{"id":1901,"date":"2017-02-04T13:03:23","date_gmt":"2017-02-04T18:03:23","guid":{"rendered":"http:\/\/blogs.ams.org\/phdplus\/?p=1901"},"modified":"2017-02-04T13:03:23","modified_gmt":"2017-02-04T18:03:23","slug":"icermania","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/phdplus\/2017\/02\/04\/icermania\/","title":{"rendered":"ICERMania\u00a0\u00a0\u00a0"},"content":{"rendered":"
\"2computers\"<\/a>

Chris Rasmussen, Alejandra Alvarado, Angelos Koutsianas, and Mckenzie West working hard at ICERM in January. \u00a0As Mckenzie demonstrates, sometimes it just takes two computers to get\u00a0a job done.<\/p><\/div>\n

*What do you with six mathematicians, spread across the world, who all want to solve the same problem but have never all been in the same room?\u00a0 If that problem involves computational or experimental mathematics, one great answer is Collaborate@ICERM.<\/p>\n

ICERM is the Institute for Computational and Experimental Research in Mathematics<\/a>, associated with Brown University in Providence, RI.\u00a0 It is the newest of the eight NSF-funded Mathematical Institutes<\/a>, and like many of the others it runs semester-length thematic programs and large workshops throughout the year.\u00a0 However, ICERM also has a program for small groups to apply for funding to meet at the Institute to collaborate on projects that are in line with the Institute\u2019s mission, which is \u201cto support and broaden the relationship between mathematics and computation and, specifically, to expand the use of computational and experimental methods in mathematics, to support theoretical advances related to computation, and to address problems posed by the existence and use of the computer through mathematical tools, research and innovation.\u201d\u00a0 Collaborate@ICERM<\/a> takes applications from groups of 3\u20136 people to meet at ICERM during the summer or over winter break.\u00a0 The American Institute of Mathematics has a similar program in SQuaREs<\/a>.<\/p>\n

\"Christelle<\/a>

Christelle Vincent and Mckenzie West fixing something major during a collaboration at ICERM.<\/p><\/div>\n

My collaborator Chris Rasmussen<\/a> and I found ourselves in the above-described situation.[ii]<\/a>\u00a0 We had started working on a project in 2012, one component of which\u00a0forced us to write code to solve the equation x+y=1, where x and y are S-units in a number field K.\u00a0 In simple terms, S is a set that contains some primes of K, and the S-units are just the elements of K generated by these primes, their inverses, and the units of the field. \u00a0So the {2,3}-units of the rational numbers are just the set of positive and negative fractions that only involve powers of 2 and 3.\u00a0 In this world, (1\/3, 2\/3) is a solution to the equation.\u00a0 By Alan Baker<\/a>\u2019s work, it turns out that there are only finitely many solutions for any given K and S, and their size can be bounded, a fact that leads to may interesting finiteness results in number theory.\u00a0 Often, enumerative problems rely on actually finding all of the solutions for a given number field, a problem referred to as \u201csolving the S-unit equation\u201d. \u00a0This is a difficult computational problem, involving work of Baker and Wustholz, Smart, De Weger, and many others.\u00a0 In our project, Chris and I wrote code in SageMath<\/a> (also known as Sage) to solve the S-unit equation in the fields we needed, which meant that we didn\u2019t need to implement all parts of the general method.\u00a0 We wanted to generalize our code and incorporate it into future distributions of Sage so that others did not have to repeat our suffering.\u00a0 I led a group at a Sage Days<\/a> workshop in 2014 to start this implementation.\u00a0 The group, Alejandra Alvarado<\/a>, Christelle Vincent<\/a>, McKenzie West<\/a> and I, made some progress, but didn\u2019t get to the most difficult parts of the generalization.\u00a0 We planned to work on it after the workshop but made no progress–it was nobody\u2019s primary focus and we were all spread out across the country.\u00a0 In spring 2016, Chris and I heard about the Collaborate@ICERM program and put in an application.\u00a0 We had just heard that our proposal was accepted when Christelle came into contact with Angelos Koutsianas<\/a>, a recent PhD from Warwick who had also needed to solve some special cases of the S-unit equation for his thesis project.\u00a0 He also had Sage code that did a big part, but not all, of the general method.\u00a0 Talking to Angelos, we realized that between the two projects there was a complete implementation\u2014we just needed to bring it all together!\u00a0 We were luckily able to add Angelos to the ICERM group, and he was luckily available for the week we\u2019d chosen in January.<\/p>\n

\"Christelle<\/a>

Christelle Vincent, Angelos Koutsianas, me, Chris Ramsussen, Mckenzie West, and Alejandra Alvarado<\/p><\/div>\n

So that\u2019s how it happened that the six of us came to Providence in early January, all having worked on this same difficult problem but never having been in the same room before.\u00a0 First of all, it was a joy to meet Angelos, who had been working mostly on his own on essentially the same problem as Chris and I, and so had dealt with many of the same confusions and difficulties that we had faced.\u00a0 How great to meet someone who has shared the same extremely obscure forms of suffering!\u00a0 We often found ourselves laughing out loud at a shared aggravation with a strangely phrased passage of a source paper.\u00a0 And it was wonderful to work with everybody else again\u2014I love this group!\u00a0 We all bring different skills to the work, and everybody has something very important to contribute.\u00a0 This project would not have been as successful if any one of us had been missing.\u00a0 And, given that we came from Oregon, Illinois, Vermont, Connecticut, Pennsylvania, and the Netherlands\/Greece, this never would have happened without the support of ICERM.<\/p>\n

So this whole blog post is in some ways a giant thank you to ICERM.\u00a0 They did everything possible to support us, beginning with an application process in which we were helped directly both by the previous director Jill Pipher<\/a> and the current director Brendan Hassett<\/a>.\u00a0 The staff at ICERM are extremely helpful and the facilities are world-class. ICERM gave us offices and let us use the entire space for the week our collaboration, as well as installing Sage on the computers for us.\u00a0 ICERM has great technological resources<\/a> at its disposal, as befits an institute focused on computation and experimentation.\u00a0 They arranged for us to go out to a \u201cconference dinner\u201d. \u00a0On top of it all, the place is beautiful.\u00a0 ICERM takes up the top two floors of a building beside the river in downtown Providence.\u00a0 Three walls of the lecture hall and some walls in other parts of the facility are enormous floor to ceiling windows looking out over the town.\u00a0 I gave an introductory talk to the group in this lecture hall and it felt like I was in a movie or something.<\/p>\n

\"Chris<\/a>

Chris giving a talk in the amazing lecture hall. This photo does not do it justice, but at least you can see the crazy glass wall whiteboards.<\/p><\/div>\n

\"Workshop<\/a>

Workshop dinner!<\/p><\/div>\n

In the end we didn\u2019t need advanced computing resources as much as we just needed to be in the same room.\u00a0 After we all got reacquainted with the project, we brainstormed tasks and made a wall of post-it notes to organize our efforts.\u00a0 Then everybody chose projects from the wall and we got started.\u00a0 Of course things didn\u2019t go entirely as planned\u2014several days of work in translating some code into a lower-level language in hopes of speeding it up actually ended up slowing it down (!!).\u00a0 There were annoying bugs, like an inexplicable clash of data types that had to be fixed in every tiny part of the code.\u00a0 We worked hard all week, hoping to have a complete working function by 4:30 on Friday, when our friend and expert in the field Bjorn Poonen<\/a> was planning to stop by and see our progress.\u00a0 When Bjorn arrived we were really close, but hadn\u2019t quite assembled everything.\u00a0 We were tired and hungry and bummed that we didn\u2019t have anything to show, but victory was too close to give up, so we went back to work.\u00a0 Two hours later we were done.\u00a0 We chose a field, and a set of primes, hit enter, and it worked! Bjorn had been hanging out working on an index for his book, so we even got to show off our function. Victory!\u00a0 Celebration!! \u00a0Then we all headed out for pretzels and beer.<\/p>\n

\"Mckenzie<\/a>

Mckenzie West, Christelle Vincent, Bjorn Poonen, Alejandra Alvarado, and Angelos Koutsianas. \u00a0Thanks for cheering us on, Bjorn!<\/p><\/div>\n

This collaboration was a tremendously positive experience for me, and we produced something that will be useful for a lot of other people.\u00a0 A victory all around.\u00a0 However, it did bring up the question of how much the rest of the mathematical world (like, say, a tenure committee) will care that we created this implementation.\u00a0 We are very proud of our work, and I will list the code as a product on my CV, but will it \u201ccount\u201d in the same way that a paper would?\u00a0 I have talked to several computational mathematicians who have been frustrated that their computational and implementation work is not measurable in the same way that other mathematical work might be.\u00a0 Our group decided to look into journals that publish articles on interesting and substantial implementations.\u00a0 There seem to be a few possibilities, but it is hard to find a good fit for work like this.\u00a0 (Which brings me to something I plan to write about next time\u2014what journals are out there for work that is a little different from the norm, or that are especially friendly for early career people?)\u00a0 Whether we find a venue to publish the code or not, we should be submitting it soon for review to be eventually incorporated into SageMath.\u00a0 So I\u2019ll just make a pitch right here\u2014check upcoming Sage releases for the awesome function solve_S_unit_equation(K, S)!!\u00a0 Um, just don\u2019t make S too big, or you might be waiting a while.\u00a0 I mean it works, but come on, we\u2019re not miracle workers.<\/p>\n

What experiences have you had collaborating at an institute?\u00a0 Are there any other programs that you would suggest for collaborations like this?\u00a0 Let us know in the comments!<\/p>\n

\n

[*]<\/a>\u00a0A note on the title: Okay, I guess a second institute-mania<\/a> post is starting a dangerous trend.\u00a0 Now I will have to try to visit all the math institutes and come up with some kind of mania theme.\u00a0 For MSRI I will have to do some crossword style title, and who knows where that will lead\u2026<\/p>\n

[ii]<\/a> Long paragraph about how we got there follows\u2014feel free to skip on if you just want to hear about how awesome ICERM is.<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

*What do you with six mathematicians, spread across the world, who all want to solve the same problem but have never all been in the same room?\u00a0 If that problem involves computational or experimental mathematics, one great answer is Collaborate@ICERM. … Continue reading →<\/span><\/a><\/p>\n

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