrad at Moscow State who is coming to Warwick too. And the usual ones: Logvinenko, Prokhorov, Costya Shramov… From Moscow I went to Madrid for Christmas and New Years and on the first I went back to Edinburgh.

In April I will travel a bit more. I will go to Barcelona for a week on holidays. From there I will go to Cambridge, to some workshop organised by Vanya and Julius Ross, a former lecturer of mine. Then, Paris for a school in Tropical Geometry and then back to Edinburgh. It will be a month or nearly a month altogether.

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The school was quite intense. We had two lectures a day for 3 hours altogether and a 2 hours exercise session. Brion’s lectures were in classical Geometric Invariant Theory. For those who are not expert, we could say that GIT studies quotients of varieties by groups. An algebraic variety X is always a topological space. If a group G acts algebraically on X (i.e. each element is an algebraic automorphism of the variety) in particular it acts homeomorphically and the quotient X/G is another topological space. However in general it is not a variety, and even if it is, hardly ever one obtains a very interesting one (the dimension usually decreases a lot, for instance).

The main problem for it to be a variety has to do with its ring of regular functions R. Since G acts on X, it naturally acts on R. The ring of regular functions of X/G should be R^G, i.e. those elements in R which are invariant by G. This is a subring of R, but surprisingly it is not necessarily finitely generated. In the second ICM, David Hilbert presented a list of ‘fundamental’ problems in Mathematics. Proving that R^G was finitely generated was one of them, the number 14th. However it turned out to be false, as Nagata proved in the 60s. The field of Geometric Invariant Theory (GIT) was therefore initiated by Hilbert and developed by several people. However the two main contributors would be Nagata, and very especially David Mumford, who established criteria to decide when R^G was finitely generated and when Spec R^G coincided with X/G (obviously he also did this in more general settings such as projective and abstract varieties), as well as which subsets of X admitted these quotients. Brion managed to explain these ideas very clearly. I should add what was Mumford’s motivation to develop these techniques. This is the subject of moduli theory, in which one would like to parametrise a set of objects, for instance all curves of a given genus, creating a variety with them. But then those objects which are equivalent (for instance via isomorphism) should be parametrised just by one point. For this it is necessary to have a group acting on the variety and the quotient should be the desired parametrisation.

The subject of Ressayre lectures had to do with applications of GIT to Schubert calculus problems and I found it harder to follow, although he gave a very good explanation of how the Hilbert-Mumford criterion works.

The school had a nice break on Wednesday afternoon, when we were taken to see a Viking village (obviously reconstructed) some amazing views of the Baltic Sea and a conservation park with European deers and bisons. At least 3 of the nights there was a bonfire and the last one we had a barbecue in which we were cooking our sausages in a stick. The last night we also had a walk into the beach (it was just 15 minutes away) and we were very impressed by the amount of stars we could see, something impossible in Britain.

I could not finish the Lukecin experience without mentioning an amusing moment. There is no access to Internet in Lukecin, neither in the pension nor in the cafe. However at some point someone found out that there is a massive hotel complex in the middle of the forest which had unprotected Internet. It was certainly hilarious to see a bunch of people in the middle of nowhere, using their batteries as much as possible in order to send e-mails, check the news and talk to their families.

All in all it has been a great experience. I hope I have another chance to go to a school in there. Jarek Wisniewski, the organiser promised another one for next year. He was wearing a ‘BIG AND NEF’ t-shirt. I keep my fingers crossed

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Last post happen longer than 2 weeks ago. Since then I have left Trento, had a crazy 20 hours trip to Poland where I attended a school in Geometric Invariant Theory, visited Berlin and arrived to Rijeka (Fiume) in Croatia. The principal reason for this is the lack of Internet and a bit my lack of will. In the next few days will try to post a few summary posts with those things which impressed me the most. I will start with the last days in Trento.

The school went better and better and I am very happy I attended in the end. Di Rocco’s lectures went more advanced and by the 4th I actually learned many things and asked myself some questions I could not answer straight away. The thing I remember the best was her treatment of toric fibrations.

Regarding the tropical geometry lectures I actually got quite interested. Mikhalkin had started very vaguely with some sort of motivation from Thermodynamics that barely anyone could understand but when the real mathematics started tropical geometry actually became very beautiful. He first introduced tropical varieties in a ‘differential’ fashion, as manifolds whose group was similar to that of affine manifolds. However instead of having GL(n,R) semidirect product with R^n the rotations were defined over the integers (if I remember right). This only worked for smooth tropical varieties and he quickly defined tropical curves in a more ‘semiring’ style. Then he developed an intersection theory for planar tropical curves which axiomatically works in the same fashion as the usual intersection theory for curves on surfaces. Moreover he described ‘modifications’ over curves. I did not quite get this, but I think it would resemble blow-ups if it wasn’t because rather than adding a ‘dimension n-1’ subvariety it was adding a ‘branch’ to the tropical curve. Finally he also explained how to ‘tropicalise’ complex varieties. Altogether a very interesting introduction, not only due to all the covered material, but also because it makes one feel sorry for not spending some time in it. It seems to me that many concepts from classical geometry, in particular in small dimensions are yet to be explored and must not differ much from the usual complex setting. I wonder for instance if there is such a thing as a ‘minimal model’ theory or an ‘ampleness’ concept for surfaces. If there is some regret is the fact that there is not a proper reference yet. Mikhalkin himself has been working on a Tropical Geometry book for a few years now (near 5 I believe) but every now and then he decides to give it another point of view and he starts from scratch. I really look forward to it.

The school finished before lunch on Friday and then a workshop started where researchers (many of them senior) presented their work. Unfortunately apart from Diane Maclagan’s talk, I found most of them hard to follow or out of my interest. In the evening there was a marvellous conference dinner with really fancy Italian food. The day after I could not attend to the closure of the conference because I had to leave towards Poland.

The trip itself was really long. I was lucky enough to enjoy the company of my room-mate Pau till the evening. Trento is very bad connected. We left at midday and arrived at 5 to Bergamo Airport where I took my flight at 7. After arriving to Poznan, I had to wait more than 4 hours to take the train at 2.30 in the morning. I arrived to Szczecin nearly at 6am. Few people speak English in Poland and finding my way to Lukecin was not easy, but in the end I manage to catch a bus which dropped me in the conference site and I finally could go to sleep at 9am.

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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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The original plan is:

- 9th of September: Bologna.
- 10th-17th of September: Tropical and Toric Geometry in Trento.
- 17th of September: Cross-Europe trip: Train or bus to Bergamo, Flight to Poznan. Train to Szczecin. Bus to Lukecin. This should take the whole night.
- 18th-24th of September. Geometric Invariant Theory: Old and New in Lukecin.
- 24th-28th: Holidays start wow! Train to Berlin. Stay there for 4 days.
- 28th-8th: Flight to Venice, Train to Trieste. Bus to Rijeka. Enjoy north-croatian Autumn rain

So the trip started already. In Bologna I stayed with one Couchsurfer who is about to start his second PhD in classic languages in the UK. Different.

In the plane and later on the train to Trento (which took an hour longer than scheduled) I had the chance of going slowly (i.e. doing all exercises) through Kollar’s Singularities of Pairs, as Costya once put it, the article that all birationalist should read at least once.

In Trento I arrived in the middle of some medieval festival. The views from my room are AMAZING!

Also, I am lucky to be sharing the room with Pau, a catalan student, so I can discuss maths in Spanish

I’m going to buy some stuff now. The classes start in the afternoon.

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On Monday I made my way to New York via Hartford. I had to take a Megabus, since the trains either weren’t running or had ran out of tickets. I finally could meet Blanca and I had a wonderful couple of days in Manhattan. The city was completely different from what I could remember. For a start half of the people in the island seemed to speak Spanish and in fact they would jump from English to Spanish and back to English several times in the middle of a conversation. I was also amazed at how expensive everything was (felt like London). At the beginnning I didn’t like the city and I felt the negativity that Blanca had told me about: the unhappy people, the poverty, everyone in a rush, the feeling that you were paying for breathing… But then I started to feel that as long as you kept yourself away from the bad things and focus on the exceptionality of NYC, there was a lot to take. First of all not all the houses are terrible and Blanca’s is a good example of that. There is lots of culture for little or no money (museums, street shows) and lots of different magic places to enjoy such as parks or certain not-so-expensive restaurants. And there are lots of nice shops to be, like a massive Toys ‘r’ us with a riding wheel inside! You don’t need to buy, just wondering around is a nice feeling, once I had not experienced in a while.

Maybe I would not like to live in Manhattan, but I can enjoy it for a while.

On the flight back home something noticeable happened. They had overbooked (even when they had checked everyone in) and they needed to get rid of one passenger. They looked for a volunteer and they found one. I was surprised of that but the Irish guy sat next to me on the plane told me that they would pay for another flight back for him and for an other one (probably return one) anytime he wanted the year after. If something like this happens again to me I may offer myself

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