Terence Tao: 2015 Breakthrough Prize in Mathematics Symposium

so it's time to start the afternoon session so we'll be having five speakers this afternoon three before the break and then two after the break so our first speaker this afternoon is Terry Tao from UCLA and around ten years ago Terry Thao together with Ben Greene at Oxford they proved this tremendous theorem that there are arbitrarily long arithmetic progressions and primes and that was a kind of classical collaboration with two people and in more recent times it's become possible to have collaborations among many many more people at once thanks to communication on the Internet and so Terry is going to tell us about polymath projects massively collaborative online mathematics unique experience actually this whole event yeah so this talk is not really about mathematics per se but it's about the practice of mathematics and a new rather the new way to do mathematics which I don't think will supersede traditional mathematics at least not for many many years but it's it is one newer alternative way of doing mathematics which is which will be useful for certain types of projects and these are these polymath projects so these are projects which are massively collaborative and in the sciences we now have master collaborations is now very common so so for example in physics we have these these massive collaborations like the discover the Higgs boson is the key paper has 3000 authors attached to it the Atlas collaboration and this was recognized instantly by their fundamental physics prize - two years back or in the life sciences we know we have for example the human genome project which has a mere 237 authors but many of those authors are Institute's which themselves have you know hundreds of people behind them so certainly in the sciences we have really big collaborative projects now but not so much in mathematics so in mathematics it's always traditionally been you know one to all the research people working in isolation and yeah if you compare like just crude statistics like no more quarters in a mathematical paper so I think you know it's it's it's means the number of code the average number of orcas in a paper in mathematics is slowly increased from like 1.1 in 1940 almost nobody collaborated in 1942 about 1.6 in 1998 and a little bit higher than that now I think in the life sciences you have is now four or five I don't have the physics numbers available and the percentage of single author papers has also gone down from 90% to like 50% today so I mean it's slowly getting more collaborative but nowhere near as much as sciences okay so and you know many of the great achievements in mathematics even today I still I still basically mostly one or two people working pretty much in isolation until they're close to completion and then they announced the result after it's all done so you know the famous examples include of course Andrew Wiles who worked for seven years on films Last Theorem pretty much in isolation and he told me but one or two people what he was doing until he finally had not spoken 93 though he did collaborate later which a tailor to repair a gap in the first version of the proof and then a bit more recently a cappella man of course 2.0 a conjecture and the geometrization conjecture also worked alone for many years before an arson was out and then sharing it with everybody and then last year and closer to my own area Jung Hyung also worked for three years on his own you know actually within rather difficult economic circumstances actually before he had his breakthrough was out on burner gas treating primes so there are certainly yeah it's certainly still the case that many of the big achievements in mathematics are it's basically one or two repetitions you know but then there are other types of mathematical contributions which are maybe not so much solving a single massive problem but we're for which there are we certainly do collaboratively I mean famous examples include Nicolas Bourbaki did you know they did many things but one of one of the things about work you did was they they introduced this very influential set of textbooks laying that laying down the foundations of mathematics properly and I mean in some cases some of these textbooks have become superseded with more modern treatments but still some of some of these these books which were written you know by by dozens of mathematicians collaborating under the pseudonym Bourbaki are still very still very influential and another famous example is the classification of finite simple groups known as the enormous theorem that you know we now have this list of I think oh all finite simple groups are one of nineteen families I don't know like five infinite families and and then like nineteen strata coops or something and it's it's a massive massive theorem it's it's there's no single place where you can find the proof it's yeah it's tens of thousands of pages over several hundred articles there is the over like fifty years to to prove the whole thing there's now a project underway to to simplify the classification but still the classification will still be I think about a thousand pages long okay so there are some really collaborative efforts but not not nearly as much as in the sciences and I think one of the reasons for this is that in a science is often the bottleneck is not so much the theory not knowing how to do something often we know how to do it is we don't have the resources you know we need a super collider oh we need this huge investment in how to sequence to sequence genes or something and so often the theory may have been developed for decades but it's only at some point where the technology advances to the point where you can actually do something and and there you really need massive resources of massive amounts of money manpower or physical equipment but in math it's it's it's very rarely like that math if we know how to do something in theory we can probably just do it it's not so much money or manpower which is the problem you know so a typical example is the Riemann hypothesis you know it's not a matter of you know if we even had a greater Institute which had a billion dollar budget to solve the women about the sister and we hired 100 of the best number theorists to do to work full-time on this I don't think we would move the needle that much on this and it would not be a good use of resources it's it's it's a problem in which we don't even know how to do it in theory and and most of mathematics is like that well at least most of the really difficult hunts were promising method I mean there's certainly at lower levels there are things that we know how to do and it's just a matter of getting someone to sit down and actually compute everything it's a great graduate student problems actually to get started yeah but but to make the big breakthroughs often we don't have you know what to do in theory so nevertheless so that of course this is the commercial wisdom so back in 2009 Timothy cowers asked on his blog whether whether this could change whether we could actually do actual mathematical research online in a massively collaborative way crowdsourcing it and so he just posed on his blog just as a question it wasn't well basically musing as to as to whether yeah so instead of having one or two people working in isolation you know have many many people work and the idea so in a traditional research you you work alone or in a pairs or maybe three people and you don't share what you've done until you're almost finished until you've polished up yo yo you're pre-printed you've all the mistakes you can find and so forth you know because you don't want to embarrass yourself and the paradigm that Tim was proposing was quite different where you have lots lots of people each working you know at any given time that each work could be 5 10 minutes half an hour on one SQL problem and if they had an idea that they would say okay this might work I'm trusting it out there maybe it's rubbish and then hopefully someone else will step in and say yeah I think that's rubbish or I think oh yeah that might work maybe if you change it this way you know so to actually share sort of the sort of war mathematical thought rather than the sort of polished type of of thinking that you do there you see in in some traditional research projects all right so yeah basically act like a big online brainstorming session if you wish right so yeah this was quote Isaiah if a large group of mathematicians could connect their brains efficiently they could perhaps soap problems very efficiently as well okay so there was a he there was a really positive reaction to his blog post and said you know we should we should do this why don't we why don't we actually find a good problem and get it started so then within two or three weeks so I think he wasn't planning to actually start one of these things he was just musing but he got so much positive feedback that he okay I will try to organize one and after some discussion they settled on a first problem as a test case something an approach to what's called the density house Jewett problem okay so this theorem is I I decided to not put too much mathematics into this particular talk so I didn't actually give a formal statement of what the Tennessee House cure theorem is but one way to phrase it involves of higher-dimensional tic-tac-toe games okay so so here is a three dimensional four by four by four tic-tac-toe game norson crosses and you win even if you get a straight line in any one of the direction it's possible diagonal hyper diagonal whatever and roughly speaking what the density house-cured theorem says is that if you're playing at a techno game game and the dimension of the game is high enough so this is a three dimensional game imagine like a three million dimensional game four by four by four by four five and you'll be playing eventually one of the two players has to win C this is not true for the standard tic-tac-toe take back to a three by three you can draw but it turns out in high enough dimension you cannot end in a draw that that someone will have to win first and in fact even long before you fill up most of the board if you dimension is high enough even after filling up say 1% of the board already someone has to win and so no matter how you play and so this is what roughly speaking this is the didn't see how Stewart theorem as phrase that just sounds like some some sort of random comment or your fact it turns out to be a very deep theorem and it implies a lot of other theorems in income at works so just to give one example Brian mentioned and the work I did have been green proving that the crimes can have arbitrary long ethnic questions that's essentially a corollary of this theorem plus some other stuff ok but yeah said that it is it is a very deep theorem and it's yes one the deepest results in here occurred Ramsey theory and until Tim Gower's silence project there was only one proof of a theorem and it was very complicated it involved an argument of first important catch Nelson which transferred the problem from coma Toric's to ergodic theory but been a very exotic non standard form of cough cough aquatic theory and and even that theorem was was difficult to prove so there was a there was a difficult paper which very few people I hear ed in completion to prove this theorem and that proof gave no bounds whatsoever so I said that if you dimension is high enough then certain things happen we don't it gave no bound as to as to as to what when that happened so forth so it was not a very satisfactory proof it was great it was a great paper but it was a good candidate for trying to improve it so the project yes so the idea I think if I go back a bit initially the Tim was very cautious yeah so he packed he specifically said I have a opinion yeah it is not the case that the English project is to find the common oil proof of the density how's your theorem he was initially much more and ambitious you know so this was the which is a comment or a theorem but it only had this difficult I've got a clear proof and so it was natural to look for a comment or proof and many people hadn't and it no one found one and so but he initially had just the modest aim of trying to work out why all the standard approaches to to proving this in common Charlie did not work and get some better understanding of all of it but the project really took off I mean it start off slowly that maybe the first day this movie you know maybe ten comments on on his first blog post but the people got interested I I started working on that many other people who started working on it and we tried all kinds of things we eventually found some model problems to work on and which we solve the model problem I got very excited and and after about a month of really intense activity I think they're like a a dozen blog posts and and in each one there's like a hundred or two hundred comments by maybe two dozen mathematicians you know of across the spectrum so you know so Tim and I but but some ergodic theorists some coma Tory lists and some how much human number theorists who amateur mathematician to contribute ISM for example some some some variable computer experiments and we actually found a comet or proof and yeah and we actually wrote it up and and and Tim McKay send it to the annals of mathematics we decided to do you not to actually have three thousand others whatever like like we do in physics or life sciences but we made up a pseudonym the HDH a polymath for density I was cured and I got published in the in the annals as yeah and it we found a simple new proof since then actually one of the participants my former student Tim Austin found another I gotta give a proof which is also much simpler than the first book has Nelson true but you know this was you know forty pages and very self-contained and it gave a good bound in everything so in fact actually in many ways if you want the shortest self-contained proof my theorem a bit on ethnic progression you would use actually this this is part of the argument okay so that was the first party Mathur was quite successful and so of course it was natural to to keep going so we we started the polymath project I guess it was I mean it that's deep I it was that's the official think that we recorded it's not it was not a sort of a really high budget thing in fact we have a zero budget I think you know it was all run with sort of off-the-shelf you know free software blogs and wiki's and so forth but you know we we ran a few more these projects we've had a few things most of them so there's been nine polymath projects so far plus some things that are like poly which we think that we didn't officially call polymath but but there was sort of like polymath we had this thing called mini polymath projects where we would take something like an Olympiad problem sewing which so known to have a solution and you just get a I get a whole bunch of you were to crowdsource the solution and the actual Olympia publicity you have three hours to solve three problems and we have one problem it turns out that actually with the crowdsourcing we were actually slower then there then a good Olympiad student it takes like 24 hours for a solution to emerge but so these are these were fun experiments so yeah we had a bunch of projects unfortunately most of them actually turned out not to be as successful as the as the first one I mean they usually start the same way there's a big burst of activity at the beginning where lots of people are very enthusiastic they they propose all kinds of things and then slowly like different avenues get eliminated some people drop out when they realize that their favorite method is not actually they were committed successful and unfortunate sometimes we eliminate everything and sort of all the approaches we try we hit a brick wall and and then people get frustrated and so Peters out so uh so some have and then sometimes we could partial results but but but we can't get anywhere new to we should not go so I think after nine maybe only three or four have really been being you could categorize the successes I mean we're still trying to work out what works and what doesn't but the most recent successful one and the one that I was really heavily involved in was last year was probably math eight and so this this was the the project on burning gaps between crimes so as I said II sang-joong last year and I think March of last year had this great advance on the twin prime conjecture so patroon graph conjecture says that there are infinitely many many pairs of primes who which differ by exactly two and we can't prove that and until recently we couldn't even prove that infinite many pairs of primes whose difference was bounded but eaten was the first to do that with this really nice argument recombining a lot of previous arguments in in a very ingenious way and he got if they many pairs of primes whose difference who differed by 70 million what's the first finite bound that he got now when he was writing it he knew that that this was not not the best possible this is just what what he wanted to write a nice clean paper and so this was the band that came out I think he said this was out it's not optimal there's another number 3.5 million that is involved it's also crude and there's certain ways to relax it and to a place this is bound by a value as small as possible is an open problem that would not be discussed in this paper so he didn't he as far as he's concerned he was done but this this this proved to be quite irresistible to all kinds of meth addictions you know when you have a really high profile rows out like this and you say no this can be improved but I won't do it you know then it's yeah so so pretty quickly and and and you know because we're in the modern era it's just on the internet people basically all over the mathematical internet and there's now many many sites discussing researched of mathematics started chipping things away so I think a week afterwards a friend of mine mark Duke oh yeah well he put this one equation the first equation here the optimize this into yeah you five minutes work you can change this this the 70 million to 683 million hee hee hee jokes it that you know he it's why I mentioned this in some blog was the something he said there you know I've always wanted you to to mention some of my work on your blog it just maybe not that one he's done more substantial work in mathematics but yeah and then I mean Tim Trojan from a new it made another observation that uses 60 million and then Scott Morrison who I think works in topological field theory you know did another tweak he just had some he's got a programming and he got fifty nine million and then I then I started getting interested yeah and they said how can i connect you squeeze the 58 and then okay and then on scotsblog actually people started conjuring all kinds of improvements there Scott soon afterwards it I can do 57 and at some point you okay yeah then we start optimizing other parts of Jung's paper and say hey yeah now I can get whoever 50 million now forty two million the next day okay yeah sooner like 13 million okay 300,000 so forth okay so that was a four point nine billion suppose so I'm so spontaneously people just started because you know this is in contrast is sort of regular mathematics where you have to work for three months six months a year before you get up before you you can have time you can publish you know this type of thing you can work with ten minutes okay and then you can get the board record for this problem very briefly before someone else steps in it became very addictive actually you know I mean I wanted I came back you know so I was working a much more longer-term project which I kept still not come back to but it was so much easier to make progress on this problem then the other thing I was working on so at this point it was slowly clear that we should do this more systematically maybe we did do it that what you didn't want to see happen was it like some huge scramble where even the archive different people were posting every day I think I can get you know 3.4 million and so there will be a huge mess so I actually proposed that we we make a formal polymath project because we've done things like this before and so yeah so we propose a project to well most ostensibly to improve the bound which we were already doing but also to to actually understand what was going on to to actually figure out how young segments works simplify it and how to connect it to the rest of literature and just to digest it basically and so this sensitives already going this people physically immediately said yes let's do a polymath project and yeah so this ran for about a year and it was really quite intense I mean I pretty much do it for me at least it took up pretty much most my research time for the last four months actually you know and I mean it was the progress just kept you know at times it was improving day by day so for example this is the wiki page where it shows the timeline of improvements the the most and H is the bound of gaps between crimes you know so it starts over 70 million and it just kept going down is it day by day often it will just drop because of our various improvements in various parts of the argument one nice thing about John's argument which was well suited for this polymath style project was it they put the arguments split into like four or five pretty much independent pieces and the final bound was basically you just optimized various parameters connecting these pieces together and so you could you could we spin the groups where each person each group worked on one part of the project and sort of optimize that bound and ever and as soon as they got a improvement that the next group so jumped on that and plug it into their machine and we got this assembly line you know you you make an advance somewhere that just propagates you know five hours later you have a new vote record and so forth it was it was great yeah and so you know so this lasted for a year those of those that it was a quite a complicated story I'll have to take a whole hour to explain what happened here so there's some other methods outside the project also made progress and then they joined the project lots of things happen eventually we got down to a pound of 246 and we published that and that's that's where it stands we can maybe with a lot of computer power maybe shave a few maybe 240 or something but it's we were having significant difficulties getting below there and we know that none of what we can do can ever get below 6 we now have a very concrete obstruction the block sister getting below 6 but we got down to 46 so that was that yeah so it worked really well for us so some of the things that worked well for us for this particular project that we didn't always have in other projects so firstly there was an intense interest in as possible so many of you may already heard about about some of the story certainly in our experience whenever one of us went to a university to do for to give a talk or something there was always somebody who would come up and say what supports the worst world record where you at now okay the people were following it you know I mean it's a yeah it was like a sporting school or something yeah as I said before it was very much other we could split it into smaller problems and you didn't need expertise so there was a lot of difficult mathematics and chunked work like for example they he used the leans working women on the Riemann hypothesis for varieties which not which very few people actually know in full detail but that's the only part of them that only involve part of the argument and there are other people you know not even a number theory you know the who who could contribute one last thing about this about this particular project was that we had a very easy metric to measure progress which is this bout and pronouncements by Ben green that this is a bound which was guaranteed to terminate because there was a decreasing natural number so by the world ordering principle you always stop and the thing is I'm and Alistair I mean if the contact the community of the participants was it was really positive people were very generous for the time and effort and they were willing to make mistakes in public and just share you know things that that didn't quite work and so forth and you know it takes a little getting used to to two-putt is being like this but when everyone else is doing it to it it was a very open and frank discussion and this was very important yeah and we didn't use fancy software really I mean we basically used technology or just available off-the-shelf it could be that every custom design you've spent a couple million dollars or something because some really fancy software maybe we could do things slightly more efficiently than we did but at this point it doesn't really make that's not the bottleneck the real bottleneck actually is getting people to run the things actually yeah so I mean it was really I mean that there was certain serious advantages to to to operating in this fashion like if you needed a computational task like someone reset would do it for you and someone set it up and within hours it was it you could just music I wish you know somebody could compute this and someone were like you do it at some point we needed a web page set up to to collect all these things called code admissible tuples and and and one of us used program for a few days and actually set up this really nice webpage do you know and you know these are things which we're just working by yourself or which other people you know you wouldnt have the expertise to do things like this but there any sort of speed bump that we encountered you know well like if it was a paper that that looked relevant but we didn't have access to it you know within within 20 minutes someone say here's here's a link you know to do this yeah we also got lots of great contributions from people who are not in the area but they were following I mean the day they just wanted to do - to follow so you know I we had a really important contribution from someone who's not a number theorist he worked in a numeric analysis but at some point we needed to solve a variational problem and he had that was his expertise and so he contributed this this is he said we shouldn't be doing this variable in this way you should do it this way you should use what's called - quite the Credo subspace method and I was much I was a much more efficient method so we just kept getting all these contributions like this which if you're doing this traditionally you've probably never even think to ask didn't America analyst for example yeah yeah so yeah so April 12 but but sometimes these things don't work so well I think they do work well if you have you need to work in a project which is fairly accessible to really broad area of the mathematical community if it's a very specialized area and and only 20 people in the world you really know enough that area to action be contributions and at this pointless to have a polymath project so so far most of the projects that work well I have been in areas like remote or X number theory or and like nomads area at least and computer science where with at least the questions are fairly accessible and for which he considered the problems up into toy problems things that work on in a very much other way one problem we still have is that we don't have a good way of recognizing the contribution so as I said we agreed to to write these papers under pseudonyms because we couldn't figure out how to design authorship you know like if if someone makes one comment out of 1,000 and it's it's somewhat somewhat frivolous is this person a contributor or not and on the wiki page we have lists of a food conscious of a soft weed it's a self signup sheet that if you feel like you're a co-author you you put your name on this webpage and we just rely on the honor system to keep everyone honest but for the the but for more publications we use a pseudonym which has got some problems in fact I I put one of these polymath papers on my CV and then the administration sent her back saying you can't do this because you're not the author of this paper so yeah you have to put it under other other activities rather than than actual formal complications so this is potentially a property for junior people for which these sort of citations it's all very important but you got you got a lot of informal recognition you mean you didn't get a lot of formal recognition so a lot of the participants are this particular project you know they reported that everywhere they went we were recognized someone recognized them as someone who was participating because they can follow the blog they can see all the comments so you get a lot of informal technology off of work here but not not so traditional a formal recognition unfortunately yeah and yeah as I said when the big things that you really need someone to moderate you know in the beginning was of naive we thought that that that these projects were so spontaneously self-organized into into the right direction it but this happen you need someone to keep guiding the discussion every so often you make a new blog post summarizing what happened before and trying to keep pushing the direction in the right way this is very very time-consuming and we don't have many people who are willing to do that and so I don't think there's gonna be that many projects is I mean we're still maybe turned out one of one every year or two but it's not a high-volume thing and currently we don't I mean these projects we all the ones that have worked I've worked in cases where we sort of knew - some example to try that there were some things we could do that those that it didn't require a complete breakthrough out of nowhere I don't think we know how to crowdsource anything like that but in an area where there's lots of things to try and what you need is manpower did haven't tried on one by one all the different things and to see what works what doesn't and because of callate all information together this is the type of thing where polymath would work very well yeah so I mean it's still nowhere near the scope you know so this this is nothing like like the Higgs or the genome project but I do think we will see more maybe not polymath per se but I think I think this is part of the future that we're gonna see a lot more collaborative projects maybe more systemically organized than these sort of ad hoc things that we're doing but yeah I think one day we're going to have big mass for saying we were your big science I think thank you very much [Applause] so we have five minutes for sir questions anybody have any Jacob yes how do those get written ah right so I repeat what the question oh so the question is how do these polymath papers get written yes all so I said it took 12 months to do this polymath project I think the actual research time for pie mf8 was maybe 2 or 3 months and by far the longest period of time was the writing and so what often happens is that somebody has to volunteer to do to do a large share of the work to start the skeleton going I just start farming our sections for people to write we used things like Dropbox and subversion and actually what got pretty well I mean we had a lot of very energetic people who wanted to see the whole thing to conclusion and one thing that was very good you get a lot of proof readers for you know so so we have maybe four or five people were doing the bottle of writing but a lot of people were some checking different sections line by line so but it but that takes minutes it'll take a long time to finish it for everyone signs off on it but it does yeah we're quite proud of with them with the writing of that yeah at the end in biology or physics say but it's hard for mathematicians to understand what the math problem is and if like different people come and try to explain it maybe getting a lot of information from different people could help mathematicians find out what the math problem is or if there is any problem to think about right yeah so the question is whether you could use this format for really interdisciplinary activities that's a great question we Tim Gower's once proposed actually I think they the origin of life as a possible method yeah this was this didn't go over so well but well I think you do need a common base so we have had projects which like the pipe eight it span different areas of mathematics there was number theory there was numeric analysis there's some combinatory X but you everyone participating we all came from ethical background and so we had this of common language that we can speak I think with if you want a really collaborative interdisciplinary project there may be more of a culture barrier I think it's it's worth trying yeah you need the right problem though yeah any other Michael the laureates have donated $500,000 to a fund that will allow students to continue to study in their home countries rather than study in in the main universities it occurs to me that that's all if this is with respect to the brain drain this is only one of the problems after they finish studying is they want to continue to do research they find themselves isolated the only people in their countries who actually are familiar with the with the material this seems to be not necessarily as only solving unsolved problems or improving on problems not the only possible application of this this kind of collaboration it would also be a way to it he allow people remain in their home countries and yet be part of an international collaboration not quite sure what the question is but yes I was some of the feedback I suddenly got from these projects that was that for many students but pretty those not working in top-tier universities this was one of the few places where they could see was this seem ethical research behind the curtain Chucky's not to see the finished products in published journals but but this of how the sausage is made you know the yet that you try lots of dead ends and and and you don't you don't know they did it until you try them and you keep you keep eliminating certain things and making progress elsewhere and even you know really top people make a lot of mistakes initially before they find sort of the right approach so a lot of like a lot of graduate students I got a lot of feedback from undergraduate and graduates saying this is very revealing so I could imagine particularly for students in the developing world maybe or for for people not served in the center activity of elite universities yeah they could there could be some some value in in having these these open so mathematical projects and beyond just of the research aspect yes although again there's this issue of assigning credit and so forth which we haven't solved yeah maybe if there's some formal way to a credit to say yeah there could be some way to do have some computing encouragement for the participation but I mean with these projects that was so important though that that we don't just sort of said you know round up 100 random people and say okay you're all course of work in this project I mean it's important that people volunteer themselves and it's not sometimes not obvious that the expertise is needed until the party reaches a certain point and then and and and then if they find out that what's needed at that what they do is exactly what's needed so I mean I it's not like you could sign up one of these things and expect that you could contribute without unless you really knew the first that there was a good fit between what your expertise and one of these problems one more question [Applause]

Channel: Breakthrough

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Keywords: Terence Tao, Mathematics, UCLA, Breakthrough Prize

Id: elWIDVI6b18

Length: 38min 29sec (2309 seconds)

Published: Thu Dec 04 2014