New Theories Reveal the Nature of Numbers

this program is brought to you by Emory University you so I'd like to begin by saying if you're worried about this at all I'm going to have a great time standing here giving this lecture and I view this as a very public lecture so from time to time I'm going to say things that are a little bit inaccurate perhaps completely absurd but I think by the end of the lecture you'll have a very clear picture of what it is that we've done uh two of my three collaborators are here tonight Yann is lecturing in South Korea so he is unable to be here but he very much wishes he could be here I'd like to have Amanda and Zack Kent stand there are two of my most recent postdocs Amanda's now a tenure track professor at Yale Zak is in his first year here along with me as a postdoc here at Emory and what I've quickly understood is he like makes best friends with everybody in the universe within within seconds so this lecture is entitled adding and counting and quite frankly there's very little I have to explain about the subject of this talk other than to find what I mean by what it means to add numbers what it then means to count was about as simple as it could get a number theory but to start I'm gonna invite you into so to speak my world my world is not a normal world I literally think about numbers all the time I think about functions called modular forms all the time and rather than tell you now maybe even off at all what that means let me tell you about how you could think about my world by means of an analogy so tonight we're going to be talking about numbers numbers that are obtained by adding and counting these numbers will be very very big and for the better part of the last 15 years the set of numbers I'm going to talk about has been like my universe so what do you see you see here a big universe and what I want you to think of is things don't get bigger than the universe the universe contains everything and because it contains everything it's clear that this has to be a pretty complex system which on denial has to be studied it starts out looking like this doesn't make any sense except they do look like numbers you started reading from the top top left you see the numbers 1 1 2 3 5 which may look to you like the Fibonacci sequence do green 1 plus 1 is 2 1 plus 2 is 3 and 2 plus 3 is 5 but then the pattern breaks down because lo and behold 3 & 5 do not add up to 7 and if I were to leave this slide up there and you were to study some of the numbers you can start playing around and literally start looking for some patterns I'm not going to give you the time to do that but I think you could imagine why at least somebody who likes numbers might do that something like studying numbers on the band and baseball cards so that sequence of numbers has been studied by lots of mathematicians lots of mathematicians going back centuries here are some of the names you will hear Euler Ramanujan party Watson Rademacher Atkin Dyson and Andrews I was going to leave this slide at that but I put this face with a question mark because I figured the best way for me to explain what it is I've been talked about is to start with someone that I think you all know you are maybe my age maybe you grew up watching Sesame Street if you did it maybe you still know who this character isn't a lower right-hand count the corner he is the count from Sesame Street and if there's one thing that he really likes to do its count so now I've explained what the adding is there were halfway there I explained the adding is Lari so that universe of numbers that sea of numbers that I'm speaking about is based in mere child's play which can be thought of as mathematical child's play so let's start with an easy observation it is indeed easy that four equals three plus one equals two plus two equals two plus one plus one equals one plus one plus one nobody's going to dispute that if so probably shouldn't have gotten into ever end that's all right and what I want you to do that's the adding part I want you to count the number of ways in which we wrote down the number for if you do that you get well what do you get five that's awesome you get P of four is five there's very little else to explain about this in this talk about the object of study this function P essentially just counts the number of ways we have there are or breaking up a number as a song one more example if we consider the number five and if we write down all the ways of getting five as a sum here they are you count there's seven of them so the value of this function P the value at five is the number seven alright so I'm like many math talks I tend to give certainly everyone's with me alright so to explain this universe of numbers the ones that start with one one two so innocently and end up with some giant numbers here with who knows what for the toss you'll notice that there are two numbers in rent the numbers five and seven and that elementary observation that we just started with adding up to get four and adding up to get five explain those two little numbers the numbers five and seven so this sequence of numbers in order are just the answers to the question in how many ways can I add up numbers to get four then five then six so on and so forth and we'd like to go off to infinity okay this is the subject of partitions just the question about how to break up a number in terms of sums and it turns out this subject plays this doesn't come as a surprise many rolls I don't really have time to explain this the particular task of adding and counting that I described plays an important role in algebra combinatorics and number theory played an important role in the early development of computers and has played a role in prime ality testing although we know that for example P of four is five let's just look carefully at some specific numbers because after all we're talking about adding and counting so I should show you some numbers so P of 2 is 2 P of 4 is 5 and the partitions of eight is 22 okay you probably can't count up 1 to 22 very quickly but you can imagine that's probably right so what I want you to do is let's think about the number 16 and make a mental guess of what you think the number of partitions 16 should be good and I could ask did you guess 231 this audience already knows too including but if you're like the student in one of my classes you might be impressed about 230 one's a little bit bigger than that not and by the way for the undergraduate students who've chosen I want to thank you for coming I recognize you have a choice I understand it's a basketball game at 8 o'clock probably starting now I will be done in time so that you can go see the second half go Eagles all right but if we were to list some more and I chose to list the values of this function at the first several powers of two you start getting these numbers that race off to well they raised off to infinity I've chosen to list the numbers of powers of two for a number of reasons but the most important reason is one it clearly indicates these numbers clearly indicate the very rapid growth of the partition numbers and at the same time I've chosen them so that you can try to interpolate or guess what must be the properties of these numbers because they start to sketch out what looks like a very beautiful curve ok and if you were us trying to figure out the asymptotics how big these partitions and numbers would tend to get this would be a very very good clue very good clue and the question and this is a famous one many many people have used this very specific question to give lectures not very different from this can we calculate numbers like P of 200 the number of partitions of 200 I don't think anybody here would agree that to money was a large number 200 pennies I can imagine 200 pennies I can even imagine County two hundred pennies wouldn't want to do it but it's not a big number but can we calculate the number of partitions of and I'd like to say well as I presented the subject how would you count the partitions of 200 well if we were to list them all we would start at 200 okay this one everybody with me 199 plus 1 there's to get 297 plus 3 but my gosh you realize that this is clearly not the right thing to do and well I'm sorry I being with the counters I couldn't resist myself the count is the only one brave enough to count but you know what's incredible about the internet what's incredible about the internet is you can find pictures like this 200 might not be so big but even account isn't brave enough to count the number of partitions of 200 honestly I take the count and I found this picture was edible okay all right so how do you calculate the number of partitions of 200 so along the way in this lecture I'm going to introduce you to some of the big theorems that define the subject some of those names are are forever linked to theorems that many of us in the room use almost on a daily basis here's the first example I'm not going to present it in a way that I'm going to present it in a way which is a little bit imprecise but I think you'll get get a very clear picture of whether what I mean here so if you want to calculate the number of partitions of 200 there is now and by the 18th century there happened to be a very beautiful device that got around a problem of actually having to where you would have to list all the partitions of 200 and then count there's a faint very famous theorem due to Euler and for these students that are here this is really the Euler that you know and this theorem says roughly speaking if you want to calculate the number of partitions of n all you have to do is know the partitions up to n minus 1 and add and subtract them in the right way it was a very precise recipe that says if you know the number of partitions of n minus 1 and of n minus 2 and some of the other known partition numbers up to that point if you can choose your signs in the right way you would be able to calculate P of N and surely that beats in the worst case that would mean adding something like two hundred numbers although it's much better than that and the idea of adding two hundred numbers is much easier than trying to calculate and count the number of partitions of 200 so why is 200 very special or because it allows me to tell you about a particular point in time in the history of number theory around 1915 the first 200 partition values were calculated basically using this method and it turns out that if you were able to tell me that the number of partition numbers of 200 was this it would be right and if you were to count I really regret to say that even if you had the luxury of all the 200 other partitions in front of you it's hard to imagine counting and actually getting the right answer that's obviously not something that you do now what this theorem says is that we have a procedure what we call a recursive procedure that can be used to calculate values of this function but it has one clear obvious difficulties it's not a flaw it always gives the right answer but it's not optimal and I'll explain why it's not optimal moment but I think it's important to say that that theorems like that the Stearman Euler is often referred to as the pentagonal number theorem is one of the prototypical identities in this subject you work in the subject you're very happy when you find an identity like that because it then means that you've got some deeper insight into some combinatorial structures which may be very easy to define but very difficult to manipulate this theory as I said lives on went on to be perfected and added to by people like Jacobi Ramanujan Watson and Andrews and like I said tomorrow novio in our lecture devoted to this subject but as you can see if calculated the number of partitions of 200 required knowing the number of partitions of numbers up to 199 maybe you can do the calculation but what if you wanted to calculate the number dishes of a thousand even with the help of a computer so here's a picture I found this also on the internet and it says in the title of the slide would be is there a better way to compute P all right so as is typical in math and just a scientific method if you want to address up a conjecture or test a hypothesis and then you find that it's very difficult to it to make much headway one of the first things you do is try to ask a simpler question because perhaps perhaps an answer to the simpler questions sheds light on the problem this is indeed what happened in the case of this problem the problem of calculating the partition numbers and in this case the simpler question would be well can we just approximate the partition numbers can we forget about exactly calculating the partition numbers when we estimate them and be close this was done in 1918 about 1918 the experts in the audience every year that I write down in the talk was plus or minus 3 I'll be right somewhere and so I don't know of 1918 is right but in 1918 it was a fabulous theorem roughly at that time by Hardy and Ramanujan that sense for large numbers n P of n is very close to this expression on the right so it's basically a sub exponential function this is the e that you know from wherever you know at 2.71 whatsoever and what this theorem says is that it doesn't say that P of n is equal to the numbers on the right-hand side because I'm sure you can imagine that plugging in numbers grand rarely give you ordinary counting numbers what this theorem says is that if you were to divide the right-hand side into the left and let n be really really large you'd be writing down numbers that look like one they get closer and closer to one so let me be completely explicit about what I mean by that so if we define aprox n to be this formula of party and Ramanujan if we did that and calculated the relevant quantities for n being 10 20 30 40 50 and so we'll be able to fill in this table so in the second column these are the real partition numbers so uh listen list there at this this row that's red so if we wanted to add and count up to 30 it turns out there really are five thousand six hundred four ways of doing that I assure you that's correct and if you plug in N equals thirty into that crazy formula of party and Ramanujan you'll get a number that looks like six thousand eight hundred and the point is if you were to divide P of n into this you get a number like 1.0 8 5 from what you're supposed to deduce that the formula party and Ramanujan is close when it over counted by roughly eight and a half percent what's actually true is that all of the numbers given by the tardy Ramanujan formula always greatly exceed the partition numbers in fact the difference goes to infinity but because the partition numbers grow so rapidly it turns out that despite the fact the distance goes to infinity but these are all bigger and everyone see that these are all bigger that distance doesn't grow at a sufficiently large rate to prohibit the partition numbers from swampthing that growth so for example by 50 you're only over counting by six and a half percent and the amazing statement about Hardy Ramanujan is that the larger the numbers and get the percentage by which you're over counting decreases despite the fact you were way off in terms of absolute terms but this is a huge theorem because if I were to ask you how many partitions there are of 50 well was that on the slide remember what did they go up to 5050 that was on slide you could use that formula and get fairly close the right answer but you would be closed that particular theorem marks a very important time in the history of analytic number theory what Hardy and Ramanujan did to achieve to obtain that formula is they invented a method called a circle method not going to explain what that means here except the state this is really important and what I want to say about that is this method is a fundamental tool in number theory and it still goes on continuously today to be a fundamental tool and in many areas of math this is not the back waters of mathematics this is a central as you can get all right now let's fast forward there was a really big theorem by a German mathematician named Rademacher he was a long time professor at the University of Pennsylvania and he found a way to bootstrap the Hardy Ramanujan proof and you got this he proved in the early 40s a formula that says if N is a positive integer that P of n is equal to that is an honest - good - equal means that the two expressions on the opposite sides of the equal sign are the same just so that we're all on the same page so the reason I want to say that that way is if I were to ask anyone of you what P of n men you know write the number of ways and add up numbers to get n what is that what about that do you not get what you don't get is you don't know how to figure that out what about the right hand side maybe I should ask do you get well all right let's look at it you would understand the two pi is 3.14 okay you would get back 24 n minus 1 to the negative street force power you get that and then everything else on the right hand side determines requires the real explanation I'm not going to explain all of them except to say this means we're summing up infinitely many numbers each one of those numbers is already very difficult to describe it depends on a Bessel function and a function called capital AEK which is called a kloosterman song and experts will tell you even at the best universities Princeton and Stanford the top analytic number theory are very interested in understanding the properties of the numbers a K of n not in a sum but one at a time how do they behave okay so what I want you to gather is the left-hand side P of n is a number we totally understand but this formula Rademacher gives these perfectly fine integers that grow very quickly express them as an infinite sum of numbers that are kind of varied to explain now what do I have to say this is a big theorem is it a theorem that you can use and that's the question that you have to ask going from an approximation to an exact formula at that point in time was a major advance the question is can one use that formula to calculate P of N and yes you can and how would you do it the idea would be instead of adding up infinitely many numbers can you find a point beyond which you can pretend that there are no other numbers in the sum truncate that is and then round the number that you did get to the nearest integer and have confidence that you've got the right answer that would be great that is exactly true you can do that but the way I phrased it as a question is really how far can we go in this crazy formula before we are guaranteed that truncating and rounding gives you the right answer so this formula is proven in 1943 we have answers to this question but what I want to say is not only is this a good question this is a question which is still the subject of very much research so in a recent paper in fact this paper might not even yet be in Printz was recent paper by Riyadh Missouri who is an outstanding number theorist of Texas and Amanda who's here in the front row they study the problem of if I truncate this infinite formula from Guatemala after a certain number of steps how far away am i from the right answer but let me remind you how close do you have to be to round and know you have the right answer you have to be within one-half right good to be with wins in 1/2 how big are these partition numbers they are huge so to be within 1/2 of a huge number is basically like saying you already wrote down the partition numbers beforehand anyway and that's that's not quite accurate but I think you get the sense that this is uh this would be a challenge I want to make crystal clear why this is a challenge does anyone argue with the fact that P of 1 is 1 so I want to show you how we're going to show that P of 1 is 1 using this formula and if it's hard to do if it's hard to show P of 1 is 1 with this formula that will illustrate that there really are theoretical issues that mathematicians have to address in order to make this effective so let's prove but P of 1 is 1 despite the fact that the moment you write 1 you're done is forget that we're that smart so here is a table capital piece of capital n of 1 means I'm going to take the Rademacher formula and forget there any other terms app after the capital nth term so that first row says 1 and it has a second number 1 point 1 3 3 5 5 and we feel very good because white one point 1 3 3 5 5 is pretty close to 1 if you had a second term to get 1.00 2 9 6 and you're really thrilled pretty good I think your calculators might even mistake that for the number 1 okay well I want to continue and let's go down to the tenth approximation and let's now forget for the moment that this number is close to 1 and let's look at what the error is this is one point zero zero six three three and I want you to compare one point zero zero 6 3 3 to the second number one point zero zero two nine six which one is closer to 1 which one's closer to 1 the second approximation or the tenth one the second one now for those of you who are students we were all students one so we can always say we are students but for 4 students here at Emory we teach you subjects that are called calculus and analysis and we talk about convergence so on and so forth and maybe in most of the examples that we teach you to to work with most of the examples that you might come across in a homework set it's typically the case that the tenth approximation is closer to the correct answer than the second one all right it turns out that these these nasty Kloosterman sums that I said people at Stanford and Princeton study have this tendency to occasionally poorly behaved and points in the direction that repels cancellation I don't really want to go much more into that but instances like this are actually not that uncommon where for some crazy reason far on down the line in the set of truncations you get much further away from the actual answer than you expect to be and this is the nature of the beast this is the nature the difficulty that Amanda and Riyadh met to cope with and some of their theories really depend on the truth of the Riemann hypothesis so it's really a very tricky business and this is just to show that P of 1 is 1 all right so what do you know about p1 is 1 maybe you have to count but there was no adding ok so what if you also have to add alright so let me tell you what we've done all right what we've done is this and for experts I apologize that there will only be one slide any technical deals details whatsoever but our theorem looks something like this detects the introductions of this paper is on though is on the aim website but a theorem says something like this it says that we have found a beautiful function called capital P it's a single function when I write down capital P pretend to me for me but it's something like the sine function that you'd come across in trigonometry it's that fundamental for us for this problem we have found beautiful function capital P that has the property that P of n is equal to and remember what we agreed upon is the definition of equal P of n is equal to 1 over 24 n minus 1 there's nothing hard about that multiplied by some what are the contents of the sum we are going to be plugging into this single beautiful function P some values some points I call alpha and you know what these finite numbers they're not difficult to calculate it'll turn out that they are all algebraic numbers unlike these numbers that appear in Rademacher formula involving Kloosterman sums and vessel functions which have which are presumably almost always transcendental to calculate P of n all I have to do is add up a finite number of values of this one single function is very beautiful magical function which will only return 13 pretty numbers no decimal approximation is necessary no truncating no rounding no nothing you evaluate these numbers and add up so I have to say a few words about what the alphas are they're actually not very difficult so we're going to show you an example which will more or less suggest to you concretely what the alphas are the alphas will turn out to be roots to quadratic equations just like you see in algebra ax squared plus BX plus C and it will turn out that the formula as I'm going to describe it it's completely effective it's completely efficient and it actually gives much more than computing the partition numbers all right so I know that's not very technical so let me prove to you that we can compute pr1 so it turns out that to calculate P of 1 remember there was this 1 over 24 n minus 1 out front that's us 123rd it turns out that all I have to do is evaluate this function P at three special points you see a whole lot of 23 is there does anyone agree all right so if you want to evaluate n my formula will involve 24 n minus ones all over the place in a very systematic way just like you see lots of 23 s and lots of minus 23 s that's a very easy thing therefore for me to explain to you what this formula is I just have to explain what these denominators are and what these other numbers and the numerators are it's a little bit technical but it's not that bad is there anything about 12 24 and 36 that you don't get they are multiples of 12 and they're the first few multiples of 12 and for some reason I choose to stop at 36 I'm not going to explain this for some reason but that's part of the beast the numbers like negative 1 negative 13 and negative 25 what's common about them to go from one to the next you're just subtracting 12 each time so the rules that determine how to choose these alphas are just a decorated version of what are the multiples of 12 up to some point what's 24 n minus 1 and then how do i plug these numbers into P if I made it sound like it's that simple and you told got it all right so if you evaluate this function at these three points you get these numbers this number beta I apologize if that looks awful but if I could write it down it's an integer plus some integer time to score 269 there's not anything that you can't get about that as a number and if you evaluate this function at 3 3 points you get these 3 numbers what can you tell me about these 2 terms here that one and that one you don't have to know beta is they cancel what about these three numbers well what's the relationship between 6 and 12 12 is play 6 there's that plus and 2 minuses so what are those three numbers in the middle to the cancel if everything with the ugly number is beta cancel what are you left with the red numbers what are all the red numbers one-third and every one thirds what you get and what were we trying to get we're just trying to show that P of 1 is 1 so what I want to say is I don't view this as a heroic proof that P of 1 is 1 but I think you all see that this is dramatically different from the idea of adding up infinitely many different numbers having to worry about things like 1 point 0 0 6 3 3 this is something very different because when you work this out there's always this beautiful cancellation all right it's a finite thing this function P is amazing all right so because of that I think we can argue that that's a formula for P of n all right so for experts in the audience who know about quite a bit about this sort of subject I have to say a few words this special function P has the property that all of those values are of course algebraic and because all those algebraic for each number and you can consider the polynomials who whose roots are precisely that set of numbers I have to show you these polynomials for n from 1 to 4 there are a bunch of numbers in red but by my theorem what are the numbers in red they are the partition numbers so what my theorem is my theorem with Yann Grenier is that not only am i computing for you the partition numbers because they just happen to be the second coefficients of these polynomials I found you some amazing polynomials these polynomials probably encode much more information this is actually the source of the formula instead of trying to calculate the partition function back at a min Palo Alto we're trying to understand what it means for modular functions that are not whole amorphic to generate Hilbert class fields and we did this with very little success for experts this is like what is the general theory of the j function so what we've proven that implies this theorem is a statement about moss forms non homomorphic moss forms with nonzero eigen value and we have a notion for what it means for them to be algebraic and we have a notion in a general theorem that determines when its values of points like these are algebraic and for experts you know we work very hard for one week trying to answer that question and it turns out this is a beautiful example of that theorem so the comments your experts this function P is a weak moss form with eigenvalue minus two if they're not an expert enjoy some Greek letters they're experts this function arises the theta lift of a derivative of a mock theta function and like I said the splitting fields of these polynomials they generate very nice Galois groups the coefficients of these polynomials are smaller than the coefficients of the ones that one gets for the J function so I would like to propose these perhaps as a suitable alternative certainly for programs like sage for computing quantities lateness B I think that's all I want to say about that so the last 20 minutes this is me but what you really want to see I'm going to talk about the yang and the yang is about the fractals turns out that the fractals have nothing to do with our exact formula but it has everything to do with the exact exact formula that we were thinking about a whole circle of ideas that happened to be someone related to each other and it just so happened that we these theorems within the short period of time the theorem about to describe to you is what came first so it's explained the relevance and and and the I don't say the beauty of the number theory but I guess I did the beauty of the number theory let me walk you very quickly through some elementary questions I could ask how often is P of n even so we can let PR 2 of n be the proportion of the first capital and many values that are even so if Capital n is 2 P of 0 is 1 P of 1 is 1 and P of 2 is 2 so out of those three values one third and over being and so that's really all I'm talking about how often are these values even and if you compute you get numbers like this amazing out of the first 1 million values roughly fifty point zero four percent of these numbers are even by the way so roughly what percent would be odds 104 maybe to worry about 0.04 but you get the point anybody want to hazard a guess that's what the conjecture is is this red number for large numbers what we have the values of this function should be even and roughly half should be odds what's surprising in number theory is some of the simplest estate and simplest to conjecture statements like this are among the hardest to prove and I will say I don't think we have a very good idea of how to even begin to attack this conjecture despite the fact it looks like these numbers are more than randomly distributed they really want to be 50/50 all right anybody want to guess here's another question how often is P of n a multiple of three make the mental guess and you might be very pleased to see these numbers well depending on how you're trained is a statistician you may or may not think of 0.33 1 as being close to 1/3 but I assure you that's really pretty close to 1/3 and you could conjecture then that among other things that one-third of the partition number should be multiples of three and again this is a theorem which we have very little idea about we do not know why this is true although we can observe that these times these happen to be true does everybody see the pattern 3 was prime to his prime through his prime and I'm gonna fly so how often is P of n a multiple of five what number am I looking for alright you get surprised so we remind you of what I'm saying among the first 2,000 values of this function roughly thirty four point six percent or multiples of five does that look anything like one-fifth I just looked like anything like a function fraction you know other than stayed 346 over a thousand okay so the question there's a cement well I don't have time to go in directly this is a great question so I'm not going to answer this question but I'm going to tell you about why we know these probabilities are very far from 1/5 there's a very elementary answer for it's very beautiful it turns out that the sequence of numbers that just arises from adenine counting possesses some very very strange patterns unexpected patterns patterns and certainly should not expect based on how I decided to give this talk so what you see here are the values of the partition function at numbers like 4 9 14 so on and so forth and if we count the number of partitions of these numbers because the numbers in this right hand column and what's true about all of those numbers they're all multiples of 5 what you observe is this this is what I want you to see that starting at for every 5th partition numbers of multiple of 5 which if true means that for free I got a head start at least one-fifth of the partition numbers are multiple thye that this is true forever it is true this is this was one of my favorite theorems of Ramanujan proved he proved among other things that for every number n the number of partitions of 5 n plus 4 is always a multiple of 5 and similar statements statements when you divide the partition numbers by 7 and 11 some very strange reason numbers of the form 5 n plus 4 7 n plus 5 + 11 n plus 6 have this forced rigid divisibility reasons you cannot see from the adding in the counter alright so Ramanujan did prove that theorem Ramanujan if you don't know the story you should look it up on Wikipedia Ramanujan was quite an enigmatic figure he was an untrained Indian mathematician who is perhaps most well known or came to stories perhaps most well-known because of his relationship with gh Rd who is a distinguished analytic number theorist at Cambridge at the time so you'll read about that story I won't tell it now but if you know the story you will know that the some of the great stories about Ramanujan involve his words there's no books and the letters that he's written so it's an X period I want to tell you read from you read for you a piece of part of one of them on engines last papers in this pavement this was a published paper this wasn't a notebook it's a little bit different from some of the other stories you'll hear about mysterious words of Ramanujan this was in a published journal Ramanujan writes I have proved these three confluences and then he goes on to say that there appear to be corresponding properties he didn't boldface that that was mine doing fear so there appeared to be corresponding properties in which the moduli numbers like five seven eleven where they appear to be powers of five seven or eleven and he goes on to say and no simple properties for any of moduli involving crimes other than these three you're saying five seven eleven have more special properties than those that are just exhibited here but then as a story goes for Prime's larger than eleven nothing simple seems to be out there so there's two parts to this claim that we as number theorists have wanted to figure out ever since Ramanujan first discovered these beautiful divisibility properties so the history is amazing it's great to live in a time when so many people that done great work before you because you get to learn from great works in terms of Ramanujan himself Watson a napkin together over the course of I guess these few decades prove the theorem like this for us that work in number theory nor who think about modular forms and who even think about operators of modular forms in a way that's not too different from what goes into the proof or from Oz Last Theorem in a way I can make precise I don't have time to do it now really love this theorem what does this theorem say it says something like you give me a power of five like five to the hundredth and I will find some number so that all the partitioner numbers are the very very special sequence despite the fact I don't know what they are they're gonna get very big but those numbers will all have to be multiples of that particular power of five so for example if M is two it turns out that this expression reads 25 M plus 24 so starting at 24 every twenty-fifth partition number is a multiple of 25 so that's like a lifting or refinement of what it was on the previous slide and this is almost certainly what Ramanujan meant when he said and there are corresponding properties for the powers of five seven and eleven that theorem alone that has inspired work in many different directions there's the arithmetic Direction I guess that's why the direction I've worked most in where you get Kelo representations and the length but perhaps more important part which came first is this work inspired by a very famous paper by Freeman Dyson in 1944 which is so beautifully written that it really should be on everybody's read list certainly in my math 250 class I think I could spend three days just going on and on and on about just the language the words that came out of this man's mind put on papers very funny I will do that and what Dyson was suggesting is that well maybe it is true that the partition numbers have these striking properties after all Ramanujan and others threw them but why are they true is there a way of seeing from the partitions themselves that they naturally form groups and that these groups have all have equal size when constraints these certain progressions and this is a thread in number theory that has borne quite a bit of fruit people like Atkin Swinton Dyer andrew is Garvin Kim and Stanton have worked along this direction and I can say that for many of the confluences that are covered by some of the powers of five seven and eleven we have a very clear picture literally what's going on at the level of adding and counting literally adding economy we know how to form piles of size five whenever you're writing down partitions in the form 5m was really a very beautiful direction in which the subject has developed but the mystery that's remained from Ramanujan Zen an enigmatic Pope would be this sort of question what did Ramanujan mean when he said that there are no simple properties for any moduli involving crimes other than these three I know you're waiting for fractal but we have to do it the right way so you can see where we came from before I show you our theorem so before I can do that I got to tell you what's known about this particular question so we know a lot about the primes five seven and eleven and as the students in my class know there are infinitely many Prime's so we've got a long way to go after studying five seven and 11 for the small Prime's like 2 & 3 there's been a huge recent development this is probably I don't know how to say enough good words this is a great deal by an Austrian mathematician who was unable to be here for this conference because I frankly dropped the ball I didn't invite him with sufficient time to earn a visa to come to this country enough said about that I don't know what you think about visas and all a lot but that really just should not happen in academia raju proved a great theorem that says that you cannot find formulas patterns like a n plus b for what's the partition function as always even or for which it is always a multiple of three it is not possible Ramanujan said there are no simple properties other than the primes Phi of seven and eleven and I offer you this theorem of Raju which confirms that for two and three and which four experts we were always very disappointed with our inability to say anything at these two primes so this is a very big result and like I said I'm very sorry you could not be here so more on the mystery what about the primes bigger than five seven eleven a number of years ago worked by Scott oghren and Matt Boylan really went a huge way to answering this question very beautiful theorem they prove that the only patterns ln+ a where the partition numbers always must be multiples of L are the patterns the pairs la namely that those listed here 5 4 7 5 and 11 6 you see three numbers and ranked those numbers happened to be 5 7 11 what they proved is despite the fact we can't compute all the partition numbers at once they were able to detect the properties like this could never happen again these we can argue our simple statement that there are no other simple properties I offer you this theorem simple is a very funny words if I call you simple so not so simple properties but 10 years ago I proved the theorem that says there are some not so simple properties and it turns out four primes like 17 to 31 these partition numbers are always multiples of these primes 17 19 23 29 and 31 there are rules of course that define these I don't want to get in the tool now I could present them in a way so it looks less monstrous but that would be defeating the point I wish to make these numbers are huge you would never guess that starting at 1 million 120 2838 every whatever that is the partition number would have to be a multiple of 17 and even if you could prove that you have to then ask is that cool the only Kovac is numbers are big or instead is it advertising that we really don't yet have yet earned the right to understand this function ten years ago I would have never said the second statement I'm very happy with these so these monstrosities really come as the result of theories times sort of apologize if your name was there and you won't and I didn't get to say these monsters were discovered basically because we live in a time we live in a time when we have been offered the opportunity to apply a big machines to simple questions so I won't read these except to say these three bulleted items really are and can be thought of as big machines that we've decided to employ to study a very simple problem namely how in the world do we add and then count all right it's actually a little bit humbling to say I've got to bring out all this how many of you have seen the movie Indiana Jones there's a very famous fight scene where this guy comes out with this big sword and he's a small full of myself if we want to add and count it really is kind of humbling that just for the simple task of adding and counting I have to bring out these big guns that would take you years of math to begin to understand that's really humbling and maybe that sounds good maybe that reveals we've learned something but what does it really advertise it really advertised that there's a lot and perhaps more that we do not understand so I like it it to this American allergy what is the Eagle Nebula it is your image it is our image very famous now already made so the formation of stars in the universe when this news came out very famous going to Barnes and Noble there's probably four or five books across the hall with this image on the cover are certainly quite prominent it is because of the invention of the Hubble telescope that we have earned the right to see this very singular beautiful behavior but does that mean we understand the universe course not alright so the reality is this the vast majority the partition numbers don't fit into the work of Rahman engines confluences or these monsters we can be delighted that we're writing down big numbers it'd be something like I can remember 50 digits of pi maybe you can do 100 and apologize for maybe something here can do a thousand it's like it's like artificial sports it's not revealing we've understood anything about these numbers whatsoever so the fundamental question really is and has always been what are the properties of these numbers why are we always talking about numbers of multiples of five do we not care about when they're not multiples of five of course not so here is the answer the answer I'm going to show you in by means of a movie because I want to show you this movie maybe you already know it and then we're gonna talk about what you see in this movie and then I and then I'm gonna explain the theorem that's related to this question all right I think you've probably all seen something like this before and maybe we could sit here for 10 minutes keep looking at it thank you do that but maybe we'll look at it a second time remember that remember that maybe you want to remember that - all right so let's see if I can stop this before it starts again and if you don't remember me but I remember how to use this machine now you're great so that is the famous image that resembles well this is a very famous image it's a it's a it's a visual it's a video of all of us zooming into the bowels of the Mandelbrot set it's a fractal we're zooming in into space which is arbitrarily small we're twisting around you see everybody well maybe you didn't twist around maybe the movie did it for you we're gonna do some of that number theoretically in a moment but you zoomed in in this infinite space that's in decimal a small this infinitely complex but what you saw or what I want you to gather from watching that movie is that you saw a complicated structure that's actually very simple to define inductively or recursively so if you don't know that that's true I assure you this is true this is something that people and dynamics do all the time they iterate some map over and over again generate amazingly complicated pictures and this is what's what generates something like this and what did you really see what what did it what is it that I wanted you to observe I wanted to observe that that famous image for example that famous weird image you've got to see over and over again that is this repeat itself similarity but you have the self similarity to arbitrarily small scales that goes on forever the theorem that we have proven is a different kind of fractal we're not using geometry where we're defining distance based on well maybe this is three inches apart I have to define for you the geometry that goes into defining a fractal in this sense that's what the next few slides are about but the theorem is that the partition numbers are letting a fractal and that's what I'm going to define for you for every prime at least five that image of zooming in to this sex I'm going to explain how we can do that with the partition numbers and then explain how we see the same patterns over and over again and at the same time see them with increased resolution if I achieved all of that that's exactly what I mean by this theorem so let me just say exactly what I just said again I need to define himself in simple inductive structure and I'd have to show you what self similarity means namely if I give you a number what does it mean to say that oh by the way I actually saw that before what does that mean and it turns out that I only need you to remember this slide this slide might look bad but if you look carefully it's not too bad there's some peas and some ELLs and some ends nevermind that on the slide for every number N and for every prime now we have two sequences so what is common about every one of the pea functions in this these two rows is of ELLs and ends re are the same they're not moving what do you see in the first row what is the only feature that distinguishes the numbers as you walk from left to right in this first role they are the exponents and what is true about the exponents the exponents on L that's L to the first that's L to the third that's what I can't reach that I that's L to the fifth L to the seventh these numbers are generated if I give you an L and you give you an end all I'm gonna do is move from left to right by increasing the exponents all I'm doing an experts will tell you that while one of these sequence is always trivial it's always zero that's true and we're just going to study the sequence which is interested in when it's not always zero so what zooming in will mean my idea assuming into the heart of the Mandelbrot set will be like starting in the left and walking to the right in pi8 number theory this zooming makes sense so when you zoom into the Mandelbrot set you can talk about consuming it's negative infinity if you use that geometry what this means oddly enough we'll be suing to the number one twenty-fourth so in the Atlantic topology there's a number very special 124th which i don't have time to explain is very important but for us for the partition numbers going to minus infinity means going to 124 so it's a very precise sense for the X and now let me tell you what the Ramanujan confluences say again because then it will allow me to explain what we've proven so there's two sequences each these sequences are determined once you give me an L or an M so I'm going to give you five and I'm gonna let n be 19 and of those two sequences on that last slide only the top one makes any sense and if you're to plug in for Ln n you would get P of four is five the second term would be P of nine which is this number so on and so forth this is a way of stringing together the partition numbers this is how we're gonna define zooming into the heart zooming in to 124 you know what what these numbers are actually aren't really that important for studying this fractal what's really important of these facts fact number one it was true but every number in this infinite sequence when you write them down every one of them is a multiple of five if every one of them is multiple of five what do they all have in common well they're all multiples of five that's similar right right if you pick numbers that have had at random how likely is that they will always make all be multiples of five that's not very likely that's a nice statement that's what managed improve that so here's a marked number two apart from the very first number which is five you know it's true about all the other numbers they're all multiples of 25 apart for the first number all the other ones have this extra fiving how cool does that is there anything about adding and Counting that said well of course for these crazy numbers I get Miss divisibility by 25 probably not and half like three this is what I want you to see note the word in red says eventually it says in this sequence all of the numbers are eventually multiples of any fixed power of five like five to the 1000th would you believe that after a thousand terms in this sequence everyone from then on is a multiple of five to the thousand despite the fact none of the ones before it are so what's similar if all the numbers from some point on or multiples of five to the thousand doesn't that isn't that a feature that they would share in common absolutely what is zooming in me it means walking from left to right what is increasing resolution means means we've earned the right to divide by larger and larger powers of L and see the same patterns despite the fact that early on you can't even begin to see that pattern doesn't that resemble what we saw on the Mandelbrot set okay I can make this all very precise an Atlantic topology but that's what I want you to see so this is our theorem for Prime's out of these five the following is true for every L for every power about two the M these sequences are all periodic there is a point beyond which you see a certain number of numbers divide them by L to the M to get some remainders those remainders are repeated over and over again and you don't even have to compute any of those numbers to know that those remainders repeat over and over again it's not true that they're all multiples of 5 but what the remainders are when you divide by those numbers are inescapable they are constrained the periods are the same for all then the way I presented this theorem seemed to suggest that I give you an L or you give me an L you give me an N then I do this calculation the theorem actually says you give me an L the theorem is true no matter what n is it doesn't depend on n at all when n is as good as any others and that's the weird part what is zooming in mean means that increasing the power of L means like now divided by 11 to the 30th and then later dividing by 11 to the 50th patterns repeat themselves dividing by larger and larger powers of L in our Atlantic topology which we typically use a number theory is exactly what we mean when we say we increase the resolution and insulin does not impact the periodicity this is not technically correct as I've stated and experts who thought about this will know what I mean but to make it correct for experts but for those of you who are just interested that's more or less true we define a notion of a house work with mentioned it's really a dimension as an elaborate module for experts and there is a dimension that lutely dictates the complexity of this process it has nothing to do with em it only has to do with the prime number that you start with in other words in these sequences it looks like they're exponents it looks like their ends what I'm telling you is none of that's relevant there's just an L you'll give me the L and everything is more or less determined in a very rigid sense so what they're on an expert language is that for every Prime at least five the partition numbers are all ethically fractal in the zooming process we calculate an upper bound for the dimension is this number and if this is the greatest integer function in two cases and it turns out if you compute this dimension for the primes five seven eleven you get zero although I didn't precisely define for you what the hell store of dimension is I'm sure you can imagine that something has to mention zero is pretty simple it's so simple you can't even put a dimension on it but you know what that means if you study our paper what dimension zero means is that the Ramanujan confluences are true those special properties the corresponding properties for five seven four powers five seven eleven is essentially equivalent of the statement that the dimension is fractal is zero because nothing could have happened except increasing divisibility by powers of these special Prime's these are the only Prime's which dimension is 0 for every other prime the dimension is positive and that's really the mathematical way of saying realizing Ramanujan Sande enigmatic quote namely that apart from the primes 5 7 11 there should be no simple properties because none of us would ever agree that a fractal is a simple thing fractal is about as complicated as they can get within the framework in which we're studying these numbers fractal of dimension 0 is simple that's what Ramanujan was talking about I think the mystery is not pollute me clear for other primes there's as fractals and now we want to go to work figuring out what the properties of those fractals happen to be all right I'm going long I think I owe you a couple examples and then we'll conclude if you calculate that formula it turns out the dimension is 1 for a few primes for the primes like 13 to 31 I owe it to you to explain what that really means in terms of the numbers tomorrow John Webb I saw a preliminary version of his lecture so for experts you're going to see some very startling things you can see some very startling tables worth and examples where the dimension is like 4 & 5 so it's easy for experts you have something very nice to look forward to tomorrow and John you are not going to let me down right okay we're in a Celtic well yes sir so when the dimension is 1 what does it mean to me in the dimension is 1 it means you see one partition number the next term in the sequence is exactly determined by its predecessor means there's no choice so that means this partition number is nothing but this partition number multiplied by some factor this factor is like twisting around when you home in on the Mandelbrot set right that's exactly what you want to be thinking does this number depend on M it does but the dimension being one depends on nothing so you can plug in B 1 B 2 whatever in this multiplicative rule holds so in other words the period is 1 so let me just show you some very simple numbers numbers that you would never have a prayer of calculating without the help of a very strong computer program let's just let M be 1 mod 13k is a free variable so no matter what K is my theorem will say this you can plug in for K and n whatever you want this is a true statement by the way 13 to the 2k plus 1 is a big number even if K is 1 right 13 cubed that only wanted to do like I don't need to show you that I can multiply 13 with itself 3 times but you get the idea so letting em be 1 gives you this unless the KB you want to be this and I want to pause and say I didn't have to make any of those specializations these are all free variables this is why we cover all the partition numbers what that theorem says is this multiplicative statement this is actually known but I'm using this is known by do to acting forty years ago in this special setting but I wanted to give you a flavor of what the theorem says so if were to string together these partition numbers like as a generated competent you get stuff like this you see some beautiful colored numbers so let's focus on the blue numbers 66 is the number you get if you're to plug in for n zero how many partitions are there of six if you remember or if you don't remember I'll tell you is 11 you multiply 11 by 6 well I don't tell you that but you would get 66 if you divide that by 13 you get a remainder of 1 so this explains how I go from this expression to the 66 to the 1 but let's now plug in for n the same zero and calculate the number of partitions of a thousand and seven which for some reason is what the fractal gives me tells me the number of partitions of a thousand seven is determined by P of six that's this number is there anything about this number that you can see from the number 66 the first glance no but if you divide that number by 13 you get a remainder and it's the same remainder one so that means the sequence of partition numbers if you divide them by 13 they are the same but I said you can zoom in zooming in means those increase the exponents so this is where it gets even more impressive let's look at this slide it turns out that an instance of the fractal would relate these three partition numbers with being through these three partition numbers the bottom three numbers are so huge that by here itself it's over a full page and if you these dots represent the full page eight by ten Oh 18 half by ten or whatever there's no size of piece of paper it's huge all that all I'm saying is we're going to plug in 0 1 & 2 in the spots that are analogous in both lines I can actually calculate these first three partition numbers with the help of a computer divided by 169 and I get these three meant remainders these numbers I actually could calculate the help of a computer but you can't fit on so I did it but if you divide all of those numbers by 169 13 squared and compute the remainders you'll get the same thing these are numbers in general we have no prayer of calculating but they repeat themselves you know it's amazing if I told you that if I divided a number by a 50 digit number got a remainder of seven but I told you that number wasn't as big as a 50 digit number you'd be saying well did you just say the number seven in some hard way yeah I did so the number seven in some hard way and it is for this reason that because of calculations by web was sharpened some of the things that we've done you can actually implement this to calculate very special values of the partition function without ever having to calculate them you can use our fractal to do that it doesn't work for all then of course it is it has to do with some other parameters but there are instances where that amazing fact is actually true so some comments for experts this is all part of a very general theory theories that are related to ongoing work that Zack and I and some others are doing largely trying to understand questions that Barry Mazur asked once he saw our theorem we've made some good progress about that will report on that sometime in the future not gonna form like this because it's really a specialized thing but loosely speaking we define new modules in theory called the theory of Atlantic modular forms it turns out that some standard operators that people use in the subject stabilize these modules so this generalizes some important work of Professor Gorger joy we owe him a great debt for writing papers that help us learn how to think carefully we also proved some some conjectures of acting that go back 40 years they're somewhat technical along the way when study these modules we have to define a new operator and this operator will go on to have its own kita theory for experts and it turns out that we have to study very carefully how certain operators interact with each other with respect to these modules so although we're adding and counting it is still true that we still need to invent some big machines to arrive at theorems like this so like I said Matt Boylan and his student John Webb have been working very diligently finding some very beautiful theorems and tomorrow I think 9:30 or 10:00 I don't remember but in the morning he's going to talk they will be talking about how to sharpen the zoom rate so you heard me say that eventually we we enter into a period Amanda Zack and I proved this our theorem is of course correct but it's a little our zoom rate that we were able to figure out a few months ago is a little bit slower than what boiling and Webb have done they very carefully analyzed some of the more technical parts of our paper and it's really beautiful by the way John is finishing his PhD in the spring and so if you're in a position to hire someone who I assure you this is very serious mathematics if that's not clear I haven't done my job all right so any of it I'm really happy for all of you coming I realize you've got a choice I hope we're defeating there's a case western or Carnegie Mellon I have a difficult time distinguishing the schools and that's my boy like I said you'll probably get to see the second half but I hope you can tell that thanks to the American Institute for mathematics and the National Science Foundation they made it possible for us to get together and discuss problems that we perhaps wouldn't otherwise have been trying to prove theorems about you couldn't help but show you the counts and then for fun believe it or not the Internet reveals that there's photographs like this and I'll tell you that we had a great time writing these papers was frustrating but the joy that we we all shared when we finally figured out the last bits and pieces of the papers really something that I can't begin to describe is an amazing thing we're very happy that have that happened our papers um the other day I think there maybe already there or at least most of them are there you can find at the American Institute of math website the first paper only the introduction is there because I'm the kind of a stickler for details but shortly the whole paper would be there but all the details of the statements of the theorems are available online you're interested in that and let me just remind you with yarn I prove a finite formula for P of n there's an Oracle capital P whose values or algebraic numbers they're finite number you add them up and you get the partition functions know one point zero zero six three whatever and four Prime's at least five the partition numbers are fractal they repeat themselves but they repeat themselves in a way that you're allowed to divide by larger and larger powers of L without escaping the fact that they still repeat themselves which the testament up to how quickly they grow because you wrote down a random sequence that's not allowed to happen what is self similarity mean well they're periodic these fractals have dimensions we can determine their complexity and only for the special Ramanujan Prime's are they simple and after all Ramanujan said it doesn't seem to be the case that there are any crimes for which the partition numbers are simple other than 5 7 or 11 for me that was a very big enigma with very big contributions to Scott and Matt Boyle and most recently Raju along the way and I think certainly this kind of seals the deal great so I hope one thing is very clear all I did today was talk about how that add and count you the preceding program by Emory University

Channel: Emory University

Views: 107,915

Rating: 4.8705502 out of 5

Keywords: Emory University, Ken Ono, number theory, partition numbers, adding and counting, fractals, finite formula, Leonhard Euler, Srinivasa Ramanujan, Ramanujan's congruences, G. H. Hardy, Mandelbrot set, Jan Bruinier, Amanda Folsom, Zach Kent

Id: aj4FozCSg8g

Length: 71min 14sec (4274 seconds)

Published: Thu Jan 27 2011

Reddit Comments

[u/[deleted]]

Awesome video, though I didn't quite understand what he meant when he said the partition numbers are periodic. Also, does anyone know precisely what the significance of 1/24 is? I've been introduced to the p-adics, and their symmetry makes this rather surprising.

The article on Emory's website highlighted Ono's claim that the partition function is important in real world applications (like password security). Does anyone know how or why that is?

[u/[deleted]]

The introduction is great, so simple. He introduces the partition function in a way anyone can understand, mentions Ramanujan and the Circle-Method.

So this is like a weird version of Hensel's Lemma, lifting a partition function instead of a polynomial.

Wow it has a Hausdorff dimension! And that formalizes and proves Ramanujans claim!

[u/1stfakeaccount]

Relevant (PDF)

[u/randomtriangles]

I enjoy the math and the history presented in the video, but I do wish he did more details earlier on. Just my $0.02.