WEBVTT Kind: captions Language: en 00:00:08.190 --> 00:00:11.418 What's the best method to arrange oranges to fill a box 00:00:11.418 --> 00:00:13.733 in as compact way as possible? 00:00:14.190 --> 00:00:17.174 Placing them carefully in ordered rows? 00:00:17.174 --> 00:00:19.110 Or tossing them in at random? 00:00:20.460 --> 00:00:21.870 In our three-dimensional world, 00:00:21.870 --> 00:00:25.950 the familiar grocery store stack turns out to be the optimal solution. 00:00:26.610 --> 00:00:28.440 But what about in higher dimensions? 00:00:28.770 --> 00:00:31.310 Dimension a hundred, a thousand, a jillion. 00:00:31.310 --> 00:00:33.930 So as the dimension grows and grows, what happens then? 00:00:34.140 --> 00:00:38.460 Very mysterious, what should be the correct thing to do there? 00:00:38.700 --> 00:00:41.010 This is known as the sphere packing problem. 00:00:41.430 --> 00:00:44.820 It's like the stupidest question you can think of, right? So you take a box, 00:00:45.660 --> 00:00:47.700 I give you a bunch of balls, and you say, okay, 00:00:47.700 --> 00:00:49.710 how many balls can you put in this box? 00:00:49.740 --> 00:00:54.120 How many of them can we fit in without actually letting them overlap or 00:00:54.120 --> 00:00:54.990 distorting them? 00:00:55.320 --> 00:00:57.833 An international group of young mathematicians 00:00:57.833 --> 00:01:00.334 have made a major advance on the sphere packing problem 00:01:00.334 --> 00:01:01.560 with a new proof. 00:01:01.620 --> 00:01:03.960 One that works in all dimensions. 00:01:04.740 --> 00:01:08.850 A sphere in two dimensions is a circle defined by X and Y coordinates. 00:01:09.210 --> 00:01:14.393 The densest packing is exactly what you'd imagine if you placed coins on a flat surface. 00:01:14.400 --> 00:01:16.470 You have some sort of honeycomb-type structure. 00:01:16.680 --> 00:01:21.480 You have one penny and then six around it. That's not super hard to prove. 00:01:21.990 --> 00:01:25.740 In the late 16th century, British Explorer, Sir Walter Raleigh, 00:01:25.800 --> 00:01:29.880 asked the mathematician Thomas Harriot to calculate the optimal way to stack 00:01:29.880 --> 00:01:31.590 cannonballs on the deck of a ship. 00:01:32.070 --> 00:01:35.580 Harriot later shared the problem with the astronomer Johannes Kepler. 00:01:36.480 --> 00:01:41.161 In 1611, Kepler hypothesized that the optimal three-dimensional packing 00:01:41.161 --> 00:01:42.300 would be a pyramid shape. 00:01:43.260 --> 00:01:46.197 Any sphere in the interior would touch 12 others 00:01:46.197 --> 00:01:51.480 and the spheres would occupy about 74.05% of the available space. 00:01:51.930 --> 00:01:54.360 This became known as the Kepler Conjecture. 00:01:54.630 --> 00:01:56.100 The question became, 00:01:56.280 --> 00:02:01.280 how do we understand why this is really optimal? Can we prove it? 00:02:01.560 --> 00:02:05.240 It was actually a really long standing open question in math to prove that 00:02:05.250 --> 00:02:08.940 that's the best way to do things. In A 3D, it took about 400 years. 00:02:10.560 --> 00:02:15.150 Sphere packings can serve as a sort of basic model of the structure of matter. 00:02:15.360 --> 00:02:17.760 Three dimensional spheres can represent atoms. 00:02:18.090 --> 00:02:21.270 An ordered packing might model a crystal or other solids, 00:02:21.540 --> 00:02:25.110 whereas disordered arrangements can represent liquids or gases. 00:02:25.620 --> 00:02:29.250 Sort of way of probing the phases of matter as well, an physics. 00:02:29.610 --> 00:02:32.940 To extend the problem from 3D spherers into higher dimensions, 00:02:33.150 --> 00:02:34.770 more coordinates can be added. 00:02:36.060 --> 00:02:38.610 Each just defines some point inside of a box. 00:02:38.910 --> 00:02:42.750 The sphere packing problem in higher dimensions has a range of applications 00:02:43.020 --> 00:02:46.410 including in error correcting codes used in computer science. 00:02:46.470 --> 00:02:50.310 Turns out to be deeply connected with topics in information theory. 00:02:51.030 --> 00:02:54.671 In 2016, the Ukrainian mathematician Maryna Viazovska 00:02:54.671 --> 00:02:58.470 solved the sphere packing problem in dimensions 8 and 24. 00:02:58.470 --> 00:03:00.670 Proving that in these dimensions, 00:03:00.701 --> 00:03:02.746 certain lattices are optimal. 00:03:02.746 --> 00:03:05.050 Sheater won a Fields Medal for her work. 00:03:05.680 --> 00:03:08.890 But beyond these dimensions, it remained an open question. 00:03:10.480 --> 00:03:13.541 A big question underlying this is 00:03:13.541 --> 00:03:16.030 structure versus randomness. 00:03:16.360 --> 00:03:19.330 Should the best sphere packings be crystalline and ordered? 00:03:19.330 --> 00:03:22.420 Or should they be completely random and unstructured? 00:03:22.810 --> 00:03:27.130 When Julian Sahasrabudhe assembled his working group in person for the first time 00:03:27.370 --> 00:03:30.340 solving the sphere packing problem wasn't on their agenda. 00:03:30.520 --> 00:03:33.158 We actually had a slightly different problem in mind 00:03:33.158 --> 00:03:34.600 when we got together in Cambridge. 00:03:34.600 --> 00:03:38.350 Failed to solve that problem in such a way that we realized that our failure 00:03:38.350 --> 00:03:41.320 continues to inform progress on this other problem. 00:03:41.560 --> 00:03:43.480 To arrive at their sphere packing proof. 00:03:43.720 --> 00:03:46.620 The group first turned the problem into one about a graph, 00:03:46.620 --> 00:03:50.470 which consists of points and lines known as vertices and edges. 00:03:50.680 --> 00:03:53.221 Totally forgot about geometry because geometry's confusing 00:03:53.221 --> 00:03:54.843 and then just work directly with this graph. 00:03:55.270 --> 00:03:57.040 Their goal was to construct an efficient, 00:03:57.100 --> 00:04:00.850 high-dimensional sphere packing from a collection of randomly placed points 00:04:01.000 --> 00:04:02.890 representing the centers of spheres. 00:04:03.160 --> 00:04:05.993 I need to be able to put a sphere around one center, 00:04:05.993 --> 00:04:08.560 a sphere, around the other center, and the spheres can't the overlap. 00:04:08.800 --> 00:04:13.300 If two spheres do intersect, then an edge is drawn connecting the two points. 00:04:13.600 --> 00:04:18.280 What we're very interested in is what is the largest subset of a 00:04:18.280 --> 00:04:20.140 graph that contains no edges? 00:04:20.410 --> 00:04:24.280 This subset of points without edges is called an independent set. 00:04:24.460 --> 00:04:28.300 You construct independent sets just by constructing it bit by bit, 00:04:28.360 --> 00:04:31.420 not trying to find the whole complicated object all at once. 00:04:31.900 --> 00:04:35.744 You can form an independent set by selecting points in the original graph 00:04:35.744 --> 00:04:38.620 using a random process called the Rödl nibble. 00:04:38.950 --> 00:04:40.975 You just place the balls one at a time 00:04:40.975 --> 00:04:43.743 until there is no more space to place any balls. 00:04:44.080 --> 00:04:48.370 Keep applying this process until you've built up an independent set that's large 00:04:49.920 --> 00:04:50.420 to create a good sphere packing. 00:04:50.920 --> 00:04:55.030 Our work is a very as random packing as you can get. 00:04:55.480 --> 00:04:58.570 The resulting proof broke a 75-year-old record. 00:04:59.020 --> 00:05:02.890 It gave a denser way to pack spheres in all higher dimensions. 00:05:03.040 --> 00:05:06.100 But I think it'd be very interesting if random like sphere packings are the best 00:05:06.100 --> 00:05:09.130 period, and this is really what the big question is about. 00:05:14.800 --> 00:05:17.808 Imagine a pool of prime numbers. 00:05:17.808 --> 00:05:20.890 Or a large messy collection of numbers chosen at random. 00:05:21.910 --> 00:05:24.940 Is it inevitable that patterns will appear within these sets? 00:05:25.330 --> 00:05:28.870 True randomness is impossible. No matter how random your set looks, 00:05:28.870 --> 00:05:30.700 they'll always contain this kind of structure. 00:05:30.880 --> 00:05:34.810 There's always a little pattern inside any large unstructured object. 00:05:35.080 --> 00:05:39.910 Identifying just how large sets of numbers can get before patterns must emerge 00:05:39.970 --> 00:05:42.310 is a central problem in combinatorics. 00:05:42.790 --> 00:05:46.150 Combinatorics often is seen as a study of counting. 00:05:46.270 --> 00:05:49.420 I think more broadly it has to do with the study of patterns. 00:05:49.720 --> 00:05:53.650 Patterns of equally-space numbers are called arithmetic progressions. 00:05:53.890 --> 00:05:57.220 So a three-term arithmetic progression is like 3, 7, 11. 00:05:57.920 --> 00:06:02.780 A four term arithmetic progression is like 3, 7, 11, 15, and so on. 00:06:02.990 --> 00:06:06.949 In February, three grad students made the first breakthrough in decades 00:06:06.949 --> 00:06:11.150 on one of the biggest unsolved problems involving arithmetic progressions. 00:06:11.480 --> 00:06:12.680 Generally as a graduate student, 00:06:12.680 --> 00:06:16.010 you don't try to tackle the cutting-edge problems in the field right away. 00:06:17.180 --> 00:06:20.819 In 1936, the mathematicians Paul Erdős and Pál Turán 00:06:20.819 --> 00:06:23.316 explored infinite sets that contain 00:06:23.337 --> 00:06:26.005 some non-zero fraction of the whole numbers. 00:06:26.005 --> 00:06:30.890 For example, the set of even numbers contains 50% of all whole numbers. 00:06:31.670 --> 00:06:35.390 Erdős and Turán asserted that even if the fraction is very small, 00:06:35.660 --> 00:06:37.250 as long as it's not zero, 00:06:37.580 --> 00:06:41.930 the set will be sure to contain all arbitrarily long arithmetic progressions. 00:06:43.730 --> 00:06:47.338 In 1975, he mathematician Endre Szemerédi 00:06:47.338 --> 00:06:48.706 proved the conjecture 00:06:48.706 --> 00:06:51.045 for sets containing infinitely many numbers. 00:06:51.710 --> 00:06:54.531 But what happens with finite sets of numbers? 00:06:54.531 --> 00:06:57.189 Say you start with some pool of numbers 00:06:57.189 --> 00:06:58.790 like the numbers one through 20. 00:06:59.690 --> 00:07:02.780 Now you build a set using some fraction of those numbers, 00:07:02.840 --> 00:07:04.760 which is called the density of the set. 00:07:05.420 --> 00:07:09.710 The question becomes what's the highest density or fraction of numbers you can 00:07:09.710 --> 00:07:13.730 add to your set before you accidentally include an arithmetic progression. 00:07:14.810 --> 00:07:17.420 You can start to ask questions about, okay, 00:07:17.420 --> 00:07:19.880 at what densities do we still have these patterns? 00:07:20.000 --> 00:07:21.950 What is the best avoiding set? 00:07:22.070 --> 00:07:26.930 I want to take as many elements as possible without creating, let's 00:07:26.930 --> 00:07:30.560 say, a five term arithmetic progression. How dense a set can you take? 00:07:30.890 --> 00:07:32.510 With a pool of 20 numbers 00:07:32.630 --> 00:07:36.950 the answer is 16 or 80% for a five term progression. 00:07:37.760 --> 00:07:40.490 Szemerédi proved that as your starting pool grows, 00:07:40.610 --> 00:07:44.480 the fraction of numbers you can take before arithmetic progressions appear 00:07:44.540 --> 00:07:45.620 approaches zero. 00:07:46.640 --> 00:07:49.940 Mathematicians want to know how quickly does this happen? 00:07:50.390 --> 00:07:52.816 What is the bound for the highest density set 00:07:52.816 --> 00:07:54.973 that avoids progressions of a given size? 00:07:55.250 --> 00:07:58.268 There was an interest for a long time to make the bounds for Szemerédi's theorem 00:07:58.268 --> 00:08:01.228 better and better. and for very short progressions, 00:08:01.228 --> 00:08:03.440 progressions of length three, there's been a lot of work. 00:08:03.800 --> 00:08:07.269 Last year, two computer scientists nearly solved this question 00:08:07.269 --> 00:08:08.690 for three-term progressions 00:08:08.750 --> 00:08:10.610 like 4, 8, 12. 00:08:11.180 --> 00:08:13.530 But for longer progressions, even progressions of length four. 00:08:13.760 --> 00:08:14.990 The subject was much trickier. 00:08:15.860 --> 00:08:21.140 Mehtaab Sawhney and Ashwin Sah, first met in 2017 as undergrads at MIT. 00:08:21.560 --> 00:08:24.380 Since then, the pair of become frequent collaborators. 00:08:24.440 --> 00:08:27.800 We sorta met through this shared interest in combinatorics. 00:08:27.830 --> 00:08:30.982 We wrote something like 50 papers in graduate school together. 00:08:30.982 --> 00:08:32.833 We our academic siblings. 00:08:33.050 --> 00:08:34.550 Meanwhile, James Leng, 00:08:34.640 --> 00:08:37.603 then a grad student under Terrence Tao at UCLA, 00:08:37.603 --> 00:08:40.787 was working on techniques related to Szemerédi's problem. 00:08:40.787 --> 00:08:44.060 Sah and Sawhney found out about his work. 00:08:44.150 --> 00:08:45.620 I had this new technique. 00:08:45.800 --> 00:08:49.130 James mentioned, he had some really exciting ideas in progress, 00:08:49.130 --> 00:08:50.870 made it seem like it might be possible to 00:08:50.870 --> 00:08:51.770 prove something like this. 00:08:51.770 --> 00:08:55.380 We sort of realized that the technique was even more powerful than I had 00:08:55.380 --> 00:08:56.213 originally thought. 00:08:56.670 --> 00:08:59.760 Over the course of a few months of long distance collaboration, 00:09:00.060 --> 00:09:03.960 the three young mathematicians figured out how to get a better upper bound on 00:09:03.960 --> 00:09:06.840 the size of sets with no five-term progressions. 00:09:07.170 --> 00:09:10.290 They then extended their work to progressions of any length. 00:09:10.530 --> 00:09:11.730 There was a general strategy, 00:09:11.730 --> 00:09:16.230 but sort of whether any particular bit would work was sort of very unclear. 00:09:16.260 --> 00:09:17.910 It was kind of dicey and eventually okay, 00:09:17.910 --> 00:09:19.350 it fell into place and then we were happy. 00:09:20.160 --> 00:09:23.880 The techniques they developed for the paper have already found applications in 00:09:23.880 --> 00:09:25.050 other areas of math. 00:09:25.920 --> 00:09:28.200 There were many more creative ideas in this paper. 00:09:28.980 --> 00:09:30.990 It really was a very impressive achievement, 00:09:31.290 --> 00:09:32.580 especially for three graduate students. 00:09:37.980 --> 00:09:42.062 The Langlands Program is a vast endeavor to develop what's been called a 00:09:42.062 --> 00:09:44.490 "Grand Unified Theory" of mathematics. 00:09:45.240 --> 00:09:49.560 The way mathematical objects combine and lead to these conjectures, 00:09:49.590 --> 00:09:51.742 there's something extremely appealing about it, 00:09:52.410 --> 00:09:53.730 the Langlands Program, especially in geometric Langlands. 00:09:54.360 --> 00:09:58.710 The geometric Langlands conjecture is a key component of this sweeping paradigm. 00:09:59.280 --> 00:10:02.400 And now after three decades of work, Dennis Gaitsgory 00:10:02.490 --> 00:10:05.760 along with his former student, Sam Raskin and seven others, 00:10:05.820 --> 00:10:09.000 have produced a monumental 800 page proof. 00:10:11.610 --> 00:10:16.020 What is now known as the Langlands program began in 1967 when the 00:10:16.020 --> 00:10:19.800 mathematician Robert Langlands wrote a letter to the French number theorist, 00:10:19.800 --> 00:10:21.964 André Weil, describing a plan to connect far reaching branches of math. 00:10:21.964 --> 00:10:27.193 It's a lot of parts of number theory, parts of physics, 00:10:27.420 --> 00:10:31.470 and it has kind of different compartments and different kind of corners that are 00:10:31.560 --> 00:10:32.730 operated in parallel. 00:10:33.570 --> 00:10:37.500 The Langlands Program takes its inspiration from another part of mathematics, 00:10:37.830 --> 00:10:38.952 Fourier theory, 00:10:38.952 --> 00:10:42.240 which splits complex signals into simpler components. 00:10:42.270 --> 00:10:45.360 The Fourier transform is one of these kind of basic building blocks of much of 00:10:45.360 --> 00:10:46.320 math that we simplify. 00:10:46.320 --> 00:10:49.560 We tried to think of everything in terms of these basic patterns. 00:10:50.640 --> 00:10:53.857 In 1822, the mathematician Joseph Fourier 00:10:53.857 --> 00:10:56.550 showed that any wave can be broken down into an 00:10:56.580 --> 00:11:01.833 infinite sum of sine waves using a technique now called the Fourier Transform. 00:11:01.833 --> 00:11:05.430 The Fourier Transform is like a recipe generator. 00:11:05.760 --> 00:11:09.150 You input a complicated wave and you get back its ingredients, 00:11:09.450 --> 00:11:12.120 the amplitude and frequency of each component sine wave. 00:11:12.120 --> 00:11:16.620 Fourier theory is an essential part of modern technology. 00:11:17.010 --> 00:11:19.690 Its applications range from jpeg compression 00:11:19.690 --> 00:11:22.980 and image recognition to quantum physics and MRIs. 00:11:23.400 --> 00:11:27.630 It has also opened a revolutionary new framework in pure mathematics. 00:11:27.750 --> 00:11:30.390 Our experience with Fourier theory guides 00:11:30.390 --> 00:11:33.351 many of the ways that we think about the Langlands Program 00:11:33.351 --> 00:11:35.101 and the way the subjects develops 00:11:35.101 --> 00:11:37.680 and the sort of phenomena that we're seeing. 00:11:38.580 --> 00:11:42.630 Fourier theory has two components, basic building blocks and labels. 00:11:43.050 --> 00:11:45.060 Imagine a child's toy castle. 00:11:45.540 --> 00:11:49.710 This castle can be disassembled into individual building blocks and these 00:11:49.710 --> 00:11:52.745 components can then be sorted by color into bins, 00:11:52.745 --> 00:11:53.245 and then labeled. 00:11:55.810 --> 00:11:59.299 Similarly, the Fourier transform disassembles a complex wave 00:11:59.299 --> 00:12:00.670 into individual sine waves. 00:12:00.940 --> 00:12:02.642 These are like the building blocks 00:12:02.642 --> 00:12:05.883 and each sine wave can be labeled with its frequency. 00:12:07.330 --> 00:12:10.810 To open up new connections between distant mathematical worlds, 00:12:11.140 --> 00:12:14.373 Langlands researchers look for analogies in Fourier theory 00:12:14.373 --> 00:12:15.640 in other contexts. 00:12:16.210 --> 00:12:19.457 Are there other basic building blocks that can fit into the bins, 00:12:19.457 --> 00:12:21.763 and if so, what are the labels? 00:12:22.990 --> 00:12:26.950 The geometric Langlands program is one way to answer these questions. 00:12:27.880 --> 00:12:32.320 The fundamental building blocks akin to sine waves are called eigensheaves. 00:12:32.740 --> 00:12:37.390 Eigensheaves are subsets of complex abstractions of functions called sheaves. 00:12:37.630 --> 00:12:41.890 So named because mathematicians visualize them like sheaves of wheat growing on 00:12:41.890 --> 00:12:44.050 top of other mathematical objects. 00:12:44.650 --> 00:12:49.180 The labels on the bins are something called representations of the fundamental group, 00:12:49.180 --> 00:12:52.510 descriptions of the loops that can be drawn on spheres, donuts, 00:12:52.540 --> 00:12:54.236 and other shapes. 00:12:54.236 --> 00:12:59.239 For Gaitsgory, the epic pursuit began in 1994 00:12:59.239 --> 00:13:00.664 when he first heard about the 00:13:00.664 --> 00:13:01.548 geometric Langlands program 00:13:01.560 --> 00:13:03.340 as a graduate student. 00:13:03.610 --> 00:13:07.120 I guess I understood about 15% if that, 00:13:07.810 --> 00:13:11.170 but I was kind of completely awestruck. 00:13:11.440 --> 00:13:13.630 After two decades of work on the problem, 00:13:13.810 --> 00:13:17.200 Gaitsgory finally began to see a way forward towards a proof. 00:13:17.290 --> 00:13:18.130 Until that moment, 00:13:18.130 --> 00:13:22.840 it was in some sense, like walking in the dark in the woods. 00:13:23.230 --> 00:13:26.710 From that moment on, I saw the framework. 00:13:27.070 --> 00:13:30.130 Gaitsgory drew what he called the fundamental diagram, 00:13:30.370 --> 00:13:34.810 an outline of the solution. But his diagram was missing one piece. 00:13:35.260 --> 00:13:39.220 He needed to know that every single eigensheave is contained within a special 00:13:39.220 --> 00:13:41.650 composite chief known as the Poincaré sheaf. 00:13:41.650 --> 00:13:45.790 Raskin also became hooked on the problem. 00:13:46.060 --> 00:13:51.130 Sam became a grad student a year exactly after this revelation until 2006. 00:13:51.940 --> 00:13:56.260 After finishing his PhD, Raskin continued to study the Poincaré sheaf. 00:13:57.010 --> 00:13:59.080 A Poincaré sheaf is like white light. 00:13:59.470 --> 00:14:01.720 Just as white light contains every color, 00:14:01.930 --> 00:14:04.219 mathematicians expected the Poincaré sheaf 00:14:04.219 --> 00:14:06.010 to contain every eigensheave. 00:14:06.730 --> 00:14:09.040 Even though it's hard to write down an individual eigensheaf, 00:14:09.730 --> 00:14:13.300 it turns out that you can write down very directly what happens when you 00:14:13.300 --> 00:14:14.560 amalgamate all of them. 00:14:14.860 --> 00:14:19.450 If Raskin could prove that the composite Poincaré sheaf contained all eigensheaves, 00:14:19.450 --> 00:14:19.950 then he could use this as a tool to access the individual eigensheaves, 00:14:19.950 --> 00:14:26.050 like a prism splitting white light into a rainbow. 00:14:27.850 --> 00:14:30.990 In 2022, Raskin and his graduate student 00:14:30.990 --> 00:14:35.091 finally found this proof completing Gaitsgory's fundamental diagram. 00:14:35.230 --> 00:14:39.940 They cracked this mystery and after which basically shape of the solution 00:14:39.940 --> 00:14:40.773 became clearer. 00:14:40.930 --> 00:14:42.820 Over the course of the next two years, 00:14:43.060 --> 00:14:46.125 Gaitsgory and Raskin led a team that wrote five papers 00:14:46.125 --> 00:14:48.910 that proved the Geometric Langlands conjecture. 00:14:49.420 --> 00:14:50.410 They've built a whole world. 00:14:51.290 --> 00:14:54.230 It took a long time to figure out what's the right edifice to build and what 00:14:54.230 --> 00:14:56.884 materials are available and how to stack them up. 00:14:56.884 --> 00:14:58.725 Of course, mathematics is infinite. 00:14:58.725 --> 00:15:02.450 Once you solve these, some new paradigms appear.