 So first of all, I'd like to take this opportunity to thank OIST for the invitations, not only to visit again, as Rico mentioned, I'm here until September 1st, but also to speak. I'm very grateful for both opportunities. So what I want to talk about today are really sort of two separate worlds in some sense. One is not very, and this other is this notion of modularity or a multiple forms. I should say this is not a math seminar talk, OK? So I gave a very detailed math seminar talk on Friday, June 24th here at OIST. And if you're interested in any of those details, just ask me, and I'm happy to share my slides. But what I want to do today is to actually give you much more of an overview of these two pictures and then talk about some very recent work. So by very, I actually mean there was a preprint that was posted, which is exactly when we're going to talk about the very end of this talk about five hours ago on the archive. That was a really good timing. And really with that preprint and then also the details I'll talk about at the very end talks about this connection with mergers and states two areas together. And really, this is in some sense kind of the start of a much broader framework that we're going to be working on for quite a number of years. So what I want to do is to start very gently, because I'm trying to make this as broad as possible. And I want to talk about what, first of all, what are knots? So informally, the way you can think about a knot, what do you do? You take a piece of string, twist it and turn it in a way you like, and then you fuse the ends. Bit more formally, the way you can think about a knot is that it's an embedding of a circle into your R3 space. And a common way of representing a knot is via a planar diagram. So what we're going to do is we're going to take this knot, which is the R3 space, project it down to the plane. And let me give you one example, and it's going to be the kind of one running example throughout this entire talk. So for example, let's write down what is called the right-handed trefoil knot. So it is a knot which looks like this. There are several things I want to explain about this particular picture. So first of all, these gaps here, these are what are called crossings. So this has three crossings. Sometimes it's called the 3-1 knot. The first knot with three crosses. In fact, it's the only knot with three crossings. The way you can think about this picture, if you've never seen knots before, is that this strand is in front. This break represents the fact that the strand is behind. If you're looking at it in free space, this is in front. This is behind. This is behind. And this is in front. A couple of other things I want to mention about knots in general, particularly this one, one thing we're going to be interested in doing is looking at the mirror image of a knot. All that means is what you do is you switch the crossings. So here, the mirror of this knot, which is called the left-handed trefoil knot, what you're going to do is that this curl, which is in front, is going to go behind. This one behind will go in front. This one in front will go behind. This will go in front. This will go behind. And this will go in front. And the other thing that we're going to be interested in is the fact this is an example of what is called an alternating knot. What that means is that, suppose let's walk along the strand. And then what happens is that the crossings go over, under, over, under, over, under. So it alternates between those two patterns. And we can see that because what we do is we go over and then you go under, around you go over and under, under and so forth. So this is an example of an alternating knot. Okay, so the fundamental, the fundamental problem in knot theory is that if I give you two knots, how can you pass or find out if they are equivalent to each other? And here what we mean by equivalent is you want to take a knot and continuously deform it into the second knot. So what you don't want to do is you don't want to snip the knot and kind of reattach it. And also you don't want to pass one strand through another. So that's what we mean by equivalent. And it turns out that this problem is highly non-trivial. And let me give you an example, well actually one of my favorite examples to hopefully convince you that trying to do this can be sometimes problematic, depending on your experience with these sort of things. So let me give you this example of a Wolfgang Hawkins, Gordian knot, a fantastic name, Wolfgang Hawkins. So here it is. This knot, if you count carefully, it has 141 crossings, supposedly 141 crossings. But if you carefully kind of untangle it, then it turns out it's actually equivalent to, yeah. So what we're going to be interested in doing is in coming up with these things called knot invariance. Some type of tests. So the idea is going to be if you have two knots which are equivalent to each other, then these knot invariance are going to be equal to each other. Yeah, go ahead. Why does this knot have even the name if it's just a knot? Oh, that's, I don't know, I don't know why. Yeah, actually to be honest with you, it's called the Gordian unknot, but I wanted to remove the word, I wanted to remove un, I don't want to tell you the answer right off. Okay, so what we're going to be interested in doing is in writing down what are called knot invariance. So the idea is that if you have two knots which are equivalent to each other, then they actually give you the same invariance. Also, keep in mind what that implies is that if the two knot invariance are not equal to each other, that implies that the two knots are not equivalent to each other. Unfortunately, there is no known, as far as I'm aware, there's no known perfect invariance. So just because you have two knot invariance which are equal to each other, that does not necessarily imply that two knots are equivalent. Okay, so that's sort of the general setup of what we're going after. Writing down knot invariance and then relating them to the second world. Or we're just still in the first world. So what I want to do, because I did view this as a very broad overview, I want to give you just a little bit of history. And the reason I want to do that is because there are certain parallels in both worlds. I like, so let me give you a very brief, incomplete history of a knot theory. And it, so actually in my abstract, you may have noticed I mentioned the book of Kells. So the book of Kells is this spiritual Irish text that goes back to the ninth century. You can look at it in Tree College, Dublin Library there. And if you look at the corners of this book, you have these ornate beautiful drawings of knots. Always a pleasure to look at these pages and they turn one page a day. Kind of neat. But it really starts from a mathematical point of view. It really starts with a Vendermal and a paper from 1771. You can also find it in Gauss's personal diaries, 1784-ish. Listing was actually a student of Gauss. Listing was interested in this property called chirality of a knot. So a knot is what's called achiral if it's equivalent to its mirror image. And it's called chiral if it's not equivalent. Listing actually conjectured that the right-handed trefoil is not equivalent to the left-handed trefoil. I'll come back to that shortly. After listing, you have William Thompson, possibly better known as Lord Kelvin. He was knighted for his work on the first transatlantic cable. James Clerk Maxwell will hear much more about Maxwell shortly. Tate, Tate's really the subject of everything. We'll talk about today. Reverend Kirkman and Charles Little. Charles Little was actually an engineer in Nebraska and they were interested in knot tabulation. I will also play a role later. And then what happened is you had Emy Orton. So Orton Orton came up with this notion of brave theory and that really put knot theory on a solid theoretical framework. And this was really a revolutionary result, revolutionary theory in knot theory. You have Alexander. Alexander came up with the first polynomial not invariant in 1928. Right in Meister, we'll see his moves. Although these moves, it turns out these moves actually were guessed ahead of time by Maxwell. He didn't know in advance. Schubert. And then possibly more modern setting. We have Conway, Vaughn Jones, Kulfman and Desilio. Okay, so amongst all these people, list of names, the one person I really want to focus on is Tate. Peter Guthrie Tate, let's show you the picture, this scholarly looking picture. So Peter Guthrie Tate was born on April 28th in 1831 in Dalkeith, Scotland, 15 miles south of Edinburgh, at least one person knows what that is. So at age six, his father died and then his family actually moved to Edinburgh to live with their uncle, okay? And it was really the uncle who instilled, I guess, into Tate, this love of all sciences. And so as a student, he was quite good at all sciences. At age 16, he entered into the University of Edinburgh and then one year later, he moved on to Cambridge. So at age 17, he entered Peter House College in Cambridge and at the age of 20, he became what is called Singer-Rangler in the Mathematical Tripos. So if you don't, for the non-nominations, what that means is that he scored the highest amongst this series of extremely brutal exams, mathematical exams, and at the time, he was the youngest person to have done so. Two years later, in 1854, he actually became a professor at Queen's College in Belfast. At the time, this was a relatively new university. It was only established in 1845. And also, I should mention that it was during his time in Belfast that he actually began a correspondence with William Rowan Hamilton. And you can actually still read their letters today. Tate was a huge proponent of Hamilton's quaternion. Well, his body might have been in Belfast, but his heart was in Edinburgh. And so five years later, a professorship actually opened up and he got the professorship and became the chair of natural philosophy in Edinburgh. Interestingly enough, he beat out Maxwell. And the reason he beat out Maxwell was not because he was a better scientist. It was actually because he was a better teacher. And when he got the job, it was actually reported in the local newspaper. And here's what the newspaper actually said. So in the Courant, what they said is there is another quality which is desirable in a professor and a university like ours. And that is the power of oral exposition, sitting on supposition of imperfect knowledge or even total ignorance on the part of the pupils. So I should mention between 1852 and 1854, he was actually a tutor for those two years. Okay, so he came to Edinburgh. He was busy teaching and publishing in many areas in science. He was also very interested in conducting various types of experiments. And the experiment, which is gonna connect everything back to knots of this one. So in early 1867, he showed Thompson this experiment. I'm gonna show you the pictures and explain it. So what he did was he had this box and then he cut a hole in one end of the box then he wrapped a towel around the other side. And then inside the box was this mixture of ammonia solution and a dish of salt and sulfuric acid. So he created this noxious fumes and then what he did is he hit the back of the box and these smoke rings will come out. Thompson at the time was thinking about atoms. And so what happened is that these smoke rings would cross with each other and the structure would stay the same. And actually watching this gave Thompson the idea that atoms were knotted boricies in the ether. So if you don't know what the ether is, that's good. The ether was this conjectural fluid which permeated the universe. So after that, this theory, even though that theory was debunked completely, it did give Tate the idea to start drawing knots, not tabulation because what he wanted to do was to create a periodic table of knots because if the atoms are created of knots, we should keep tracking right down all possible knot configurations. And so that's what he did. He began down the road of knot tabulation. And let me show you one page from something he did. So there are many remarkable things about this picture. So the first thing is he did have help from Kirkman, Reverend Kirkman and Charles Little. And this picture here, this goes back to, I think, 1855 and this is the alternating knots with 10 crossings. The remarkable thing is, so first of all, he drew them all and he was guided by just intuition. The remarkable thing is there are no mistakes. This is the same table that we used today when we look at alternating knots. So there should be 123 of them. Other interesting thing about this picture is that he made a series in conjecture just based on writing these down. And I'm not gonna tell you all three of the conjectures but I do wanna share one of them because it's gonna come up just a little bit later. So Tate's first conjecture just says that, okay, a reduced, so that's a slightly technical term. It just means that you reduce the, you, sorry, you get rid of what are called the nougatary crossings. If you have kind of a curl, you wanna uncurl it. Diagram of an alternating knot has the fewest possible crossings. And so what that means is if you write down an alternating knot with, say, four crossings, there's no way you can somehow rewrite it magically and it becomes three crossings. That's the minimal number of crossings. And what this reduced condition does is it just takes, you remove crossings that would kind of artificially inflate the number of crossings. That's his first conjecture. Well, tabulation is the rate. You can tabulate all you like. And in fact, I think as of 2020, all knots up to 19 crossings have been tabulated. So they're, I think for 19 crossings, there's something of the order of 350 million knots. So that's not gonna give you very far in terms of coming up with some general theory but maybe you can spot patterns there. So the first result, which sort of leads you to a general theory is the following. It really starts with Roy DeMeister. So in 1927, what he showed is that what you can really do if you want to show that two knots are equivalent to each other is just look at their diagrams, okay? So it's like K and K prime be two knots with diagrams D and D prime. And then we'll say, okay, so K is equivalent to K prime and R three. If and only if, well, all you have to do is just look at their diagrams. And the diagrams are related to each other by sequence of trivial moves in the plane, that's all that means, plus a series of three moves, okay? These three moves are denoted R one, R two and R three, Roy DeMeister move one, Roy DeMeister move two and Roy DeMeister move three. And they're given as follows. So the way you think about, let me explain this picture. So what you do is this dotted circle means outside of that circle, the knot stays the same. So what you're doing is you're looking at the picture locally, okay? And so locally, the idea is that if you take this single strand and then have it be equivalent to this curl, really you say, think about it the other way. Also I should have mentioned I really should have also included the, you can curl it the other way and you want it to be equivalent to the single strand. That's Roy DeMeister move one, Roy DeMeister move two. Just says that if you have two strands, that's thought to be equivalent to these crossing. Also drawn at the other way, crossing at the other way. And the most difficult one is Roy DeMeister move three. The way, one way to think about it is that you have this, these two crossings in the back, this X, you have this single strand which is in front and then it crosses over. Cross like this, you cross it over to the left and it becomes this way. And it's exactly if you want to unknot booking Hawkins knot, which we do is do these Roy DeMeister moves and see how to get it Okay, so the point here is that if you want to come up with some sort of knot invariant, what you have to make sure is that it satisfies Roy DeMeister moves one, two, and three, has to obey or respect these, exactly what this theorem is saying, okay? So what I want to do is let me take a little bit of time and fast forward, because of the whole history. Let me fast forward to about 1984, 1985. And in 1984, Vaughn Jones passed away. He discovered a new polynomial knot invariant. So this was an amazing discovery because this was only the second known polynomial invariant after Alexander in 1928. And it was really for this work that he was awarded the Fields Medal. I think 1990 was the, I see him was in Kielto. And so that's just absolutely phenomenal development. Jones polynomial is related to a wide variety of topics, almost none of which I'll be able to talk about today. And what I decided to do, I'm just gonna show you. And then just ask you to sort of one example. So here it is. The Jones polynomial, the way you think about it is that it's actually what is called a Laurent polynomial. So that means you're allowed to have the exponents to be positive or negative integers, positive integers. And here's the construction. So what are these all this mess? So the point is this thing here is what's called a Kauffman bracket. It's a map which send, if you're given a diagram, then what you do is it sends it to a Laurent polynomial and a variable A. This thing here, it turns out it's not a knot invariant. It satisfies the right of my source moves two and three, but not one. And you can fix it by multiplying, sort of adding the appropriate pre-factor here. Function is what's called the right. It just counts the number of positive crossings minus the number of negative crossings. And then once you do it, once you fix it, then you can check that this polynomial really is invariant under right of my source moves one, two, and three. So it is an invariant of oriented knots. Okay, so that's a very sort of technical thing, but let me invite you, if you wanna go back and anybody's interested in doing this by hand, you can, if you compute the Jones polynomial for the trifle, the right-handed trifle that we saw at the very beginning, it turns out this is just q to the minus one plus q to the minus three minus q to the minus four. It's also mentioned that if you computed the Jones polynomial for the left-handed trifle, then what you get is that it's q plus q cubed minus q to the fourth. The point there is that since those two polynomials are not equal to each other, that proves Listing's structure, right? Because the polynomials are not, the knot invariants are not the same, so that means the two knots cannot be equivalent. So that's the Jones polynomial. So what we're gonna be interested in doing, let me mention a couple of nice things about the Jones polynomial, and then what we're gonna do is we're gonna sort of generalize the situation. Why is the Jones polynomial? I mean, you saw this construction, difficult, but you can do examples. And the reason the Jones polynomial is of interest, it's because it's actually was used to solve Tate's conjectures. So in 1900 years later, so in 1885, this is when these conjectures were made, what we were gonna do is we for Jones come up with this 99 years later, and then you can prove it. So Kauffman, Moisuki, and Thistlewaith proved the first two of the conjectures using properties of the Jones polynomial, and then four years later, Manasco and Thistlewaith proved the third conjecture, which is about piping, okay? So there are many things we do know about the Jones polynomial, but there are actually also many things we don't know about the Jones polynomial. One of the more philosophical point of view is you could say, why does it exist? But the question that sort of piques my interest is the following, we don't know the answer to this question. Is there some non-trivial knot? What I mean is it's not the un-knot with the Jones polynomial equal to one. We prove that there are infill and many knots, inequivalent knots with the same Jones polynomial, but we don't know what this question is. Both equal to one. Any best results in this direction, one of our favorite results, is this fairly recent one of Sakura and Tunza from 2021 that says, suppose you have a knot with the trivial Jones polynomial, then there's only one of two situations that could happen. So you could either, it could be the un-knot, which has Jones polynomial one, or it has at least 25% so it just depends whether or not you're optimistic or not, okay? So that's kind of a nice result. What we're gonna be interested in, especially as it links to the second world, is this generalization of the Jones polynomial. So the moral here is that the Jones polynomial is one Laurent polynomial. So what we're gonna do is we're gonna embed this one polynomial into a sequence of Laurent polynomials. The idea is gonna be the following. Suppose we have k is a knot, and then what we're gonna do is we're gonna talk about this thing called the inf-colored Jones polynomial. I am certainly not going to define what this is. The way you should think about this is that this in, this is a natural number. This is a sequence of Laurent polynomials, okay? And let me just show you one example of what do these things actually look like. So if you go back to the trefoil, it looks like this. So you have q to the one minus n, you have the sum n greater than or equal to zero, q to the minus n capital N, and then you have the symbol q to the one minus n, little n, where all the symbol means it's just a very compact way of running out of product. This is what's called a Q-Pockheimer symbol. And so this just means that whenever you see the symbol, it's just one minus a, one minus aq, but the thought all the way up to one minus a, q to the n minus one. So the a stays fixed, and you're just increasing the powers of q. So all of this notation just means that everywhere you see an a, you substitute in q to the one minus n. A couple of comments about this, well, in general, and also about this example, you could look at this example and complain because you said, wait a minute, you just said the color Jones polynomial was a polynomial. And this, the way you've written it, it looks like it's an infinite sum. It's not an infinite sum, it really is finite. Because what happens, you can check, is that this Q-Pockheimer symbol q to the one minus capital N to the n, this is zero if little n is greater than or equal to capital N. So what that means is this actually truncates at capital N minus one. So it really is a polynomial, and it really is a little wrong. It means you could have negative exponents because of you have q to the minus little n capital N. You pick up negative powers of q here. So it really is a polynomial. The other thing I invite you to check, well, let me make a general statement. The reason this is a generalization is that it turns out that if you take capital N equal to, so the second color Jones polynomial, that actually recovers the Jones polynomial. And I invite you to check that if you take n equals two and compute all of this out, see that you really do get back to Jones polynomial for the right hand trip or that we saw in the previous, or check. Related to the open question, there are other integers that you don't know if there are any not responding to, or. That's a good question, that I don't know. That's a good question. You change one to seven. Yeah. Yeah, that I don't know. Okay. Yeah, that's a good question. Oh, if you pick and prove this as non-trivial. Yeah, that I don't know, I don't know. This goes as you. Can I ask again, what was the third tape conjecture? Third tape conjecture? Okay, very good. So it's about flanking. So what you do is, I think it says that if you have, if you're given two, if you're given two reduced alternating diagrams for the same prime knots, actually, I won't talk about knots today, then they must be related to each other by a series of flapping moves. If you have two regions and you have a crossing and then flip it, so you can take one and then you do some flights and get to the second one. You can do this for any two alternating diagrams for the same alternating knots. Very good. Thank you. You already know what, you probably already know what the second one is. Okay. The second one, I think it says that, I think it's a car roll alternating knots at zero ride, if I remember correctly. Maybe I should also mention just out of it, well, okay, I won't say that. Anyway, I'm tempted to say it though. No, okay. Okay, so what we're gonna do is we're gonna cut. So this is the first world, okay? So what I wanna do is I wanna switch gears completely and start talking about modular forms. And then at the end, I wanna connect the two worlds together with some fairly recent work. So first of all, what is a modular form? So whenever non-methodition asked me, what is a modular form? It can be kind of a tricky business, but I always sort of, I like this quote. So there are five fundamental operations of arithmetic. Addition, subtraction, multiplication, division, modular forms. So this is a quote to apparently do the Martin Eichler. And the reason what makes this quote interesting is that the funny thing is modular forms have this tendency to appear in a wide, variety of topics, places where you would certainly not expect it to appear. So I'll give you the formal definition to help you if that's fine. If it doesn't, that's fine. I'll show you. I wanted to really kind of focus on the applications. The modular form of weight K, K at least from the beginning is an integer. What is it? It's a whole more function on the complex upper half plane such that it transforms very beautifully under the action of SLTZ. So you have this F of this Mobius transformation, AZ plus B over CZB. You have this piece out front. And that's what I mean by the nice transformation probably. And you want this to be true for all points in the complex upper half plane and all two by two matrices in SLTZ. There's actually, well, and there's another sort of slightly technical condition, but I'm not sure about that. But the point here is that what this implies if you chose the matrix, say 1101, you would have F of one plus C is equal to F of Z. That means the function is one periodic. And that means that it has a Fourier expansion. And at the point that here is that these Fourier expansion, if you write up the coefficients in this Fourier expansion on a variable Q, Q here is each of the two pi I, D with the complex upper half plane, then these coefficients have this tendency to contain a wealth of arithmetic information. And that's where these type of objects appear. So they depends on what your problem is. Sometimes these module forms can pop up. And so let me just talk about some of the applications, certainly not all, but I guess in some sense kind of the most famous one is from us last year. From us last year, it was this statement going back to conjecture from ongoing back to 1637. It says there are no positive integer solutions X, Y, Z to the equation, definite equation X to the Z plus Y to the Z is equal to X to the, sorry, X to the N plus Y to the N is equal to Z to the N for any N greater than equal to three. Stated in 1637, but it's finally proved by Wiles and company in the early 90s, after a considerable amount of work. And it opened up a brand new avenue of automotive forms. They also occur in mathematical physics, string theory in particular, algebraic geometry. So for example, if you have some variety and you want to count points on this variety of our finite field, a lot of times that point count can be packaged into a generating function and that generating function gives you exactly a modular form. They also occur in combinatorics. We're gonna see that momentarily. And then as Rico sort of previewed a little bit earlier, it's sort of great because about 48 hours to 48 hours and 36 minutes ago, Marina Diazovka actually won the Fields Medal for her work on the spearpacking problem. So I'm not gonna talk about the spearpacking problem. Watch her do it, experts. And if you look at her proof for how she did it, it actually uses modular forms. So if you want, so it's a remarkable piece of work. That's sort of the key. To be strictly, to be very precise, you use something called quasi-modular form. That's another important avenue. So what I want to talk about today in the sort of the world of modularity is I wanna talk about one particular mathematician and the reason why I'm sort of choosing this mathematician to talk about is because this person's work is applicable to sort of what's happening now with this connection between knots and modularity. So this mathematician, again, I don't assume you've ever heard of who this person is. So this person I wanna talk about is Srinivasa Ramanujan. So Ramanujan was born on December 22nd, 1887. So here's his famous passport photo coming into England. He was born actually in Erode, India, which is I think about 400 kilometers southwest of Madras, which is not Chennai. He was a talented student in all areas. That was until about age 15 or 16. And then you ran across this book by GS Kar and titled a synopsis of elementary results in pure mathematics. And this is a horrible book, right? Because if you haven't seen it, why is it horrible? So it's horrible because there are about 5,000 formulas just listed with little note proof, right? And unfortunately, he followed this style, but he would write down his discoveries in notebooks and then he wouldn't really say how he did it. So once he saw this book, he became in some sense kind of obsessed and he ignored all his other studies. He actually flunked out of college, but in 1912, he was able to secure a job at the clerk in the Port Madras Accounts Department. He was extremely fortunate because in this accounting department, his coworkers actually had a strong background in mathematics and they spotted his talent. So what they did is they encouraged him to write to mathematicians the time in the UK. And so he did. He wrote to, I don't know how many, I know he wrote to someone in University College London. No, Imperial College, and they ignored him. And then he wrote to someone else and they were very drug-atory, they were not exactly helpful, it's fine. He finally wrote a letter to Hardy, G.H. Hardy in Cambridge. And that's the first letter. The first letter, I'm just gonna read parts of it. He wrote to Hardy in January 16th, 1913. And in it, he states, following, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Office of Madras. I'm not trodden through the conventional regular course, which is followed on the University course, but I'm striking out a new path for myself. I made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as quote-unquote startling. So in the beginning, Hardy didn't quite believe these results. And so he worked with Littlewood and then they went through the letter. I mean, the letter is sort of a menagerie of results in number theory. I mean, if you look at it, it has like series evaluations, interval evaluations, it has some topics in number theory, like if you wanna count the number of primes up to x, getting asymptotics, you know, count the number of n, national number is n, which are the sum of two squares, getting asymptotics there, which is all over the place. But what they looked at the results and they classified them into three categories. The first category was already known. The second category was possibly new and potentially interesting. And the third category was new and important. And based on going through that letter, Hardy wrote back. So Hardy wrote back and that started a correspondence and they had invited Ramonaging to come to Trinity, Cambridge, and he did. So this is very brief, but for the next five years, from 1914 to 1919, he wrote 30 papers, seven with Hardy, and it was a very broad, very wide variety of topics. So I'm certainly not gonna go through all this. We'll talk a little bit about partitions. Unfortunately, I won't talk about Q3s at all. Finally in 19, oops, 18, he was recognized for his work. So he was named a fellow of Trinity College in Cambridge and then he was elected a fellow of the Royal Society of London. He was ill actually. So in fact, his last two years in Cambridge, he was under nursing help for a time. And this might be due in part to the fact that he was actually a strict vegetarian during wartime in Britain. So in an effort to improve his health, he actually went back to India. Really, it's February 1919. Unfortunately, his health did not improve. And then one year later, he died until 26, 1920, at the age of 32. So that's a very brief synopsis. Okay, so what I want to do to sort of highlight the fact that modular forms can sometimes tell you something surprising is I wanna talk about one of his favorite problems and that was about partitions. So let me talk a little bit about the partition functions. So I mean the partition function and combinatorics, so apologies, not partition function physics, they're all physics. So what is a partition? So a partition is, well, what we're gonna do is we're gonna take some natural number and basically ask ourselves, how can we sort of break it up? So quickly speaking, what a partition is a non-decreasing sequence of positive integers whose sum is n. And what all we want to do is for a given natural number n, let's keep track of how many partitions you could have, okay? So let's do some examples and then show you what we're managing solve. So let's count the number of partitions of four. Four, one, two, two's in ones. Better have all ones and hit. So we have P of four is five, there are five partitions of four. Do another one quickly. So P of five is seven and that's just because you have five, you have four plus one, three's in twos, three's in ones, two's in ones and then let me just let you check on the P of six is actually 11, okay? And this is all fine and good for small values of the, you can compute small values by hand but if you want to compute large values, this becomes quite onerous quickly. So let me just say a word about larger values. So Percy Alexander McMahon who's actually in the Broyish Broyal Altillery, he actually computed a table for all values of partition function up to 200, okay? And here's what he computed. So he computed that P of 200 is a little bit short of four trillion. Well, probably one thing that everybody can agree on is what he did not do. At least I hope. Okay, so what did he do? But what he, okay. So what he didn't do, what he did do is he used the fact that there is actually a recursive formula for the partition function. So he used this recursion. And so the point, couple points here. So what you do can do is express P of n in terms of lower values, these numbers, diagonal numbers. And what you can do is you can define P of a negative integer to be zero. So that means the sum truncates. And so what he did is he computed small values and then you build up and you get up to this larger value. I should also mention that this is just a consequence of something called the Euler's pentagonal number theorem. Okay, so that you can compute values there. And what he did is he published a table of values and then Ramanjan saw this table and then he rearranged the table and then he spotted this pattern. So what he did was, so P of zero will define to be one and P of one is one, P of two is two, P of three is three and P of four, that's what we saw before, that's five. Then you have the next row, P of five, P of six, P of seven, P of eight, P of nine, four values. And then what he spotted is what you see in red, maybe I shouldn't have colored it in red, that's fine. What he noticed is that if you compute P of four, you have nine and these other values in the last column, that they're actually all divisible by five. And then what he did is he noticed that you actually have more divisibility properties than just five. So he looked at more values and then here's his quote. So in 1919, he said I proved a number of arithmetic properties of P of n, in particular that P of five n plus four, so you look at that arithmetic progression, this is divisible by five, so this is for all natural numbers in, P of seven n plus five is divisible by seven, again this is for all natural numbers. And then I've since found another method which enables me to prove all of these properties and a variety of others of which striking is P of 11 n plus six, this is divisible by 11, again this is for all natural numbers. Here's for the best part, he says this is it. So it appears there are no equally simple properties for any other modular involving prime other than these three. So what he means is that if you're trying to search for a remodeling type congruence for the prime 13, it's never gonna happen. You're not gonna have P of 13 n plus some congruence classes zero module of 13. Place 13 by any prime greater than 13. And that was his conjecture based on data. So let's fast forward 84 years later after the advent of modular forms and Scott Algren and Matt Boylan paper appeared in Vincione, it proved the conjecture true. And a key aspect of this proof is the fact that if you look at the generating function for the partition functions, this is an example of modular form. They go from there. It's approved by contradiction. Beautiful paper, short review. Okay, so that's his congruences. And also that's a sort of another use of modular forms. We've seen the first letter, but what I really, really wanna get to is the last letter. In some sense that ties back to knots. So the last letter, so three months before his death and on April 26th, 1920, he wrote, he only wrote one letter when we went back to India to Hardy. And this is it. So this letter, everybody can see it. It says, if we consider a theta function and transform, okay, Euler and should go there. Euler and form. And then he writes down some examples of Q series and determine the matter of singularities that these are the roots of unity. We know beautifully the asymptotic form function. And then the point, so it goes on and then he writes down some asymptotic properties. So what are called Q hyper geometric series. So what was the point in this letter? So in this letter, he introduces this notion of what is called what he called a mock data function. So what he says is I'm extremely sorry for not writing you a single letter up to now. I've discovered very interesting functions with a recently called mock data functions. Why do you call them mock? Well, roughly speaking the way you can think about it is that data functions, you can think of these, these are examples of sort of classical modular forms. And these other functions mock or mimic asymptotic properties of these classical modular forms. Rough way to say it. And in this, let me show you an example of, okay, they enter into mathematics as you play as ordinary data functions. Let's see the letter with some examples. So he wrote down 17 examples. Damn, what do you call it mock data function? Obviously he wrote down 17 examples of mock data functions of certain orders. But he never said what order meant. That this is order three, this is order five. So let me give you an example. So F of Q, it looks like this. This is what's called a third order mock data function. We kind of now know, have some idea about what that means related to the level of modular form. So this is one plus Q over one plus Q squared, Q to the fourth, one plus Q squared, one plus Q squared squared. And you can write this more succinctly in terms of the notation we saw before. This is an impenetra sum, n equals zero to infinity, Q to the n squared minus Q to the n squared. So there's 17 examples there. And his last address is President of the London Mathematical Society. Watson wrote down three more, three, and then George Andrews. He found two more in Romanians, quote unquote, lost notebooks. So the reason why I'm putting these words in quotes is because it was neither lost nor notebook, okay? It was not lost because it was actually in the personal belongings of Watson. And it was certainly not a notebook. It was about 500 loose sheets of paper in which you had formulas just scribbled all over all of them. And so Andrews and Bruce, George Andrews and Bruce Berndt have written five beautiful books going through all of these pages, writing down all of these formulas and then finding proofs for them if they're correct. If they're not correct, fixing them, then finding the proof. It's a monumental piece of work to go through all of this. Why are these things interesting? Especially with people in mathematics and then also outside of mathematics. Well, the point here is that there was this belief that they should be connected to what I talked about before a lot of the forms. And that's the reason why they should be connected is that for the self-feeling is that these satisfy really nice identities. So you have one mock data function plus another mock data function is equal to some infinite product which is a modular form. And those identities are reminiscent of identities in the classical theory of modular forms. So that's why there was some feeling that they should be, you should be able to relate the two theories together. So this is 19, we have to fast forward again and wait 82 years. So Dutchman enters the picture. Saunders-Vegers in his 2002 PhD thesis solves it all basically. And he came up with this framework which Nano explains these 22 mock data functions but all the ones that people wrote down since then how they fit into this area of modular forms. So let me explain very briefly how this works. What you do is the missing pieces of following. Modular, these mock data functions by themselves do not transform like a modular form. What you have to do is you have to add this error term, this piece, what is called a non-holomorphic integral. This piece which is called a mock-modular form plus the non-holomorphic integral it does transform like a modular form. This whole completed thing is what's called a weak-moss form. And this was a stunning development. I mean, yeah, so I'm a little bit biased but I dare say this is probably one of the most important PhD pieces in pure mathematics in the last 20 years. Let me give you a very crude, so apologies for this, but a very crude estimate of how important this thing is, how important this thesis is. So since 2007, there's a little bit of gap. There's about five years before people understood the importance of his PhD thesis that might be due in part to the fact that so he get his PhD thesis in 2002, he was a post-doc in Bonn for a year and then he quit, left mathematics altogether. But he was encouraged to come back, he's back. And so since then there's been about 200 papers, but what I did, all I did, this is a very rough and definitely an underestimate, all I did was I went into math or cognitive, I typed in mock modular form and 170 papers come up and then I checked math signets and there's 30 papers which are not on there. So that must be at least 200. And there's been a lot of conferences. The one that's coming up here in, I think Newton Institute is at the end of August, I'm looking forward to hopefully watching some of the videos for non-mathematicians that might. If you want to read a very beautiful survey of work, I would strongly advise you to read Amanda Folson's perspective on mock modular forms. It really gives you an overview of why these things are important, historical background and also connections to areas outside of mathematics. So for example, there is some connection to what are called what, quantum degeneracies of black holes and somehow what you can do is you can package that information into a generating function and that turns out to be an example of a mock modular form. I'm hoping somebody who knows more about black holes can explain that. Okay, she also mentioned, if you want to hear more about this story, Miranda Chang, who is a string theorist at the University of Amsterdam, she has done spectacular work relating a mock modular forms to moonshine, watching her talks on YouTube, pastic talks. So that's mock modular form and mock data functions. So what's been going on very recently, so very recently, within the last 10 years, Don Zagie, who incidentally was the PhD advisor of both Saunders figures and Mariamma. He's a permanent member, well, he's in many places. He's in Bonn and also in Trieste and also in China. He came up with this more relaxed notion of a modular form. So let me very briefly say what this is, what he defined it to be a quantum modular form of weight K. This is a function on the rationals now, so it's not on the complex upper half plane, such that for all points in SL2Z, you have what you do as you look at what's called this error and modularity. So the point here is that if this was an ordinary modular form, this difference would be zero. But we don't want it to be zero necessarily, what you want is you want this function R gamma to have what are called nice properties. So they could be C infinity or it could be real analytic. And the point is that when you're strict to the rationals, you have this nice transformation form. And since the advent of this notion, there have been, so you can tweak this definition, you can, the weight doesn't have to be an integer, it can be half integer, it doesn't have to be an all of Q, it can be on subsets, it doesn't have to be on all of SL2Z, it can be on concrete uppers. All right, so let's go back all the way to the beginning. What is the connection between sort of this notion of modularity and those not invariants that talked about the color Jones polynomial. And that's what the recent result is about. So let's go back to the right-handed trip oil and it's mirror image, the left-handed trip oil. So what you can do for each one of these knots is you can compute the color Jones polynomial. That's some Laurent polynomial. And then what you can do from this color Jones polynomial is if you let Q be a root of unity, you can extract an expression. So this F of Q, which is the sum from the ingredient for the zero of this quack or some of Q sub n. So in fact, what I invite you to do is to compare this expression with that formula that I showed you before for the color Jones polynomial of the trip oil and see that they really do match when Q is a root of unity up to the pre-factor. And then it turns out also if you look at the color Jones polynomial for the mirror image, so this knot invariant, what you can do is you can play the same game and you can extract a series here, U of Q, something like this. And the point is both pieces of modularity turn up. So the theorem of Don Lugge says in this first case, so here I'm being quite loose here because you have to appropriately normalize this function, but this turns out to be a weight three-hands quantum argument form. So it's quantum, this thing which comes from a knot invariant is this new type of modular form. In the second case, it's a weight one half mock module form. So it's something which born out of these letter, this last letter from party to Ramanujan, we have this modern perspective and now we have the proof that this is a mock module form. What I wanna make here is that all of this work is for one particular knot. And so very recently, what we've been able to do is to extend this. So as opposed to looking at one knot, you can actually look at an infinite family of knots. So what we're gonna do is to look at the family of what are called torus knots. These are knots which live on the surface of a torus. Torus knots are indexed by two coprime positive integers, P and Q, roughly speaking P counts a number of times you go through the hole, Q counts a number of times you go around. So you can check that the trefoil knot, T of three comma two is this. So that picture that I'm drawing here on the surface of this torus, it's exactly the first picture we saw on the first slide. So we weren't just interested in looking at this one knot, but we actually are interested in looking at an infinite family. So it turns out what you can do is you can compute an explicit expression for the color Jones polynomial for this infinite family three comma two the T. When T is one, you get back right in a trefoil and you can do something similar from this beautiful formula due to Isaac Conan in Paris. You can extract an F of T function and then what we've been able to show, this joint work with Ankush Goswami at the University of Nottingham, is that this infinite family is also a way to have quite a much more appropriate linear normalize. And so this result, especially in also connection to the original two results, it's part of a much bigger framework and it's really that bigger framework that we're gonna go after. That's what we're gonna span at least next week. Five, 10 years on. And here's the picture. The general picture looks like this. So what you do is you have your knot K and you can go in one of two directions. It depends on how the color Jones polynomial looks like. So if it looks like it's a psychotomic expansion, then what you can do is you extract this U function and then you get a much more controller form. If you go down this way and your color Jones polynomial looks like it has a non-psychotomic expansion, what does that mean? And then you can extract an f of q and then we supposedly get a quantum module form. And this has been shown whose picture has been sort of verified in the following cases. Well, that's what I just showed you of Zagie, a three of bright-handed trefoil. We know that we can go down this way and we get a quantum module form. Result of Kazaura Kami who's in Kushu who's gonna come visit me here in August. And Jeremy Lovejoy, they showed that you can go up this way. That's the result I showed you. You get the quantum module form. For this infinite family of torsnot two comma two M plus one, Kami showed that he wrote down the non-psychotomic expansion and you get a quantum module form. And then very recently, Eric Mortensen, I talked about this in the seminar and I said, Eric hurry up. He did. So apparently enjoying work with Eric Mortensen and Sanders-Vegers. They went the other way. So the point is actually Kami and Lovejoy wrote down an explicit expression for the psychotomic expansion for the color Jones polynomial. But what they did not necessarily, what they didn't show is that it was a quantum module form. But these guys did. And it's today. So they posted it on the archive today. I just saw it this morning. And they show that it's a, it's actually, I strictly speaking, it's got a mixed quantum module. But that's that picture. Join work with Ankush Goswami. We have this side of the picture. We have no idea about this side of the picture. So there's an obstruction, which I'm not going to really talk about. I'm currently working with a couple of Japanese mathematicians I'm talking to here, but at least then also. And so the picture, all of this story is for torus knots. But there's a deep result of thirst and what says that knots can be classified into three types, phenylite, torus, which is here and also hyperbolic. And so a natural question is that for these other two types of knots, do we have a picture like this or do we have something that you changed? And that is a long-term ongoing project that we're going to continue to work on. I basically need an army of students. I need an army of grassroots students. Get this going. And finally. Any questions? Also from online? Maybe I'll ask one question first. So what do you mean by this quantum module? Yeah, thank you. It has to do, yeah, so why is it called quantum? So one reason it's called quantum is because of the construction of the color John's polynomial. Okay, so the reason is that one way you can construct the color John's polynomial is to use quantum groups. So basically what you do is you look at the braid that gives you the braid representation of a given knot. And then what you do is you color the crossings by irreducible representations of this quantum group in UQSL2. And that's why he used the word quantum. Because the fundamental example comes from the color John's polynomial. At least that's my understanding. Thank you. Do these weights associated with these forms can they act as like an invariant on the knots you're plugging into them? Not that I'm aware of. Yeah, I don't know if they smell. Classify the knots you're getting. Yeah, that I don't know. See, okay. I wonder if you could help the peons and the audience like an engineer. I wonder if this comes up in circuit board design where you have to have the different circuits on them and then maybe these nods become more important in that environment. I don't know. I was thinking about the brain because you have these massive axonial buses going back and forth. I wonder maybe that's also, the brain has to handle it to keep things functioning and instead of an epileptic bed or something. But I'm just thinking. Yeah, I don't know. I have one thing. Even if in the near future you end up fully seeing this picture which connects knot invariants with modular forms. Is there expectation how this would improve the understanding of knots in general or vice versa? Yeah, that's an excellent question. I mean, the only thing I can say, yeah, so the answer is gonna be big, so apologies, that the only thing I can think of at the top of my head is that there's something called the volume conjecture. And the volume conjecture connects the color Jones polynomial to something called the hyperbolic volume of the knot. And that conjecture in of itself is sort of amazing because on one hand you have this Laurent polynomial and somehow this Laurent polynomial knows information about the geometry of the knot. And this conjecture has been verified, for example, for torus knots, because in that case, the hyperbolic volume of torus knots is zero. So you get it right away. But one thing that the connection could ask is that the reason we're getting modularity so far is because all we're doing is looking at torus knots. Is that somehow related to this hyperbolic volume zero property? And that's what I don't know. I don't quite know or understand basically what happens when you get outside this picture. So for example, if you look at the four one knot, if you look at the figure eight, then there is some very recent and beautiful work of several square philitis and Don Zaghe where they talk about a generalization of the volume conjecture and talk about another type of modular property. But it's, they have worked out some examples. But I don't think this is a general theory yet. So somehow this polynomial gives you both pieces of information. It gives you modularity and also it gives you information about the geometry of the knot. So that's what I would sort of say. I shouldn't also mention there. Yeah, so this whole story is where capital N is fixed. It's actually a completely different story. This could have been a completely different talk if you actually let in vary. So if you let in vary, it turns out for alternating knots that the coefficients are stabilized. And a lot of what you can do is you can write down a power series which is built out of the coefficients which stabilize. And a lot of times these power series also turned out to be modular forms. And so that's a phenomenon we also do not understand whatsoever. Why is that happening? And how does it relate back to somehow the geometry of the original knot? So what I would say is that both things are completely wide open. A very naive question, maybe, but you showed in the beginning this guardian knot and then you said it's equivalent to the unknot. But is there any relevance to this particular knot? No, no, I just wanted to... The reason I gave you that example is to show you that if I just give you some random knot, terminating its equivalent classes can be kind of tricky business. Okay. Which is sort of for illustrative purposes only. What is this duality that you're pointing to? Oh, I did not explain this at all. Thank you very much. So it turns out that these two things are related. And the reason why they are related to each other is because the Jones polynomial, in some sense, kind of satisfies a duality. So what happens is that if you compute the colored Jones polynomial for your knot and you replace Q by Q inverse, then you get the colored Jones polynomial of the mirror image. And so what that means is that when you extract this expression and you extract this U function, they also are related to each other via that duality. And that was a property which was first written down by a paper of Ken Ono and some students in an REU paper. But that duality is what their word can actually be explained just using that property of the colored Jones polynomial. And so what we don't understand, I mean, these are sort of two separate pictures in some sense, right? Because to prove that this is a mach modular form and that the properties that you use to prove this is a mach modular form is completely different than trying to prove this is a quantum modular form. And we don't understand why that's true. So for here, you have to get what's called a hecotype expansion. And that's exactly what Eric Mortensen and his figures do, and that's why they get this thing here. But that tells you no information whatsoever about if you go down in the duality to write down the F of Q, proving that that's a quantum modular form. It's a completely different proof. That's also something we would like to sort of connect to each other. Yeah, let me also mention, well, yeah, let me also mention there is some indication of how to go back and forth. So for the Trefoil knot, you can use what is called a Q-hybrid geometric series transformation and you can actually go back and forth between that F of Q that I wrote down and the U of Q that I wrote down. There's a way to transform one into the other using Q series techniques. And so one thing that we're going to try to look at is see is that transformation that works in the space case for Trefoil knots that somehow generalize to other families of knots. Outside of tourist knots. For example, if you look at the 401 knot and you look at its psychothermic expansion, look at its non-technical expansion for the color-jones polynomial, can you somehow transform one to the other and then you'd have the right expansion? And the reason why that's important is, for example, in cases in which we don't know how to do this one, right? This is all dependent. The reason we don't have a U here is because we don't know how to compute the psychothermic expansion. But we do know how to compute the non-psychothermic expansion. So if we had some way of starting here and then flipping and going up the duality and getting the psychothermic expansion here, then we'd be in business and we could prove that it's a Mach-Modular form. But we don't know the transformation property at that general. Sorry, so is there a duality between Mach-Modular form and Quantum-Modular form? I mean, Mach-Modular forms, there is a way, it's an example of a Quantum-Modular form. You can, there's a way to explain it internally. It gives you a Quantum-Modular form. But my point is that proving that this type of thing is a Mach-Modular form is completely different than showing that something like this is a Quantum-Modular form. Sort of two separate techniques that you use to prove each case. And one does not seem to help the other. That's what we're trying to understand. Why is that happening? So it seems like you're saying it's hopeless to expect that one can prove is if the upper stairs, if the upstairs is a Mach-Modular form, then proving the downstairs thing is a Quantum-Modular form. It's not clear to me how to do that. I don't wanna see a Hopeless, right, it's not clear. We have to wait a hundred years. Any other questions also online, maybe somebody? Basically, this Quantum-Modular form can be obtained from the color transformation in a sort of Laplace transform process, like there's something called the higher gear transform, which is similar version. Does this relate to this? So a bit more about this, what type of transform this? So the higher gear transform is basically taking the variable Q and replacing it with some of the lambda in the end. So we get the variable N instead of the Q. I guess, I don't know. And it has some nice factories, ability properties when we're talking about Taurus nodes, but then not for other kinds of nodes. So I thought maybe this kind of relates to this construction here. Yeah, is that related to matrix models? Or is it? Yes, it is related to matrix models. So basically there is something called the node matrix model where the Jones polynomial is used as a character. And then, yeah, this higher gear transform kind of defines if we can construct a matrix model out of Jones polynomial over nodes. Okay, so the answer, I don't know, but I would like to see if there's a way to extract modular information from this matrix model interpretation. One comment I wanna make, please clarify this if I screw up, is there a matrix model known for non-Taurus nodes, like for twist nodes? No, there is none. Because of the, yeah, they don't factorize nicely in the case of non-Taurus nodes. So I guess the factor is ability is kind of like a requirement for constructing a matrix model. Yeah, we should talk some more. We need to find out who this person is. Okay, yeah, let's definitely talk some more offline. Is that okay? Yeah, very interesting, very interesting. Any other questions? Are you? So your answer earlier about the why quantum and basically said something about quantum SL2 only and some representations and stuff. A lot of quantum groups out there and a lot of different representations of different quantum groups as well is like, and a lot of not invariants. Like if you get anything, why, everything's like Jones polynomial, Jones polynomial, QSL2, blah, blah, blah. Why not study some other invariant? You could, so the only, yeah, so the only other invariant that I'm aware of in which you can extract modular properties is a thing called the Witten or should you can derive invariant. Something called a, there's a unified version of it. And in that case, that, so what's the story there? That is an invariant of three manifolds, I think. And so the idea there is what you do is you take your knot and you do certain rational surgeries and that gives you a three manifold and then you compute this unified WRT of that three manifold and in certain cases you can actually prove it's a Mach modular form. And it's the same sort of, you have to get this piece. You have to be able to write down this hecotype expansion exactly with Morton's and Beggar's there and then you get that as the Mach modular form. So that's the only case this unified WRT invariant, that's the only case in which it's been proven that you, there's this connection to modularity. I don't know if anything has been proven in cases like for example, the colored home fly or these other situations, these other constructions of knot invariance coming from other quantities that I don't know, I'm not aware of. Any urgent last question? Okay, then let's thank the Robert again.