 I'm going to bring up someone who will introduce tonight's speaker and our introducer tonight works for Google Brain Robotics. Did I get that right? And that sounds pretty exciting and so he'll tell you a little bit about our speaker but what I also want to mention about our speaker is he's not just a wonderful mathematician but he's a very generously spirited person who you will hear from our introducer about their relationship but I will also just say on a personal note Joel Spencer tonight's speaker was a mentor for one of my three children and helped her develop a love of mathematics at a very early age and so I'm very grateful to Joel and very delighted that he's here tonight but first let me turn it over to Kristoff who will share his own experiences and stories about Joel. Thank you. Thank you very much. Good evening. It's a great pleasure to introduce Professor Spencer and I gave one a short like introduction at 4 p.m. session so I promise for those who attend this one it will be I will try to make it different. I mean I cannot like change it completely because I mean Professor Spencer is simply a brilliant mathematician and a wonderful person but I want to talk about a couple of different things that I didn't mention before so yeah just like a Cindy like said like I work like in Google Brain Robotics New York but I want to tell my story how Professor Spencer changed my life and it was like quite some time ago when I moved to New York and summer internship actually like at IBM not Google that time and I met Professor Spencer I was very interested in mathematics in random graph theory but of course I didn't have like any results yet he was the one who encouraged me to apply for the PhD program at top American universities and I pretty much followed his suggestion I ended up at Columbia University wonderful PhD program with Maria Shudnowsky who is a very good close friend of Professor Spencer another wonderful mathematician and so pretty much like from the very beginning of my career as a researcher Professor Spencer played a very important role as an advisor like in terms of mathematical support but also like giving like very useful life advice which is like even like more important so he's definitely a wonderful mathematician I mean in the giant like in graph theory and combinatorics I remember reading papers and books like we've proofs move results from random graph theory out by Professor Spencer before I met him so it was like a great pleasure like to work with him in person like at the very beginning like when I moved to New York and then we started like talking about mathematics working he was never officially my advisor but I consider him my academic grandfather so so I was like working I spent quite some time working like during my PhD on one of the Erdesh conjectures actually Erdesh Hainal conjecture Professor Spencer was a very close collaborator of of Paul Erdesh it was wonderful to talk with him about this stuff lots of craft theory he actually inspired like one of the papers that that was like we later like published pretty much predicting an existence of the very specific family of graphs with some very weird properties it turns out those graphs exist and we put it in a paper so it was like wonderful to talk with him about mathematics but really the very important thing is that he taught me how to think about mathematics think about it as an art it's an art right like if you if you really work on that you think about it this way you look for beautiful ideas you you know if you work like as a mathematician you never like think about you have like those ideas like come like in very unexpected moments and and you know do not necessarily need to have like a piece of paper like with you to to think about them so so he taught me like how to how to think them how to come up like with really nice beautiful concepts and and I was like really like very lucky to to have him as my mentor you know like as people like say like when you stand on the shoulders of giants you always see more and I was very lucky and I'm very lucky to talk about mathematics to work on that with Professor Spencer and he's definitely a giant of mathematics and ladies and gentlemen Professor Joel Spencer and his beautiful mathematics right now. Thank you Cindy and thank you Kristoff for those very kind words you know hearing this this famous phrase of Isaac Newton if I have seen a bit further it is because I have stood on the shoulders of giants and Peter Winkler a mathematician at Dartmouth has a very nice parody of this that's quite appropriate since I'll be talking about the Hungarian mathematician Paul Erdisch he likes to say if I have seen a bit further it is because I have stood on the shoulders of Hungarians and in the mathematical world I think there's there's some truth to that there's some truth to that Leopold Kroniker the great 19th century German mathematician said it best the Ganzitsalen hat der Liebe got gemocked Alessandra is mentioned or in its most beautiful translation God created the integers the rest is the work of a man the work of man the work of mathematicians the work of Paul Erdisch Erdisch was the quintessential 20th century mathematician 1913 to 1997 he was a man with prodigious gifts his mathematical theorems and conjectures were enormously influential and indeed the influence has only increased over time he was also extraordinarily fortunate he was fortunate that he was born in 1913 in Budapest Hungary and the Hungarian community at that time and even today but really especially this was a particularly golden period where they gave tremendous support to those youngsters that had conceptual abilities that had mathematical or scientific abilities there is a long list of people from that era John von Neumann Edward Teller George Polia Leo Salard a bit later let me mention my colleague and friend Peter Lacks there's a wonderful joke that John von Neumann was chairing the advisory committee to the United States atomic energy commission and he looked around the table and he said oh Einstein is not here today now we can speak Hungarian apocryphal but who knows which brings me to a non-mathematical joke but one of my favorite language jokes so I just love to tell it and it concerns Billy Wilder some of you are too young Billy Wilder was a great German movie director who then became a great American movie director among his films is Irma Ladoos so I'm really dating myself here yeah and some like it hot is also very good that's right many many great films and the story is that he had a Hollywood party and he invited everyone including the Hungarian screenwriters and you may not know that the Hungarians had a tremendous role in the screenwriting in those golden days the 50s 60s into the 70s in Hollywood including most notably in the 40s with the writing of Casablanca but many many other films and so the screenwriters came in a group and as they often did they moved into a corner of his palatial house and just started talking to each other in Hungarian well Billy Wilder just got totally incensed and he ran up to them and said stop it stop it you are in America now speak German to good story I hope it's true because it's a great story so Erdisch was a Choda Jerich a child prodigy and this was something that was greatly prized and still greatly prized in Hungary indeed we see it in every land we see people that are child prodigies and he was he was one one of his earliest memories I remember him telling me about this was of taking 250 from 100 and saying that the answer was 150 below zero it's quite a conceptual leap and by high school he was doing research mathematics there was a key paper that was joint work with George Sekeresh and Esther Klein and it was worked on when he was 19 years old the three of them would walk the beautiful hills of Budapest and talk about mathematics it's very melancholy to read the description George Sekeresh wrote about it some 50 years later because we know the horrors that were to overtake their world and indeed the world as a whole and there they were not he was the youngster at age 19 blissfully unaware of the future also at age 19 Erdesh's fame spread well beyond Hungary in 1852 Chebyshev chebyshev proved that there's always a prime between any number and it's double well except for one and two so between two and four there's three between ten and twenty there's eleven and thirteen and seventeen and nineteen so often there are many many more than one but he gave a mathematical proof that there's always a prime between a number and it's double then Erdesh in so it would be 1932 so 80 years later he proved the same result but he gave a much more beautiful proof a more elegant proof more what he called and I'll get back to this the book proof and it was immortalized with a marvelous doggerel chebyshev said it and I'll say it again there's always a prime between n and 2n and with that he became he entered the world stage of mathematics at the age of 19 and he remained there proving in conjecturing until his his death in 1997 indeed he was at a conference in Prague and he was planning to go the next day to another conference now I'm forgetting I think it was Estonia but I'm not quite sure and that's when he passed away so as we like to say he died with his boots on he had a really wonderful wonderful life one really wonders just what it is in a culture that leads to great mathematics to my mind the abilities are everywhere but certain cultures really foster this intellectual thinking this and in particular the mathematical thinking and many people have wondered why is it that this time in Hungary was was so good for mathematics and there are all kinds of theories but I'll give you my humorous theory about why it is which is that when the young so I I'm what's called Tistelet Magyar Honorary Hungarian because I started going in 1976 and I never stopped and it's just become this great enriching of my life really wonderful part of my life and so I go to people's houses and see their families and I see them with their with their babies and when they look at their babies and they smile they don't cry out yeah I did she knows isn't that a beautiful baby instead they cry out yeah I don't coach isn't that a clever baby and there's something there that the cleverness so it's a half joke but who knows what's there but it really is true that there's something that the culture prizes the cleverness one can think of a similar thing with Raman Nuzion you know there's this wonderful book the man that knew infinity by Kanagol that was then made into a movie about Raman Nuzion and his relationship with Hardy and he grew up in India and he grew up poor but he grew up in a culture that really prized mathematical thinking I think this came comes through more in Kanagol's book than then in the movie but was definitely strongly there that the the culture very much supported this intellectual endeavor and to my mind this is a critical thing that some cultures really support mathematical intellectual thinking the idea that you don't have to be doing this in order to build a but better mousetrap you're doing this because of the the joy of the the purity of thought and Hungary had it in 1913 and India or at least his part of India had it for Raman Nuzion when he was growing up and I was also quite fortunate so I to get personal was born let's hear it Brooklyn yes okay but but sad to say my parents were the adventurous sort and they were always looking for new horizons and so they joined the great migration and they moved to Queens yes yeah so I grew up in Queens it wasn't so bad actually I have to say I had a very good I still miss and I grew up in Queens village off near the the Nassau County border I went to a wonderful public high school it's called it's still there today Martin van Buren high and I can always I'm very grateful to the Soviet Union for Sputnik I was I was just in Moscow and I told them I was so happy about Sputnik because because why I was at the perfect age and when Sputnik came all of a sudden math was cool and if you were on the math team which I would have been anyway suddenly math and science were exciting fields and it totally revolutionized for kids like me the education and so I was very very fortunate and I was also fortunate that van Buren was an excellent school though I have to say that Bronx High School of Science Brooklyn Tech Stuyvesant they always beat us they always beat us so so but sometimes we were number four in the math competitions and then I was very I think today there are many more schools but at that time those were the three schools that were in math and and science also I had a great math teacher you know you only need one and I was very lucky I had one his name was Ira Ewen he was a teacher and then principal in the New York City public school system for many many years and he showed me the beauties of mathematics and if you want to do beauty if you want to do mathematics beauty is at the very center here's here's a quotation from G. H. Hardy the man that is also in the the film that worked with Ramanujan and Hardy was considered the premier mathematician of his time at the turn of the last century 1900 1910 fact for many decades but he wrote about this and he said the mathematicians patterns like the painters or the poets must be beautiful the ideas like the colors or the words must fit together in a harmonious way beauty is the first test there is no permanent place in this world for ugly mathematics now I'm not sure whether that's literally true about ugly mathematics but it certainly is a beautiful sentiment when I was in high school my teacher Ira Ewen showed me a conjecture that had an enormous influence on me it's called the twin prime conjecture I think most people here know what a prime number is like 7 or 37 a number that can't be factored into two numbers unlike say 91 which is not prime because it's seven times 13 so otherwise the numbers are prime numbers and twin primes are simply two primes that are two apart so 11 and 13 they're both primes so those count as twin primes or 101 and 103 or 571 and 573 well thanks to Kristoff's colleagues I could go on Google and find that one and there are many many more but what is the conjecture the twin prime conjecture is that the list goes on forever that there are an infinite number of twin primes but I emphasize the word conjecture because we don't know if this is true we don't know if this is true now with computers we have enormous statistical evidence and the evidence the statistical evidence is abundantly in favor of the twin prime conjecture we can compute we can check for twin primes out to 10 billion one of our youngsters here can get on their cell phone and compute it out to 10 million by the by the end of this session and and you'll see that there are lots and lots of twin primes and this and even they seem to they this is less clear but they seem to have a pattern in their frequency that we can conjecture but all of that is conjecture that's not a proof so there are lots and lots of twin primes out to 10 billion that we computed so the number is a lot bigger than the 1 million say that was computed years ago before computers but maybe when we get to a Googleplex in three suddenly the twin prime stop and for some reason we don't understand there aren't anymore now I don't think that's going to happen in fact very few mathematicians think that's going to happen but mathematics is about proof and we don't have a proof that the twin prime conjecture is true and we don't have a proof that the twin prime conjecture is false it is indeed a conjecture as I thought about it when I was in high school well I mean I tried to do something with it but I didn't get very far so I joined the club as it were but as I thought about it I had an epiphany and I used the word deliberately the epiphany was well the twin prime conjecture is either true or it's false and this is an absolute this is an absolute so to explain what I mean let me compare mathematics to the sciences and generally mathematics is not considered a science and it has the special absoluteness think of biology that's a fine subject and nothing against biology but is it absolute well you know we could imagine another world where we had silicon-based life forms indeed sometimes at Google it seems that's already the case but but so it's not an absolute thing about biology or even think of physics which is closer so here I have to stretch a little bit more but still you could imagine a universe where instead of an inverse square law you'd have an inverse cube law for gravitation so but in mathematics here is my faith here is my faith that the twin prime conjecture is either true or false and whichever one it is we could be on another planet we could be another species we could live in a different universe in an alternate universe it would still have the truth same truth value that is my faith is that I cannot conceive of you one universe where the twin prime conjecture is true where there are an infinite number of twin primes and another universe where it's false where there only a finite number of twin primes perhaps because the God the Ganzitsalen hat the Libhagat Gamak that God created the integers the integers are there and they're there and they transcend time and space let me give a wonderful quotation about this transcendence not from a mathematician but from a incredibly great author Jorge Luis Borges South American author and he's a mathematician's favorite even though I've never found any calculus formulas in his books but he has a way of thought and he wrote a short story that I highly recommend it's called the library of Babel it's in his collection fiction ace and in this story there is the protagonist is the librarian and the librarian goes through this universal library and he's searching for what eriders will call and I'll come to it soon the book he's looking for what Borges calls the total book the book that has all knowledge in it and at the critical moment I carried this passage with me in my wallet for about 20 years it's just for me it's very moving and so the the the librarian cries out he's searching and searching and he says to me it does not seem unlikely that on some shelf of the universe there lies a total book I pray the unknown gods that some man even if only one man and though it have been thousands of years ago may have examined and read it if honor and wisdom and happiness are not for me let them be for others may heaven exist though my place be in hell let me be outraged and annihilated but may thy enormous library be justified for one instant in one being well to me this and to many mathematicians this resonates tremendously for Erdisch the Erdisch philosophy theology one could call it non deus theology but theology nonetheless was in what he called the book the book and in the book as he described it were all the theorems of mathematics and in the book in my picture on the left page there was the the theorem and on the right price page there was a proof but mathematicians know that when you have a theorem often there are many different ways of proving it but in the book this platonic structure that that Erdisch talked about there was only one proof and this was the proof that was the most beautiful proof the most sublime proof perhaps not the shortest proof but the proof with the deepest understanding and he called it the book proof and this happened very frequently that there were problems that people were working on and working on and then finally someone came up with the solution and Paul would be very happy he was very generous about mathematics for him it never mattered who proved the theorem I mean he would celebrate that person but the important thing was that there was this discovery so he would be very happy and tell everybody that the result had been proven but then he would say well now we have a proof but now let's search for the book proof let's search for the proof that shows the core ideas in the most beautiful way and this this idea of the book was a very powerful idea for all of us in Erdisch's circle I like to think that the circles I'd like to think of concentric circles around Erdisch at the inner inner circle there were there were four people there was Andres Hoinal, Vera Shoesh, Paul Turan and Ron Graham the one American this is in my mind and then there were concentric so I like to feel I was in the second cent circle it's my own hubris and then a third circle with some hundreds and then a fourth sir sir like Beijing the rings keep building up more circles and then in the end there were just these thousands and thousands of people that had the contact with Erdisch and were were deeply influenced by him I mean certainly people were influenced by his great mathematics but equally so they were influenced his the power of his personality and his spirit just spread he was somebody recently described him like an apostle from the Catholic Church and he was like that if you saw the apostle well you know then you would get the faith and then you would you would spread the news and and he was like that he was like that and certainly I feel that when I'm talking to young talent that I just hope that I'm able to do a fraction of what Paul was able to do to thousands and thousands of people mathematics has a wonderful international aspect to it and Paul was epitomized this he was constantly traveling here's a wonderful quote from a great logician and mathematician Julia Robinson she said I think of mathematicians as forming a nation of our own without distinctions of geographical origins race creed sex age or even time the mathematicians of the past and you of the future are our colleagues too all dedicated to the most beautiful of the arts and sciences I had the good fortune of knowing Julia she very sad she died quite young but I I didn't know her for a few years and she had a wonderful wonderful spirit and did great great mathematics let me give another favorite quote of mine that just for fun to get off the seriousness of this is from but my favorite movie everybody has a favorite movie no it's not Terminator 2 no although that's high up on the list it's Butch Cassidy in the Sundance Kid used to be Casablanca but I've moved on to Butch Cassidy in the Sundance Kid and it's got millions of great great acting and beautiful cinematography and it's a buddy movie and with with Redford Robert Redford and Paul Newman at the very heights of their careers and Catherine Ross playing sort of the female foil also doing a wonderful job but there's so many great lines but my favorite is at one point Paul Newman who who plays Butch Cassidy is coming up with ideas about their bank robbers about how to rob the banks and what to do and Robert Redford who plays Sundance Kid turns to him and he says you just keep right on thinking there butch that's what you're good at so that's what I think of myself sometimes that I shouldn't get too high in my day I should just keep on thinking and hope hope for the best sometimes you get answers mathematics it's a field where where you know if banging your head against the wall is something you enjoy mathematics is for you because you know you you you work on problems and the successes are wonderful but most of the time you know you you don't get success there's there's another quote by Kelvin you know the the Lord Kelvin who who is famous for the the Kelvin temperature scale but but also did not just that I mean he was a great scientist and he said one word describes my most strenuous efforts for the advancement of science and that word is failure and I've always taken to that I think you just have to see it in a positive way you have to see that you know you're struggling so like Jacob wrestling with the devil you know you know or with the angel whichever one it was I flunked Bible I'm sorry but it was the angel but you're Jacob struck wrestling with the thank you with the angel you know so you know most of the time the angel wins but still you keep on trying and and when you get successes it's it's really sweet so let me mention in terms of the twin prime conjecture well it's still open but I want to mention a success and this was just recently well recently 2012 so maybe not recently for the youngsters in the crowd but for most of us that was recent and it was done by a mathematician whose name is Yitang Zhang and in fact there are a number of documentaries about this you can go on on YouTube and look up Yitang Zhang one of them is by George Chitry and it's called Counting to Infinity and it tells his story and he had a PhD in mathematics he was at University of New Hampshire it was really quite interesting he didn't even he didn't have a full faculty position you know he was he was teaching there and people knew he was kind of a bright guy you know he was working on this number theory and he thought and thought about the twin prime conjecture and no he didn't solve it that would be a much better answer but the twin prime conjecture is that their infinite number of pairs of primes that are two apart and he proved that there are infinite number of pairs of primes that are within 60 million of each other and this was a tremendous breakthrough because the 60 million was a constant and nobody knew that there was any constant before so well 60 million is kind of a large number and since then enormous numbers of mathematicians have worked on this to try and refine his ideas and to get new ideas and he continues to work on right now I believe the current result is that there infinitely many pairs of primes that are within 210 but I may have the number wrong and it's an ongoing thing where people are doing for again it's not the twin prime conjecture that there are an infinite number of pairs of primes that are two apart but we feel that this problem is not a brick wall we feel and we may well be wrong that that there's been progress and working we're getting new techniques the new techniques are giving us some results even if they don't give the full result and so there's some hope that maybe not in my lifetime but maybe in the lifetime of some of the younger people here that the twin prime conjecture will be solved but maybe not the thing with conjectures is you just don't know you don't know erdish frequently asked problems and he had an interesting element that he would give cash prizes for his problems but you know after he died some people tried to duplicate but but it couldn't be duplicated because he would ask problems first of all they were open problems they weren't exercises they were open problems now in mathematics it's fairly easy to come up with a question that somebody will solve at lunch and it's also not so difficult to come up with a problem that nobody will solve in 500 years but erdish had a way there would be some area and he'd be working on it working on it and he would get to a point and he would have some results but he wanted but he didn't have the full solution so he would ask a question and in order to solve that question you'd have to come up with a new idea but not ten new ideas maybe one or two new ideas and so you would advance the subject and one of the things that I was very very proud of when I was quite young was I managed to solve an erdish problem and I got a check from him and it was a check for ten dollars it was it was so so nice it was really a wonderful wonderful moment I was told I had to people weren't cashing his check checks and so they it was it was a big big mess so I was told no you have to cash the check so I Xerox the check and I put it and it was on my wall for many years the Xerox of it he asked questions with various amounts let me mention one question that was his favorite question and I will slightly simplify it let's see if that's okay so this question deals with arithmetic progressions so arithmetic progressions are like 13 19 25 31 37 so the the difference is the same so that's called an arithmetic progression it can start anywhere and you look at how many we're not talking about infinite progressions we're trying but but he was interested in long arithmetic progressions and there are a lot of long arithmetic progressions and he felt that if you had a set that had a lot of numbers in it make it more precise if you had a set that had a lot of numbers in it then it would have in it long arithmetic progressions and more particular now here I'm putting in particular numbers but let me say this suppose you have a set s and it's it contains it's in the first n integers and think of n as a Googleplex as some really really large number but suppose s is a really large set that it has a third of the elements so that's got lots and lots of elements then what he said was no matter what set you pick in that set you'll get an arithmetic progression of 50 terms of 50 terms now the actual conjecture the 50 could be any particular number and the one-third could be any particular number but this is an illustrative illustrative example and he made this conjecture he made it along with his colleague Paul Turan they were great collaborator collaborators Turan was a couple of years his senior they were both great Hungarian mathematicians and they came up with this conjecture so it became known as the Airdish Turan conjecture and people had already arithmetic progressions are a very natural thing to to study when you're studying the integers and it was a very early conjecture he came up with it in the late 30s and he offered I'm not sure how much he offered originally for if I'm not even sure if he was offering money at that time but when I came on the scene he was offering a thousand dollars which and that was his his biggest biggest prize so mathematics you think well all these people have tried it and nobody succeeded I mean there must be just something impossible or maybe it's false or we don't know sort of like the way we think about the twin prime conjecture now maybe we're just not gonna have any success but it turned out then in I believe it was 1971 plus or minus a few years a Hungarian mathematician Andres Szemeredi proved this statement so n has to be a big number maybe a Googleplex I mean exactly what now how big n has to be was part of the proof but let me just say for some large number n like the Google notice that if it's a Googleplex the set is contained in the first Googleplex numbers but it's got an enormous number of numbers in the set oh sorry the the vertical lines just means size that's mathematics for size means the number of elements of s is at least a third of the numbers from one to n are in the set that's what it's saying and so this was proven by Andres Szemeredi in 1971 and it was a great great result so one of the great results in discrete mathematics very very involved proof and in 2002 Szemeredi was awarded the obel prize for this work and some other works but this this played he also this wasn't the only thing he did he was it wasn't is a great mathematician but he was so the obel prize some of you know that there's not a Nobel Prize in mathematics but in the early 2000s it was decided to create a new prize and so Norway which has some of the Norway and Sweden have shared the Nobel Prizes different ones are awarded by different countries but Norway decided to create the obel prize and I had the the wonderful good fortune to go as I a friend of Szemeredi's and we've worked together of course he solves the thousand-dollar problems I solved the ten-dollar problem so not quite at the same level but but he's a wonderful wonderful human being and I had the good fortune to go to Oslo and he was presented the prize by the King of Norway and it was really a beautiful thing and here was this problem just this beautiful beautiful problem that remained open for decades and decades and decades and then finally it was solved of course there are many other problems that remain unsolved so let me give a problem that Erdisch offered ten thousand dollars for but it still is unsolved to today so it has a little bit of notation so I'll try and do it slowly so again we're going to want arithmetic progressions and now we're going to have a subset of the integers and I'm going to make an assumption about the set so this is a little bit technical I'm going to assume that if you sum the reciprocals of all of the elements of the set that it becomes an infinite sum now some of you may have seen that if you take the integers themselves and you sum one plus a half plus a third plus a fourth plus a fifth it the sum becomes infinite it's an infinite sum on the other hand if you take the squares and you sum one plus a fourth plus a ninth plus a sixteenth plus a twenty-fifth then the sum is finite I think it's pi squared over six one of those strange things in mathematics but here the point is it's fine so sometimes the sum is infinite and sometimes the sum is finite and Erdisch is three thousand dollar conjecture or ten thousand dollar conjecture actually now I'm forgetting it's only seven it's the it it's the glory that's important not the money that if the set has this property so there have to be a lot of elements in it though it's a technical property so like the squares are too sparse so there some of the reciprocals is finite but all the integers then the sum is infinite and the conjecture of Erdisch was that if s is so big that it has this property then s necessarily contains an arithmetic progression actually not just the 50 terms but for any number any fixed any fixed number of terms after all it's an infinite set so even of a thousand terms somewhere in the set you may have to go way out far to to find it there is this progression and this remains an open problem but again there has been tremendous progress one of the key examples of the set here is the prime numbers and it's not easy to prove but it is true that the sum of the reciprocals of the primes so a half plus a third plus a fifth plus a seventh plus an eleventh plus a thirteenth plus a seventeenth plus dot that that that is an infinite sum it's not obvious after all some sums like the sum of the reciprocals of the squares is finite so it's not clear which things are finite in which thing but this was known that this sum was finite was infinite so this was a key example so in particular if you prove this conjecture one thing it would say was that the primes contain arithmetic progressions of length 50 indeed no one's found an arithmetic progression of length 50 in the primes I think the record now is about something in the 20s and this is with massive computation but there was amazing progress and in 2004 the mathematicians Terry Tao who's well again the international nature Chinese heritage Australian upbringing and he's at UCLA and Ben Green who's English English and English and in 2004 they didn't prove this conjecture but what they did prove was that the primes contain arbitrarily long arithmetic progressions so this was an open question that people had worked on for a long time and it was finally resolved by Tao and Green and in part for that Terry Tao was awarded the our our biggest award for young mathematicians for mathematicians under the age of 40 it's called the Fields Medal and he was awarded the Fields Medal and actually he did Terry Tao has done many many great things but this was one of the many great things that he did and was mentioned in in his award of the Fields Medal so in mathematics we have these problems they're open problems people struggle with them for years and years and years but sometimes we we get successes we get successes so it's nice to talk about the successes and not all the years and years with that so now let me turn to the game and Tony where are you Tony come on up Tony it's like the price is right oh brings bring some chips sorry I don't have any chips you bring some chips and we have a microphone for Tony okay so okay so Tony I'm gonna guess where you're from just by looking at you I'll guess that you're from where I was born Brooklyn no I get two guesses and then you can tell me well if it's not Brooklyn no New Jersey no I give up Manhattan okay well you've come a long way baby all right let's let's try a game and we're gonna try this game and then we'll we'll have some time and we can all play with it and we're gonna try variant one so we're gonna start with five blue chips and we're gonna have five rounds and we're gonna have two players Paul and Carol and Tony would you like me to pick or do you want to you want to say which one you want to be either one Paul Paul okay so Tony is Paul and I will be Carol and so we're gonna have five rounds and Paul is going to split the chips and then I will in we'll just split them over here so we can see them up on the board and then I will take one of the piles and my object is not to take a lot of chips because if there are two or more chips left at the end of the game then I have won the game so I am Carol oh and then the meta question for those of you that are good at anagrams is we spell Carol with an E so you might think of the anagram while we're doing the game so okay so I am Carol so you're Paul so you can split the chips okay so he puts three and two now I want him to keep chips so what I'm going to do I don't want to take chips but I have to take one of the piles so I'll take two of these but when I take a blue chips are special when I take a blue chip I have to return a red chip try it again now he takes those two and okay I will take these but I've taken two blues so I'll give you back two reds so that's at the end of two rounds we're going to have five rounds and Paul as we now call Tony is trying to get down to one chip he can't get down to zero chips because when he splits it I if one was empty I would just not take I would just take the empty one see it's impossible for you to get one chip all right you're doing this I'll take these two okay now there's you've almost won because they're just two chips ah but I will take this one so you still have two chips but there's one round left and the winner is Paul so congratulations well done so now we'll take a let's take around 10 minutes and please help yourself and I'll be walking around if you have any questions about the game I'll try and describe the rules let me end with just some erdish anecdotes and then be happy to open things up for questions but one of erdish's favorite phrases was I have no home the world is my home and he lived his life in a special way this was literally true that he would just travel from place to place to place he would come to my house with my family and he would come for at that time I was at Stony Brook so he would come out to Long Island and he would stay for three or four days and he would talk to me he would talk to his Hungarian mathematician friend Peter Seuss and many many other mathematicians at Stony Brook and he would do that for three or four days and then he would fly off to Michigan and from Michigan it was the Los Angeles and from Los Angeles it was to Melbourne and he just he just traveled from place to place and wherever he went there were people there to welcome him and welcome them him into his their homes like we did besides his mathematics he was a wonderful guest he always asked about the fan he had a prodigious memory he knew so he always asked about the children and he always brought gifts and he's also always concerned about how the family was it there was only one thing about it that about him as a guest that was a little bit difficult which was that he would wake up at five in the morning and I'd be and I heard that this happened with many other people and I'd hear this knock on my bedroom door and it would be Paul and I'll try to give it yet this heavy Hungarian accent which I'll I can speak some Hungarian but I I'll try and do the Hungarian accent in English which is Joel Joel is your brain open well it was five in the morning Paul I mean most people wake up a little bit later but no he was awake but one one further thing I want to say is that is that one thinks of Paul of course he was intensely involved in mathematics that was his profession in his life but it would be wrong to think that he was only interested in mathematics he was supremely interested in history and he was supremely interested in politics he was totally international so he knew people all over the world and so he would discuss the history of the different countries which he knew backwards and forwards and we talk about the politics not just when he was in America the politics of America but the politics of of the Soviet Union and of France and of Australia and of England and indeed of the entire world so in that sense I think it's important to think about him that yes mathematics was the center of his life but he was a man that was really interested in life and had this tremendous interest in people sometimes the popular press quite naturally emphasized his idiosyncrasies to be sure waking up at five in the morning was was one of them and so he did have them but to me I always think of him as the most social of mathematicians he had this wide set of friends all around the world and wherever he went there were people that were there to welcome him to do math with him to talk with him to laugh with him and I miss him I miss him so thank you for listening and we'll have some questions too I'm sure somebody have some questions they will ask in private of our speaker but why don't we ask them in public right now so I know you've got a question out there thanks for your talk why did we play this game sorry why did we play this game oh why did we play this game this was a version of something that that I and many other people have written a lot of papers on called the liar game where you're trying to again they're called Paul and Carol and it's like 20 questions where Paul's trying to find a number from say one to a hundred and he has 10 questions but Carol can lie with her answer but she's only allowed to lie once and it turns out that this is mathematics is filled with these things that look completely different than yet or the same this is really the liar game if you think that I'm thinking of a number from one to a hundred you just take a hundred blue chips and mark them from one to a hundred and then when you ask the question is the number less than or equal to 50 you put the numbers less than or equal to 50 on one side and the numbers greater than 50 on the other side and then when Carol says yes she keeps the numbers from one to 50 but the numbers from 51 to a hundred don't get thrown away they become red chips instead of blue chips because she's already lied for those answers so it's a little subtle but it turns out this thing with chips is the same as the liar game and then to back up yet another step my first job was at AT&T Bell Labs at Murray Hill New Jersey wonderful spot and they were interested in sending messages they're the telephone company what else are they gonna do so they want to send messages so let's think of the messages with bits zeros and ones but occasionally a bit would get garbled and a zero would come out a one and a one would come out a zero and what they wanted to do was design systems so that even though there was say one garble it would still be correctable they would have a system like these this was used most famously on the Mars one of the Mars rovers for transmitting things to earth and it turns out that that is quite this changing the zero to a one or one to a zero is like a lie and so it actually comes for me personally back to my very first job at AT&T Bell Labs there are more than one anagram to be made from the word Carol name Carol no who got the anagram you got one oh tell say what it is Oracle Oracle is the anagram you got it right maybe there are others I don't know but Oracle is the one that I'm thinking of and so Carol is like like the Greek Oracle she always answers the questions but oh so this is in the liar game format where Paul is asking the questions and Carol is answering and like the oracles in Greek mythology she always gives an answer but it's not particularly helpful so so we felt and thank you to professor Ron Rivest at MIT we were just we wanted to use Paul because it was Paul Ertis and the natural thing would have been Vera because Vera's show she was one of his closest collaborators but somehow it didn't I don't know we didn't like it or whatever and so we came up with Carol and we didn't like that either and then finally Ron Rivest said oh just add the E and like you said you get Oracle so so we stick with now the names of the players are Paul and Carol forevermore we're not changing have there not been or will there not be other vagabond mathematicians or was was this the one and only well that's a good question I mean to my mind there are a select few at this very very top level that Ertis was at so in that sense he's not unique and then there are a number of people that are total vagabonds so in that sense he's not unique but my feeling is that if you look at the intersection of the two then he is unique at least I don't know of anyone else that does such great mathematics and had such a unique lifestyle you said that you were you in the second ring yeah you have an Ertis number of two one one because it because I did write joint papers with Ertis and the rule with the Ertis number is if you write a joint paper even just one joint paper though I've written several if you write a joint paper with theirs then you have an Ertis number of one to get an Ertis number of two you just have to write a paper with somebody that's written a paper with Ertis it's her like there was an old song I mean old like before my time I danced with the man who danced with the girl who danced with the Prince of Wales it sounds like the Ertis number but actually I have to say there've been occasions when I've been talking to some mathematician and they keep saying oh we got this result oh we really should write it up and I really feel like what they really want is the Ertis number of two they're really not interested in me at all but yes my my number is one all right then let's give our speaker one more hand