“Tyrants cannot stop us from doing mathematics”
July 5, 2022 1:43 PM Subscribe
The recipients of the 2022 Fields Medal, awarded every four years to recognize outstanding mathematical achievement, have been announced: Hugo Duminil-Copin, June Huh, James Maynard, and Maryna Viazovska.
The IMU page has links to interviews and Plus Magazine! articles for each of the recipients. Quanta Magazine also has in-depth profiles: Hugo Duminil-Copin, June Huh, James Maynard, Maryna Viazovska.
Viazovska (previously) is only the second woman to receive the medal. She is originally from Ukraine, and learned that she was receiving the medal just weeks before the war began in February. Her two sisters and their children were able to flee and join her in Switzerland, but her parents remained behind to look after her grandmother, who refused to leave.
Two of this year’s medalists had previously done videos for Numberphile explaining their work at a more popular level: June Huh on the g-conjecture; James Maynard on the Twin Prime Conjecture, Large Gaps between Primes, Primes Without a 7, and Approximating Irrational Numbers.
Via Peter Woit’s blog.
The IMU page has links to interviews and Plus Magazine! articles for each of the recipients. Quanta Magazine also has in-depth profiles: Hugo Duminil-Copin, June Huh, James Maynard, Maryna Viazovska.
Viazovska (previously) is only the second woman to receive the medal. She is originally from Ukraine, and learned that she was receiving the medal just weeks before the war began in February. Her two sisters and their children were able to flee and join her in Switzerland, but her parents remained behind to look after her grandmother, who refused to leave.
Two of this year’s medalists had previously done videos for Numberphile explaining their work at a more popular level: June Huh on the g-conjecture; James Maynard on the Twin Prime Conjecture, Large Gaps between Primes, Primes Without a 7, and Approximating Irrational Numbers.
Via Peter Woit’s blog.
OMG, so totally cool that Viazovska won.
The math is WAY
But just go look at a picture of E8. That jumped out at me. E8. Still trying to grasp what Lie algebras are even about.
E8
Now packing 8 dimensional oranges is, I guess, complicated, but proving that E8 is the very best perfect way to pack oranges in an 8 dimensional space. Packing problems are, cough, non trivial, like just balls in a box is not determined or only recently. And some multidimensional problems seem to be resolved before some regular space problems, E8 has some pretty clear symmetries. But how do they wrap their minds around that stuff.
posted by sammyo at 6:41 PM on July 5, 2022 [1 favorite]
The math is WAY
But just go look at a picture of E8. That jumped out at me. E8. Still trying to grasp what Lie algebras are even about.
E8
#/media/File:E8Petrie.svg){kind=link}
Now packing 8 dimensional oranges is, I guess, complicated, but proving that E8 is the very best perfect way to pack oranges in an 8 dimensional space. Packing problems are, cough, non trivial, like just balls in a box is not determined or only recently. And some multidimensional problems seem to be resolved before some regular space problems, E8 has some pretty clear symmetries. But how do they wrap their minds around that stuff.
posted by sammyo at 6:41 PM on July 5, 2022 [1 favorite]
I saw the Quanta article on June Huh, I recommend it. He's the would-be poet who dropped out of high school, and then eventually "found" mathematics. Also, the article explains the math in a way I could understand. Good for them for doing that.
posted by storybored at 8:54 PM on July 5, 2022 [2 favorites]
posted by storybored at 8:54 PM on July 5, 2022 [2 favorites]
how do they wrap their minds around that stuff
Piece by piece by endlessly conceptually scaffolded piece.
posted by flabdablet at 10:30 PM on July 5, 2022 [2 favorites]
Piece by piece by endlessly conceptually scaffolded piece.
posted by flabdablet at 10:30 PM on July 5, 2022 [2 favorites]
Here's the picture of E8 that actually matters. Each node in the graph corresponds to a reflection which helps generate all of E8, and the edges (and non -edge) describe the relationships between the generating reflections... But the point is: E8 is enormous, but it is described by a small set of generators and the relationships between them.
posted by kaibutsu at 4:49 AM on July 6, 2022 [2 favorites]
#/media/File:Dynkin_diagram_E8.svg){kind=link}
posted by kaibutsu at 4:49 AM on July 6, 2022 [2 favorites]
Sphere packing in 8 dimensions sounds a bit like counting angels on the head of a pin, but let me make a case that it's actually a pretty natural question that you don't need to be a mathematician to appreciate.
First, what is n-dimensional space? It's space with n Cartesian coordinates, i.e., the points can be represented by lists of n numbers. But that works in reverse, too: if you have any kind of data that are lists of n numbers, you can conceive of your data as points in n-dimensional space.
Second, what is distance? Fundamentally, it's a measure of how dissimilar those lists of numbers are to each other. Euclidean distance is only one of many such measures we can cook up. The Euclidean distance from (x1, x2, ..., xn) to (y1, y2, ..., yn) is √((x1–y1)2+...+(xn–yn)2). This is a certain way to amalgamate the differences in each coordinate (x1–y1, x2–y2, etc.) into a single measure of differentness between lists. You can change the formula to amalgamate these differences in other ways, giving more weight to how many coordinates differ or to how much they differ by, as suits whatever kind of data you're working on. There's a whole area of mathematics that deals with distance functions in general. So distance is a flexible notion, but it comes down to quantifying how different two lists of numbers are.
Third, why spheres? A sphere is the set of points within a fixed distance from a center point. Two spheres of the same radius r can coexist in a sphere packing (i.e., not overlap or touch) if and only if the distance between their centers is more than 2r. (This is easy to picture for circles, which are 2-dimensional Euclidean spheres. The rules for "distance functions in general" guarantee that this mental picture is still correct for more abstract kinds of spheres, based on any distance function in any-dimensional space.)
Thus, packing spheres means choosing points (centers) which are all more than a certain distance from their nearest neighbors. Or, fully stripping out the geometric framing, packing spheres means choosing as many lists of numbers as you can which are all sufficiently unlike each other.
Here's one practical example. Let's say you have a shop inventory and you want to assign each item a 4-digit code for cashiers to use when ringing people up. But you want the system to be robust against fat-fingering, so if the right code for a certain item is 3972 and the cashier types 3975, it should produce an error rather than ring up as a different item. This means you can't assign all possible 4-digit codes to items; you have to leave some unused. The codes you do use in your system all need to be "far enough apart" that a single manual error can't change one valid code into another. You want to make a list of such codes -- ideally as many as possible.
This is a sphere packing problem! Each code you assign to a product is surrounded by a spherical bubble of "nearby" codes it preempts from use. The underlying space is all 4-digit codes, which can be thought of as points in 4 dimensions. The distance function is up to you; you might decide that the distance between two codes is simply the number of digits where they differ, or you might also take into account distance on a numpad (so 3972 is closer to 3975 than to 3974, for instance). You can also decide how large to make the radius of the spheres, which is just the threshold for how dissimilar codes in the system have to be. Fun fact: if you force codes which are assigned to products to be at least 3 edits apart, then not only does entering a code which is 1 edit from a valid code trip an error, but you can have the system automatically correct it to the closest valid code, because any other valid code must be at least 2 edits away.
I haven't said anything about Viazovska's work, and I mostly won't, except to say something about 8 and 24 being "special" numbers of dimensions. It's tough to explain why those are special numbers for Euclidean distance, but I can make an analogy to a discrete sphere-packing problem which is easier to understand. Take the inventory code problem from above, but let's say we're working in binary, so the codes consist of just 0s and 1s. In this case, there are 2n possible n-bit codes, which we think of as forming an n-dimensional space. Let's say the distance between two codes is just the number of bits where they differ. (This whole setup -- the space and distance function -- is called the Hamming cube.)
Now let's say we want to pick codes that are a minimum distance of 3 apart. This is the same as packing spheres of radius 1, since the condition that no two spheres touch means that the distance between centers is strictly more than 2. (The distance function we chose is integer-valued, so "more than 2" = "at least 3".) Now remember that our space has a finite number of possible codes, namely 2n. How many of these belong to the sphere of radius 1 surrounding a given code? The answer is n+1; there's the code itself, plus n codes that can be gotten by flipping a single bit.
So we see that, potentially, we can pick up to 2n/(n+1) codes whose spheres don't touch. That number might not be an integer: if n+1 doesn't divide evenly into 2n, there's guaranteed to be some waste. But if n+1 does divide into 2n, there is the potential prospect of a perfect packing -- a system where every code is either chosen for use, or within distance 1 from a chosen code. This condition of 2n being divisible by n+1 is met when n = 1, 3, 7, 15, ..., so these are the special, most promising dimensions for this particular sphere packing problem. In fact, a perfect packing does exist in these dimensions -- it's called a Hamming code -- but nothing I've said here proves it; there's more work to do after establishing that it's a theoretical possibility. This is where you have to actually get into the details of how the spheres fit together.
Viazovska's work concerns a harder kind of sphere packing problem, in a continuous (not discrete) space with Euclidean distance. Here we knew that 8 and 24 were special numbers of dimensions, and we knew about the unusually good packings in these dimensions, but they aren't perfect packings; there's still empty space between the spheres. Viazovska did something amazing: she showed that these are the best possible packings, with the least wasted space.
posted by aws17576 at 11:12 AM on July 6, 2022 [10 favorites]
First, what is n-dimensional space? It's space with n Cartesian coordinates, i.e., the points can be represented by lists of n numbers. But that works in reverse, too: if you have any kind of data that are lists of n numbers, you can conceive of your data as points in n-dimensional space.
Second, what is distance? Fundamentally, it's a measure of how dissimilar those lists of numbers are to each other. Euclidean distance is only one of many such measures we can cook up. The Euclidean distance from (x1, x2, ..., xn) to (y1, y2, ..., yn) is √((x1–y1)2+...+(xn–yn)2). This is a certain way to amalgamate the differences in each coordinate (x1–y1, x2–y2, etc.) into a single measure of differentness between lists. You can change the formula to amalgamate these differences in other ways, giving more weight to how many coordinates differ or to how much they differ by, as suits whatever kind of data you're working on. There's a whole area of mathematics that deals with distance functions in general. So distance is a flexible notion, but it comes down to quantifying how different two lists of numbers are.
Third, why spheres? A sphere is the set of points within a fixed distance from a center point. Two spheres of the same radius r can coexist in a sphere packing (i.e., not overlap or touch) if and only if the distance between their centers is more than 2r. (This is easy to picture for circles, which are 2-dimensional Euclidean spheres. The rules for "distance functions in general" guarantee that this mental picture is still correct for more abstract kinds of spheres, based on any distance function in any-dimensional space.)
Thus, packing spheres means choosing points (centers) which are all more than a certain distance from their nearest neighbors. Or, fully stripping out the geometric framing, packing spheres means choosing as many lists of numbers as you can which are all sufficiently unlike each other.
Here's one practical example. Let's say you have a shop inventory and you want to assign each item a 4-digit code for cashiers to use when ringing people up. But you want the system to be robust against fat-fingering, so if the right code for a certain item is 3972 and the cashier types 3975, it should produce an error rather than ring up as a different item. This means you can't assign all possible 4-digit codes to items; you have to leave some unused. The codes you do use in your system all need to be "far enough apart" that a single manual error can't change one valid code into another. You want to make a list of such codes -- ideally as many as possible.
This is a sphere packing problem! Each code you assign to a product is surrounded by a spherical bubble of "nearby" codes it preempts from use. The underlying space is all 4-digit codes, which can be thought of as points in 4 dimensions. The distance function is up to you; you might decide that the distance between two codes is simply the number of digits where they differ, or you might also take into account distance on a numpad (so 3972 is closer to 3975 than to 3974, for instance). You can also decide how large to make the radius of the spheres, which is just the threshold for how dissimilar codes in the system have to be. Fun fact: if you force codes which are assigned to products to be at least 3 edits apart, then not only does entering a code which is 1 edit from a valid code trip an error, but you can have the system automatically correct it to the closest valid code, because any other valid code must be at least 2 edits away.
I haven't said anything about Viazovska's work, and I mostly won't, except to say something about 8 and 24 being "special" numbers of dimensions. It's tough to explain why those are special numbers for Euclidean distance, but I can make an analogy to a discrete sphere-packing problem which is easier to understand. Take the inventory code problem from above, but let's say we're working in binary, so the codes consist of just 0s and 1s. In this case, there are 2n possible n-bit codes, which we think of as forming an n-dimensional space. Let's say the distance between two codes is just the number of bits where they differ. (This whole setup -- the space and distance function -- is called the Hamming cube.)
Now let's say we want to pick codes that are a minimum distance of 3 apart. This is the same as packing spheres of radius 1, since the condition that no two spheres touch means that the distance between centers is strictly more than 2. (The distance function we chose is integer-valued, so "more than 2" = "at least 3".) Now remember that our space has a finite number of possible codes, namely 2n. How many of these belong to the sphere of radius 1 surrounding a given code? The answer is n+1; there's the code itself, plus n codes that can be gotten by flipping a single bit.
So we see that, potentially, we can pick up to 2n/(n+1) codes whose spheres don't touch. That number might not be an integer: if n+1 doesn't divide evenly into 2n, there's guaranteed to be some waste. But if n+1 does divide into 2n, there is the potential prospect of a perfect packing -- a system where every code is either chosen for use, or within distance 1 from a chosen code. This condition of 2n being divisible by n+1 is met when n = 1, 3, 7, 15, ..., so these are the special, most promising dimensions for this particular sphere packing problem. In fact, a perfect packing does exist in these dimensions -- it's called a Hamming code -- but nothing I've said here proves it; there's more work to do after establishing that it's a theoretical possibility. This is where you have to actually get into the details of how the spheres fit together.
Viazovska's work concerns a harder kind of sphere packing problem, in a continuous (not discrete) space with Euclidean distance. Here we knew that 8 and 24 were special numbers of dimensions, and we knew about the unusually good packings in these dimensions, but they aren't perfect packings; there's still empty space between the spheres. Viazovska did something amazing: she showed that these are the best possible packings, with the least wasted space.
posted by aws17576 at 11:12 AM on July 6, 2022 [10 favorites]
counterpoint: 'Mathematicians cannot stop them from doing tyranny'
posted by elkevelvet at 1:49 PM on July 7, 2022
posted by elkevelvet at 1:49 PM on July 7, 2022
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Viazovska's article notes that she is "only the second woman to receive a Fields Medal, following on from Maryam Mirzakhani who won it in 2014." That's kind of stunning.
I appreciate good explanatory writing, and Marianne Freibeger does a great job in the profile of Viazovska, using the packing of oranges as an illustration of Viazovska's work. (And I chuckled at the later comment that it's "a fruitful approach." Hah.)
I look forward to spending a little more time trying to more fully understand these medalists' ideas and work, but in the meantime, I am delighted that their work has been recognized, and delighted to get to read about it here.
Thank you so much for posting this, mubba!
posted by kristi at 6:19 PM on July 5, 2022 [4 favorites]