Source (in Korean): https://www.youtube.com/watch?v=079pj0NLDkU
Thank you so much for being here today. I think it’s a really special case to be able to know the things I wanted to say while receiving congratulations from so many people at such an event. As I was preparing for today’s lecture, I thought about various things. Actually, I have almost no experience in public lectures for people who are not mathematicians, so I thought about what topic I should talk about so that many people do not get bored with it so that they can listen to it happily. I brought this up (Setting up boundaries & relationships) as the subject of my lecture because I wanted to share my thoughts on boundaries and relationships that I felt while doing math research.
Do you have any memories of playing while looking up a dictionary as a child? I played a lot while doing this. I started playing with any word, found the definition of that word, found another word I liked among the words that appeared in the definition, and played around looking for a word I liked among them. For example, as I was preparing for today’s presentation, I looked up the two words in the title of the presentation in the dictionary and found this. “Boundary” is the limit at which things are distinguished by certain criteria. And “relationship” means that two or more people, things, phenomena, etc. are related or in relation to each other, or something like that. It was written like this. Then choose another word from the two. I’m going to pick out the word “relation” that appears in the definition of “relationship”. If you look up the word “relation” and look it up in the dictionary, it says this. “Relation” means that the position, association, connection, or status of one person or thing with regard to another or others, or something is entangled with something else. It is written like this, and I will choose “connection” from among them. I looked up what “connection” means, and “connection” is the connection of one object with another. Again, let’s find out what the “object” is. “Object” is a subjective thing to which the actions of consciousness, sense, and action are directed. You can play like this for an hour or two while doing this.
What kind of fun game can be played while doing this as an example? To define a certain word, it is not good if the word you want to define appears in the definition itself because it becomes cyclical. If you search for the definition of a word in this way, search for it again and again over and over, you will have no choice but to return to the place you once went through. Because the number of words we have, though many, is finite. So in this way, you can always make a loop like this, how quickly you can make a loop and get it back to where it was. Of course, it depends on which word you start with, which word you choose to build your path with, but you can always make a loop this way. I found such processes very interesting. However, in the end, when you think about these things, at first glance it seems silly to define the words themselves using the words you want to define, but in our daily life, the language system we have works really well, and we can use this language to create cool things – you know from experience.
So, after thinking deeply about how this happened, at a toy store that my first child happened to visit when he was little, I just bought a toy at once, saying in mind, “Is this the toy I saw? I think this toy well expresses my feelings about language.” Actually, my child doesn’t play much with it; I play well with it. It’s called “Skwish,” a toy that was quite popular in the US a few years ago, and it’s made of wood and rubber bands. It is an icosahedron supported by six staggered rods. If you look at it more closely, there are hard parts and soft parts. The hard parts represent the 6 rods and the 12 vertices of the icosahedron, but the hard parts are not connected to each other at all. It’s amazing how they are softly connected to each other only with rubber bands, and how they support each other and maintain their shape without being fixed to any floor or hard place. No matter how much a child twists and tries to do this, it will come back to its original position in no time.
I think that creating and maintaining the meaning of an icosahedron while supporting each other like this felt very similar to language to me. If you continuously follow up definitions, you will eventually come back, but it is strange that it is not empty at all but has a distinctive meaning. So, I thought, just as these 12 vertices create meaning for a relationship through rubber bands with each other, I wonder if we are creating meaning of any kind through these two actions: Setting up boundaries and relationships.
Relationships seem to be the most important element we need to create meaning. I think all meaning comes from relationships. However, there is one very important prerequisite for a relationship to exist. For there to be a relationship between you and me, there must be a boundary between you and me so there are you and me. In this regard, the boundary-making that creates the boundary between us must always be a premise before creating a relationship. I am going to use the remaining time to convey what I felt while doing math, that this very similar kind of work was going on.
Now let’s consider a similar case in a similar frame of mathematics. Many people seem to have a different image regarding mathematics. I was like that when I was young. In mathematics, never using a circular definition, thinking about why the proposition is true no matter what proposition it starts with, thinking about why the proposition that makes the proposition is true, and repeating this process over and over again, we have been taught that we are faced with a finite number of axioms that cannot be reduced to other propositions, and we often think so.
Here, like the most basic assumption of proving other propositions of axioms, such as an indivisible atom, these are finite and always result in them. In other words, it is not a structure that supports each other like the toy we saw earlier, but a tree with hundreds of thick leaves and branches is reduced and reduced, and I have greatly felt that something is fixed to a solid place by axiom, as if it returns to a single root.
However, this is in fact limited in contexts, such as “studying Euclidean geometry”, “Euclidean geometry has as an axiom that given two arbitrary points, there is a unique straight line passing through them”, and “the other everything can be exposed from these few axioms” – this seems to be the right context, but not in this limited context of Euclidean geometry or some other geometry or any algebra; in terms of mathematics as a whole, in fact, rather than the image of this tree, I think the toys we saw and the images in which our language supports each other really fit together well.
I’ll try to convince you guys. Now let’s go back in memory and think about the first section of the (very famous) Art of Mathematics for high school students. Everyone may usually have looked at that part only, and so the sides are blackened out. If you look there, you’ll probably remember at first glance that there were these kinds of paintings that were the most important. Now in the chapter for propositions and sets, when I was learning it was part of what was called common mathematics, and how these four propositions were connected and what they were called.
“If p, then q”. Its converse is “if q, then p ”, its inverse is “if not p, then not q”, and its contraposition is “if not q, then not p”. Well, in this way, I learned about proposition with its converse, inverse and contraposition, but among them, there was one that was clearly the most important. After all, the contraposition was the most important; at least that’s what I remember. This is because of the fact that: a proposition and its contraposition are equivalent. In other words, I learned that the proposition “If p, then q” and “if not q, then not p” are equivalent. This might be the most important part of that chapter.
The expression “equivalence” appears here. What does “equivalence” mean in mathematics and logic? It means “two sentences are exactly the same”, meaning that “if one statement is true, the other is also true; if one statement is false, the other is also false”; The so-called sharing of truth values.
There is a case where a proposition is equivalent to another proposition; there is a case where a proposition is not equivalent to another. Let’s think of this in view of such relationships. When you think of proof by drawing it like a cartoon, it would roughly look like this. There are at least a few propositions that you already really know. For example, let’s say here are the three true propositions you are well awared of. And we have “a proposition we want to know whether or not it is true”. What we mean by “we proved a proposition we want to know whether or not it is true” means that we repeat a finite number of reasonings from known propositions that are true – such reasoning’s correctness often depends on known true propositions. There are many, many, arrows of reasoning in this way, either there is an equivalence, or just an implication, and so on; with such arrows, going from the domain we already know to the area we want to know one by one: that’s what we call “proving” in mathematics.
From the Almighty’s viewpoint, very simple structure of the space of all the propositions would be seen. That is, there are just a point for true propositions and just a point for false ones in the space. Now, consider the space of all possible propositions, not just these 1 or 2 propositions. Let’s just think of all the propositions existing in the space. Imagine what this space would look like – that must greatly depend on who’s looking at it. In the most extreme case, if there is the Almighty, in which all causality and all equivalence are immediately obvious, the space would seem very simple. There are only two points. One point, of course, contains all the condensed infinitely many propositions like the universe before the Big Bang, but this one point contains all the propositions that are true, and the other point contains all the propositions that are false. And there is no structure other than that. Because, as I said before, equivalence in mathematics and logic means that two propositions are the same, which means that if one proposition is true, the other is true, and if one proposition is false, the other is false. The propositions at each point are equivalent to each other. In other words, all true propositions are completely the same in view of logic. Likewise, all false propositions are equivalent for the same reason, and so this space has no interesting structure when the Almighty looks at the space.
But we are not. To us, if we think about the process by which we are deriving as a proof through certain propositions in the proof a while ago, the space might look like this to us. You may have seen a small part of this, too, but all of these points are propositions, and from that point they are connected to other nearby points, a process we call reasoning. At some points, the points are close together and can be easily derived, and at some points, the points are, of course, both true, but they are so far apart that deriving them from other propositions will be very difficult and time consuming. What this means is, as if there is a concept of distance in this space, when we as humans feel when we do math in practice, it feels as if that distance is real (existing). In other words, it feels like there is a geometric structure.
To use a nonsensical analogy, this is one of the poetic images in my mind, and I always wondered if the space would always be like this. The picture you’re seeing now is a representation of the very macroscopic structure of our universe, where each brightly visible dot is a galaxy. What is interesting is that the galaxies are not homogeneously spread in the universe, but in some places there is a very vast empty space, in some places they are very well connected as if they are connected by filaments, having some geometric structures seemingly. If we think of this as a space where we have collected all of the propositions, every single point that shines brightly here will be propositions.
For example, if we highlight just two very famous true propositions, somewhere out there there will be a proposition that was painstakingly put together a few decades ago, “the Four-Color Theorem: any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.” There will also be a proposition “Fermat’s Last Theorem: no three positive integers x, y, and z satisfy the equation x^n + y^n = z^n for any integer value of n greater than 2”, which was very hard proved. But among these numerous possible true or false propositions, such as the “four-color theorem” and “Fermat’s Last Theorem” are particularly prized and particularly popular by mathematicians, so why?
Is it because knowing the speculative theorem could bring about some revolutionary change in semiconductor design next year? Or is it interesting because many people wonder whether the Riemann hypothesis is true or not, because if the Riemann hypothesis is proven, our bank’s security system could collapse, and we have to do something like this to prevent it? I don’t think so, and many mathematicians don’t. There is, however, a clear reason why these two propositions are particularly interesting over the others. What makes these theorems interesting is that if you think about the words and expressions used in the proposition, they are very familiar to us and can be easily explained. However, to prove that proposition, you have to go through a very difficult detour. I mean, I’m standing on a point in the propositional space, and I see a point right there. I want to go there from here, but it feels like it’s only 5 steps away, but I can’t get there easily no matter what I do. You can only get there by taking a very difficult detour, which means there is a huge structure blocking us from here to there. Although this structure is invisible to our eyes, like any very large empty space in this photo, its existence (like dark matters) can be inferred from the fact that we just feel it extremely difficult to prove these propositions without taking such detours. The reason why these are interesting is because it strongly suggests how we people (species of human) think. About what kinds of intuitions we have, and about what kinds of prejudices we have at the same time, we learn about ourselves by repeating these experiences. So, the ultimate goal that mathematicians want to do in the future is to understand how we humans think by drawing a map of these relationships in as much detail as possible, as many propositions are related to each other in this propositional space. I think like that.
It’s one of the questions I sometimes think about, and I think it’s similar to the discussion I just talked about. Have you ever thought about it in the following way? When children say something like “Why is this?”, then parents replied something, then children say, “Then why is that?”, then parents replied and again children say, “Why is that again?”, and such children’s questions and parents’ answers keep repeating like this; have you ever wondered why this process is so common to all children around the world? Of course, some of our rational reasoning is possible, and I don’t know the correct answer, but the first rational reasoning seems to be to cultivate the patience of mom and dad, of course, but personally, looking at my children, they don’t seem to have any interest in training mom and dad in any way, so I will say it’s a wrong answer. Secondly, it may be because children are really curious about the answer to the question and the facts that appear in the answer. But of course, I don’t think that’s the case either, because if I answer my kid, no listening without any concentration. Third, I may be vaguely too romantic, but thinking from an evolutionary biology standpoint, as we just discussed, these children are actually more interested in exploring the map of relationships than in listing a series of facts. I think like that. How this and this and this and this and this are interconnected. Why this is interesting to us is that, of course, we may be overestimating our sloppy-like kids, children’s instinct actually dimly feels (possibly in somewhat vague/indistinct ways) such relationships that are what give us meaning.
These were stories about relationships, and now let’s talk a little bit about boundaries. As I said before, boundaries seem essential if we are to create relationships. From a mathematical point of view, the sharpest and clearest boundary in mathematics lies between the so-called discrete and continuous mathematics, which in English are called discrete and continuous. Let’s talk a little bit about the boundary between the two. You probably know what it feels like to say “continuous” because you often use it in your daily life. Discrete mathematics may be a little unfamiliar, but among mathematics, discrete mathematics deals with countable sets. It is a field that studies objects that have distinct values, such as integers, graphs, and logical operations.
In modern mathematics, there are four major fields: combinatorics, analysis, algebra, and geometry. Combinatorics and algebra have a great deal of discrete mathematical aspects. In a similar context, analysis and geometry mainly deal with continuous objects. And our intuitions about discrete objects are completely different from our intuitions about continuous objects including various spaces and smooth functions. But not only in mathematics, but also in some languages and in some cultures, the distinction between these two is too clear, so there are cases where it is implemented grammatically.
For example, in English, nouns are divided into countable nouns and uncountable nouns. Do you remember? – Sungmoon English Grammar’s (used to be very famous in Korea) Chapter 2 in your memories maybe. For example, dog, tree, book, orange, bottle, coin, and cup are countable nouns. And water, happiness, intelligence, rice, love, information, and power are uncountable nouns. So, what grammar is used in relation to a distinct article for each? – it completely depends on what to use, how to use, etc. However, it is very, very, very difficult to learn, and no matter how much you learn, just think about orange and rice, for example. Orange is a countable noun; rice is an uncountable noun. We can count if there’s an orange here; even if you have three oranges, you can count them as well. You can count Rice. There may be 100 grains of rice, 112 grains, or 11.5 billion grains. By the way, they claim that rice is uncountable. In other words, they think of it as a continuous object because grains of rice are so fine; they don’t think they can count grains of rice just like, “we can’t count how many water we have”. These two distinctions are not in the physical nature of the object itself, but just depending on how we humans perceive the object, it finally becomes a countable or uncountable noun (by our perception).
There is a famous anecdote in the history of physics that shows that the boundary between the two discrete and continuous is contrived, called the duality of light; Is light a particle or a wave? Einstein had this to say about this in the 1930’s. It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do (Albert Einstein, 1938, on wave-particle duality). There’s a story like this. So I wanted to point out that the boundaries between what we think of as discrete objects and what we think of as continuous objects themselves, like all other boundaries, can be contrived. So, although we of course created such boundaries because they are useful and necessary, but knowing how to cross those boundaries when we don’t need them is very useful.
There are countless examples of this in mathematics. To introduce the two most familiar to you, how do you measure the area of a complex shape? First, we divide this continuous object into multiple grids and turn it into a discrete mathematical object, and then we count how many grids there are. The act of counting is a discrete mathematical act. And by making the grid finer and finer, more and more accurate values are derived. It goes from “continuous” to “discrete”.
The opposite is also what we always do – solving discrete mathematical problems in a continuous way. For example, there are many jelly beans here, and if someone asks you to “find how many jelly beans there are”, then we might try to count like, “1, 2, 3, 103,648, … “. It’s not easy to do that. A trick is to take a cup and count how many jelly beans are in a cup, and then see how many cups come out as if the jelly beans were just water, so you can count much easier. It goes from “discrete” to “continuous”.
Well… To take one more, slightly more complex example, this is an article I recently interviewed with the New York Times, where a reporter asked, “Give me an easier example of how the geometric intuition of discrete mathematical objects is used.” When I was in middle school, there is a puzzle from a computer game called “The 11th Hour” that I played hard with. Here, the positions of the black knights and the white knights are to be changed using the rules of chess, but if you do this recklessly, at least for me, a junior high school student at the time, the positions do not change no matter what.
However even here, if you think about it, the only thing that matters is the relationship between that square and square (in view of the knights). For example, when considering a simpler 9-square chess board rather than the previous chess board (taking account of the knights’ positions here as well), give the names like 1, 2, 3, 4, 5, 6, 7, 8, 9. If you think about the relationship in between the squares, you can directly see that the knight in the square 5 is isolated because it can’t go to anywhere else, and the other 8 squares form a loop with each other. So, one example turned into another. For example, in this other board, only the squares 1, 5 and 10 are connected to each other, and the other 7 squares are connected by a single straight line from the knight’s point of view.
If we apply this philosophy of relationship to the original problem, we see that the ten squares are arranged in this way. If you think about it from the point of view that the squares 1, 5 and 7 are connected by a straight line, and the square 9 is jagged above the square 2, and the squares 2, 6, 8, 3, 10 and 4 are adjacently connected by the straight line above. How does the problem of changing the original knight look like now? The problem has been turned into a problem where we can only move a knight to the next square; changing the problem in this way, it is easy to see the solution. Push the black knight all the way to the right end, push the other black knight all the way in the same way, push the white knight to the jagged part, put the two black knights next to the white knight at the left end, and then push the white knight in the jagged part to the right end, push the black knights to the right end next to the white knight again, put the white knight at the left end to the jagged part, push the black knights all the way to the left end, and then put the white knight at the right end next to the black knight again. In this way, it is possible to completely change the positions of the black knights and the white knights in a very intuitive way.
But if you think about it, it’s actually quite fun, and you’ll see that math isn’t about logic at all. For example, from the point of view of a computer that knows only logic, the equal sign I put here between the left and the right is really exactly the same, completely. The problem on the left and the problem on the right are completely logically equivalent. However, for us as humans to accept, one feels almost impossible and the other has a solution that is too obvious. Because, the right problem is because we can use our visual intuition. The moment you immediately understood what I meant by pushing the black knight all the way to the right, you were already using your visual intuition. This is the most primitive form of geometry. I think I said this example because this is a good example of how the original discrete mathematical problem can be helped by geometric intuition.
For the rest of the time, I’ll tell you a little bit more complex, but very briefly, as for, regarding my research, how I cross the boundaries between discrete and continuous mathematics, when I set boundaries more distinctly, and when I even break down such boundaries. It’s hard to go into details in a hurry, so I’ll give you these few references. There is a paper by Gil Kalai that he wrote briefly on ICM Proceedings about what I did (“The work of June Huh.”). Also, Andrei Okounkov wrote “Combinatorial geometry takes the lead” so that it can be read happily at a slightly more complex level, but still at the middle and high school level who are still interested in mathematics. He wrote for ICM Proceedings, and I wrote a thesis, “Combinatorics and Hodge theory”, at a slightly more professional, undergraduate and graduate level. And for those of you who are more interested in hearing explanations by voice, I am scheduled to give a Korean lecture at Seoul National University sometime next week on this topic of “Hodge Structure and Crossing Boundaries”. If you take the lecture there, I’ll explain about that in more detail.
I’m going to give you a spoiler of what I’m trying to say very quickly right now; over the past few decades, mathematicians have been moving back and forth between these four pillars of algebra, combinatorics, analysis, and geometry, from the standpoint of the Hodge structure. I’m just trying to tell you a little bit of history. In fact, if you look at some articles related to my works (giving him a hand the Fields Medal), it is often said, “He (June Huh) has opened a way to connect combination and geometry”, but in fact, such an attempt is not at all new, and many other mathematicians have done and tried to do such things that happened constantly between the six relationships that connect these four vertices, in addition to the relationships related to, so to speak, geometry, combinatorics and algebra. I’m going to tell you in 30 seconds what’s happened in a very specific framework called the Hodge structure, and what’s happened in the last 50 years about that.
Between algebra and geometry, there was a famous problem, “Weil conjectures on zeta functions over finite fields”. And Alexander Grothendieck, who laid the foundation for Algebraic Geometry, found that to solve the “Weil conjectures” it was enough to solve “Standard conjectures on algebraic cycles” between algebra and geometry; this conjecture is still unresolved. But now, a little later, for example, between algebra and combinatorics, David Kazhdan and George Lusztig (studied representation theory) conjectured that there is an intriguing structure inherent in the representation of an algebraic object called the Coxeter group. Two mathematicians, Elias and Wiliamson in 1999, solved it by discovering that in fact there is a hidden Hodge structure behind the representation theoretic object.
But what’s interesting here is that the Hodge structure they found is logically identical to the Hodge structure in Grothendieck standard conjectures. There is no logical causal relationship, but neither can one prove the other, nor vice versa, but when you look at it that way, it looks exactly the same. And a little later, for example, there were numerous links between combinatorics and geometry, and I would not shyly highlight one study in which I participated, Combinatorial Hodge Theory, published by Adiprasito, Huh, and Katz (2018). With a combinatorical object, matroid, it was possible to make a major achievement to discover that there is actually a hidden Hodge structure. Indeed, if you look at the form of the Hodge structure, the Hodge structure from the study of Elias and Wiliamson and the Hodge structure from the Grothendieck standard conjectures are of the exactly same form.
There’s no logical connection, or we haven’t seen it yet, but we can tell a similar story in the other three cases. So, if we think about what this means to us, in order to understand this universal structure, we constantly cross the boundary and break the boundary, then make new discoveries in that new area and give a name to a new object, and create new boundaries, trying to find out new relationships, and again break those boundaries, advance, give them names, and repeat such process es over and over again; there are many times when I feel like we are being asked to do that repeatedly.
This is my impatient acceptance speech when I received the Samsung Ho-Am Prize last year and was said that I had to go ahead and do it that morning finely. At that time, I said, “Personally, mathematics is a process of understanding my own prejudices and limitations, and more generally, mathematics includes a process of wondering how the human species thinks and how deeply the human species can think.” I was talking about mathematics like that; I was in a rush and made up a bunch of cool looking words without much meaning, but fortunately now coming back here from the past, I think that makes sense.
Intrinsic problems in mathematics, like solving the “Grothendieck standard conjectures” and numerous other conjectures and discovering new ones, are really important because they constantly require us to cross such boundaries (as mentioned above). And I think this is one of the important values that pure mathematics can teach us. It gives us an opportunity to transcend our own innate prejudices.
My talks ends here today. Let me say one final word. Today there are so many people here and online watching on YouTube, listening to me. Among them, there are so many young student who can make a long-term change, and also here are not a few elderly people who can make immediate effects on changes in policy. So, I would like to ask for continued support and interest in researchers who are working hard on pure mathematics or basic science in the future. As I have now been given a new role as I have received a great award and received a significant attention from many people, I will do my best to give back as much as possible the many things I have received in the Korean mathematics society, and more broadly in the Korean society. My speech ends here. Thank you for listening.