WEBVTT 00:00:00.000 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:04.320 align:middle line:90% [RUSTLING] 00:00:04.320 --> 00:00:07.200 align:middle line:90% [CLICKING] 00:00:07.200 --> 00:00:11.360 align:middle line:90% 00:00:11.360 --> 00:00:15.170 align:middle line:84% LAWRENCE GUTH: So this is the second day of our little unit 00:00:15.170 --> 00:00:17.190 align:middle line:90% on random walks, on groups. 00:00:17.190 --> 00:00:19.670 align:middle line:84% And the first day, we introduced it, 00:00:19.670 --> 00:00:24.925 align:middle line:84% and the second day, we'll describe some more modern stuff. 00:00:24.925 --> 00:00:27.300 align:middle line:84% And the second day will be a little bit more of a survey. 00:00:27.300 --> 00:00:31.340 align:middle line:84% I won't prove everything, but I'll try to-- 00:00:31.340 --> 00:00:33.230 align:middle line:84% and in the second day, we'll see how it 00:00:33.230 --> 00:00:36.470 align:middle line:90% connects to projection theory. 00:00:36.470 --> 00:00:40.555 align:middle line:84% OK, so first, let's recall where we're talking about. 00:00:40.555 --> 00:01:03.260 align:middle line:90% 00:01:03.260 --> 00:01:06.070 align:middle line:84% So we have a finite group G. And then we 00:01:06.070 --> 00:01:13.120 align:middle line:84% have a probability measure on G. And we have an operator T mu. 00:01:13.120 --> 00:01:17.200 align:middle line:84% T mu of f is f convolved with mu. 00:01:17.200 --> 00:01:20.390 align:middle line:84% And so this does a step of a random walk. 00:01:20.390 --> 00:01:23.210 align:middle line:84% So if f is the probability distribution at some moment, 00:01:23.210 --> 00:01:25.060 align:middle line:84% then you make a step of the random walk, 00:01:25.060 --> 00:01:30.256 align:middle line:84% and then T mu f is the new probability distribution. 00:01:30.256 --> 00:01:34.810 align:middle line:84% So T mu of 1 is 1, the constant function 1. 00:01:34.810 --> 00:01:39.020 align:middle line:84% And T mu maps the functions of mean 0. 00:01:39.020 --> 00:01:42.290 align:middle line:84% So these are functions who sum to 0 or have mean 0, 00:01:42.290 --> 00:01:44.710 align:middle line:90% it maps it to itself. 00:01:44.710 --> 00:01:50.880 align:middle line:84% And sigma 1 of T mu is the largest singular value of this. 00:01:50.880 --> 00:02:01.560 align:middle line:90% 00:02:01.560 --> 00:02:04.500 align:middle line:84% And we saw that sigma 1 of T mu is 00:02:04.500 --> 00:02:07.570 align:middle line:84% related to the mixing behavior of this random walk. 00:02:07.570 --> 00:02:12.330 align:middle line:84% So we always have that sigma 1 of T mu 00:02:12.330 --> 00:02:14.350 align:middle line:90% is less than or equal to 1. 00:02:14.350 --> 00:02:17.440 align:middle line:84% And we don't use a positive number. 00:02:17.440 --> 00:02:21.700 align:middle line:84% And sigma 1 of T mu is strictly less than 1. 00:02:21.700 --> 00:02:25.098 align:middle line:84% Then that gives some kind of mixing of the random walk. 00:02:25.098 --> 00:02:26.390 align:middle line:90% We made that precise last time. 00:02:26.390 --> 00:02:31.170 align:middle line:90% 00:02:31.170 --> 00:02:34.365 align:middle line:84% Next, we focused on the group SL2Fp. 00:02:34.365 --> 00:02:41.010 align:middle line:90% 00:02:41.010 --> 00:02:44.370 align:middle line:84% And one special feature of this group 00:02:44.370 --> 00:02:46.410 align:middle line:90% came from representation theory. 00:02:46.410 --> 00:02:50.790 align:middle line:84% So we had a proposition that if you 00:02:50.790 --> 00:02:57.260 align:middle line:84% have a representation of this group, 00:02:57.260 --> 00:03:01.080 align:middle line:84% so that's a unitary representation, 00:03:01.080 --> 00:03:07.570 align:middle line:84% and let's say it's non-trivial, then D has to be pretty big. 00:03:07.570 --> 00:03:12.860 align:middle line:90% 00:03:12.860 --> 00:03:18.980 align:middle line:84% And we use that to prove a mixing estimate. 00:03:18.980 --> 00:03:22.610 align:middle line:84% Mixing estimate says that if mu is a probability 00:03:22.610 --> 00:03:34.550 align:middle line:84% measure on this group, then sigma 1 of T mu 00:03:34.550 --> 00:03:38.815 align:middle line:84% is bounded by p squared times mu L2 squared. 00:03:38.815 --> 00:03:41.480 align:middle line:90% 00:03:41.480 --> 00:03:45.710 align:middle line:84% For example, if A is just a subset of SL2Fp, 00:03:45.710 --> 00:03:54.050 align:middle line:84% then there's a uniform measure on A, mu A. 00:03:54.050 --> 00:03:57.525 align:middle line:84% And it's straightforward to check that its L2 norm squared, 00:03:57.525 --> 00:04:02.650 align:middle line:84% mu A L2 norm squared is 1 over the size of A. 00:04:02.650 --> 00:04:05.770 align:middle line:84% So a corollary is that if A is significantly bigger 00:04:05.770 --> 00:04:11.890 align:middle line:84% than p squared, then the random walk using A mixes rapidly. 00:04:11.890 --> 00:04:20.200 align:middle line:84% So corollary, if for instance, A is bigger than p to the 2.1, 00:04:20.200 --> 00:04:28.450 align:middle line:84% then sigma 1 of TA, so I'll write TA is an abbreviation 00:04:28.450 --> 00:04:33.760 align:middle line:84% for T mu A, sigma 1 of TA would be smaller than something like p 00:04:33.760 --> 00:04:35.590 align:middle line:90% to the negative 1/10. 00:04:35.590 --> 00:04:37.065 align:middle line:90% So it would mix quite rapidly. 00:04:37.065 --> 00:04:40.750 align:middle line:90% 00:04:40.750 --> 00:04:42.980 align:middle line:90% So this is a nice proposition. 00:04:42.980 --> 00:04:45.200 align:middle line:84% And it's kind of sharp of its type. 00:04:45.200 --> 00:04:49.420 align:middle line:84% Because if the set A is a subgroup of SL2Fp, 00:04:49.420 --> 00:04:54.090 align:middle line:84% then the random walk will not generate, will not mix at all, 00:04:54.090 --> 00:04:56.630 align:middle line:84% and there are subgroups, or easy subgroups that 00:04:56.630 --> 00:04:58.530 align:middle line:90% have size around p squared. 00:04:58.530 --> 00:05:03.320 align:middle line:90% So we can't do better than this. 00:05:03.320 --> 00:05:09.440 align:middle line:84% So remark is that this corollary is sharp 00:05:09.440 --> 00:05:18.110 align:middle line:84% because there exists H, a proper subgroup size of H 00:05:18.110 --> 00:05:19.320 align:middle line:90% around p squared. 00:05:19.320 --> 00:05:22.590 align:middle line:84% So for instance, H is the upper triangular matrices. 00:05:22.590 --> 00:05:23.480 align:middle line:90% Yeah? 00:05:23.480 --> 00:05:25.410 align:middle line:84% AUDIENCE: If you have, for a different group, 00:05:25.410 --> 00:05:28.520 align:middle line:84% a similar proposition where you can lower bound 00:05:28.520 --> 00:05:33.960 align:middle line:84% the dimension of the nontrivial unitary representation, 00:05:33.960 --> 00:05:38.270 align:middle line:84% could you also use that as an upper bound for the sigma 1 00:05:38.270 --> 00:05:39.170 align:middle line:90% for that group? 00:05:39.170 --> 00:05:40.670 align:middle line:84% LAWRENCE GUTH: Yeah, so the question 00:05:40.670 --> 00:05:42.890 align:middle line:84% is if we had an analog of this proposition 00:05:42.890 --> 00:05:48.020 align:middle line:84% for some other group, say SLDFp, then could we 00:05:48.020 --> 00:05:50.700 align:middle line:84% do this whole story for all these other groups? 00:05:50.700 --> 00:05:51.800 align:middle line:90% Yeah, you could. 00:05:51.800 --> 00:05:55.540 align:middle line:84% That's all that we've used so far. 00:05:55.540 --> 00:05:58.390 align:middle line:90% Yeah, OK, cool. 00:05:58.390 --> 00:06:01.880 align:middle line:84% So then also last class, we stated an old theorem, 00:06:01.880 --> 00:06:03.160 align:middle line:90% which goes rather-- 00:06:03.160 --> 00:06:05.980 align:middle line:84% it's older than this proposition, but it goes rather, 00:06:05.980 --> 00:06:07.920 align:middle line:84% it does not obviously follow from it. 00:06:07.920 --> 00:06:13.900 align:middle line:90% 00:06:13.900 --> 00:06:19.000 align:middle line:84% So theorem, this theorem is essentially due to Selberg, 00:06:19.000 --> 00:06:22.060 align:middle line:84% although he didn't say it in quite this way. 00:06:22.060 --> 00:06:30.580 align:middle line:84% So here, A, so we pick A is a particular list of generators 00:06:30.580 --> 00:06:31.390 align:middle line:90% of SL2Fp. 00:06:31.390 --> 00:06:39.880 align:middle line:90% 00:06:39.880 --> 00:06:44.060 align:middle line:84% So this is perhaps the simplest set of generators of SL2Fp. 00:06:44.060 --> 00:06:45.880 align:middle line:84% But it is also somewhat special in a way 00:06:45.880 --> 00:06:48.340 align:middle line:84% that we'll talk about when we talk about the proof. 00:06:48.340 --> 00:06:55.350 align:middle line:84% So then the conclusion is that sigma 1 of TA is less than 1 00:06:55.350 --> 00:06:59.260 align:middle line:84% minus c, where c is positive and c is-- 00:06:59.260 --> 00:07:01.080 align:middle line:90% this is true for every p. 00:07:01.080 --> 00:07:04.260 align:middle line:90% So c does not depend on p. 00:07:04.260 --> 00:07:09.000 align:middle line:90% So these all mix uniformly fast. 00:07:09.000 --> 00:07:12.030 align:middle line:84% So this theorem is not an immediate corollary 00:07:12.030 --> 00:07:15.930 align:middle line:84% of this proposition because this is a much smaller set. 00:07:15.930 --> 00:07:21.460 align:middle line:84% And so today, a couple of goals in this survey. 00:07:21.460 --> 00:07:23.400 align:middle line:84% The first goal is to sketch a proof 00:07:23.400 --> 00:07:26.650 align:middle line:84% of this theorem in which this proposition plays a key role. 00:07:26.650 --> 00:07:28.210 align:middle line:90% But we'll see some other ideas. 00:07:28.210 --> 00:07:30.570 align:middle line:84% And a second goal is to discuss what 00:07:30.570 --> 00:07:33.570 align:middle line:84% happens if you replace what was special about this generating 00:07:33.570 --> 00:07:35.700 align:middle line:84% set at all, and what happens if we replace it 00:07:35.700 --> 00:07:37.025 align:middle line:90% by some other generating set. 00:07:37.025 --> 00:07:44.190 align:middle line:90% 00:07:44.190 --> 00:07:51.360 align:middle line:84% OK, so the first proof was due to Selberg, basically. 00:07:51.360 --> 00:07:53.340 align:middle line:84% But that proof was quite difficult. 00:07:53.340 --> 00:07:57.650 align:middle line:84% It used a bunch of stuff, including the Neyman hypothesis 00:07:57.650 --> 00:07:59.840 align:middle line:90% for curves over finite fields. 00:07:59.840 --> 00:08:06.890 align:middle line:84% We're going to follow a later proof which 00:08:06.890 --> 00:08:11.720 align:middle line:84% is due to Sarnak and Xua, and perhaps simplified or adjusted 00:08:11.720 --> 00:08:14.540 align:middle line:90% a bit by various people. 00:08:14.540 --> 00:08:16.690 align:middle line:90% OK, cool. 00:08:16.690 --> 00:08:19.250 align:middle line:90% 00:08:19.250 --> 00:08:25.170 align:middle line:84% All right, so this set is symmetric, 00:08:25.170 --> 00:08:29.250 align:middle line:84% meaning that if a group element is in the set, 00:08:29.250 --> 00:08:31.340 align:middle line:90% so is the inverse. 00:08:31.340 --> 00:08:37.250 align:middle line:84% And so we could say that mu is symmetric if mu of g 00:08:37.250 --> 00:08:45.510 align:middle line:84% is equal to mu of g inverse for every g in G. And T mu, 00:08:45.510 --> 00:08:50.230 align:middle line:84% actually, it's a bit nicer if mu is symmetric. 00:08:50.230 --> 00:08:51.530 align:middle line:90% And here's why. 00:08:51.530 --> 00:08:54.790 align:middle line:90% So remark, mu is symmetric. 00:08:54.790 --> 00:08:57.800 align:middle line:84% That is equivalent to saying that the operator T 00:08:57.800 --> 00:09:00.675 align:middle line:90% mu is symmetric as a matrix. 00:09:00.675 --> 00:09:06.040 align:middle line:90% 00:09:06.040 --> 00:09:11.150 align:middle line:84% And so because-- so if you think of T mu as a matrix, 00:09:11.150 --> 00:09:15.640 align:middle line:84% then it has a g1, g2 entry, and the g1, g2 entry 00:09:15.640 --> 00:09:18.290 align:middle line:90% is mu of g1, g2 inverse. 00:09:18.290 --> 00:09:21.860 align:middle line:84% So if you switch these, you're taking the inverse of that. 00:09:21.860 --> 00:09:25.270 align:middle line:84% And if mu is symmetric, that doesn't matter. 00:09:25.270 --> 00:09:27.370 align:middle line:90% So it's a symmetric matrix. 00:09:27.370 --> 00:09:29.650 align:middle line:84% So that's nice because a symmetric matrix, 00:09:29.650 --> 00:09:30.950 align:middle line:90% you can diagonalize it. 00:09:30.950 --> 00:09:32.830 align:middle line:84% And instead of just having singular values, 00:09:32.830 --> 00:09:34.840 align:middle line:90% it has eigenvalues. 00:09:34.840 --> 00:09:37.580 align:middle line:84% So once T mu is a symmetric matrix, 00:09:37.580 --> 00:09:41.650 align:middle line:84% then the first singular value of T mu to the k 00:09:41.650 --> 00:09:46.965 align:middle line:84% is the first singular value of T mu to the k. 00:09:46.965 --> 00:09:49.120 align:middle line:84% It helps to have a blackboard to say it. 00:09:49.120 --> 00:09:53.331 align:middle line:90% 00:09:53.331 --> 00:09:55.650 align:middle line:90% And so that suggests a-- 00:09:55.650 --> 00:09:56.498 align:middle line:90% this is helpful. 00:09:56.498 --> 00:09:58.290 align:middle line:84% This isn't true for non-symmetric matrices. 00:09:58.290 --> 00:09:59.832 align:middle line:84% But it's true for symmetric matrices. 00:09:59.832 --> 00:10:01.800 align:middle line:84% And it suggests a strategy of how 00:10:01.800 --> 00:10:04.180 align:middle line:84% to get a handle on sigma 1 of T mu. 00:10:04.180 --> 00:10:06.095 align:middle line:84% So we can say the following thing. 00:10:06.095 --> 00:10:08.700 align:middle line:90% 00:10:08.700 --> 00:10:11.370 align:middle line:90% So we have sigma 1. 00:10:11.370 --> 00:10:13.990 align:middle line:84% Yeah, let's also think about T mu to the k. 00:10:13.990 --> 00:10:18.360 align:middle line:84% So T mu to the k of f is f convolved with mu 00:10:18.360 --> 00:10:20.880 align:middle line:84% and then convolved with mu and so 00:10:20.880 --> 00:10:29.010 align:middle line:84% on k times, which is f convolved with mu convolved with itself 00:10:29.010 --> 00:10:30.750 align:middle line:90% k times. 00:10:30.750 --> 00:10:33.430 align:middle line:84% So that's an operator of our type. 00:10:33.430 --> 00:10:34.950 align:middle line:84% But with a new measure, the measure 00:10:34.950 --> 00:10:39.640 align:middle line:84% is mu convolved with itself k times. 00:10:39.640 --> 00:10:46.250 align:middle line:84% So therefore, sigma 1 of T mu to the k, sigma 1 T 00:10:46.250 --> 00:10:51.505 align:middle line:84% mu to the k, which is sigma 1 of T mu to the k. 00:10:51.505 --> 00:10:57.335 align:middle line:90% 00:10:57.335 --> 00:10:59.460 align:middle line:84% So if you try to explain this without a blackboard, 00:10:59.460 --> 00:11:03.363 align:middle line:84% you have to have really good ability to evoke things 00:11:03.363 --> 00:11:04.030 align:middle line:90% with your voice. 00:11:04.030 --> 00:11:06.650 align:middle line:90% 00:11:06.650 --> 00:11:11.720 align:middle line:84% All right, so mu might be the uniform measure 00:11:11.720 --> 00:11:15.620 align:middle line:84% on this set of four elements A. And if we apply this proposition 00:11:15.620 --> 00:11:19.160 align:middle line:84% directly to that set of four elements, it gives us nothing. 00:11:19.160 --> 00:11:21.180 align:middle line:90% But we can use this. 00:11:21.180 --> 00:11:27.290 align:middle line:84% And then we can try to apply the proposition to mu to the k. 00:11:27.290 --> 00:11:29.810 align:middle line:84% So if we put together what we have, 00:11:29.810 --> 00:11:34.520 align:middle line:84% we can make the following corollary, sigma of T mu 00:11:34.520 --> 00:11:35.400 align:middle line:90% to the k-- 00:11:35.400 --> 00:11:43.330 align:middle line:84% so OK, so this is if mu is a probability measure on SL2 00:11:43.330 --> 00:11:51.070 align:middle line:84% of Fp, then sigma 1 of T mu to the k 00:11:51.070 --> 00:11:53.480 align:middle line:90% is bounded by, well, this thing. 00:11:53.480 --> 00:11:59.230 align:middle line:84% And for this thing, we can apply our proposition here. 00:11:59.230 --> 00:12:07.125 align:middle line:84% So we get p squared times mu tensor k L2 squared. 00:12:07.125 --> 00:12:15.350 align:middle line:90% 00:12:15.350 --> 00:12:20.860 align:middle line:84% So for the original, if you take the Selberg measure A then 00:12:20.860 --> 00:12:23.710 align:middle line:84% for the original mu A, this thing 00:12:23.710 --> 00:12:25.730 align:middle line:84% would be much too big to be interesting. 00:12:25.730 --> 00:12:28.210 align:middle line:84% But perhaps we could find a high power k 00:12:28.210 --> 00:12:32.097 align:middle line:84% where this L2 norm is actually smaller than 1 over p squared. 00:12:32.097 --> 00:12:33.930 align:middle line:84% And that would give us an interesting bound. 00:12:33.930 --> 00:12:40.860 align:middle line:90% 00:12:40.860 --> 00:12:45.330 align:middle line:84% Cool, so all of the stuff that we'll talk about today 00:12:45.330 --> 00:12:47.670 align:middle line:90% is based on this corollary. 00:12:47.670 --> 00:12:49.480 align:middle line:84% So you can think about it like this, 00:12:49.480 --> 00:12:53.220 align:middle line:84% you start off with a probability measure mu, and we want to know, 00:12:53.220 --> 00:12:55.410 align:middle line:84% does it get really evenly mixed if you convolve it 00:12:55.410 --> 00:12:57.240 align:middle line:90% with itself a lot of times? 00:12:57.240 --> 00:13:02.550 align:middle line:84% And what we'll see here is that maybe if you convolve it 00:13:02.550 --> 00:13:08.130 align:middle line:84% with itself a bunch of times, it will get fairly well mixed, not 00:13:08.130 --> 00:13:11.310 align:middle line:84% spectacularly well mixed, but just fairly well mixed 00:13:11.310 --> 00:13:14.700 align:middle line:84% to guarantee that this L2 norm is smaller than 1 over p 00:13:14.700 --> 00:13:16.140 align:middle line:90% squared. 00:13:16.140 --> 00:13:18.960 align:middle line:84% So if it manages to be kind of roughly evenly 00:13:18.960 --> 00:13:22.480 align:middle line:84% distributed around p to the 2.1 elements in this group, 00:13:22.480 --> 00:13:26.040 align:middle line:84% that would be good enough that this would be less than 1. 00:13:26.040 --> 00:13:28.650 align:middle line:84% Once that happens, we can use this corollary 00:13:28.650 --> 00:13:31.110 align:middle line:90% and we can bound sigma 1. 00:13:31.110 --> 00:13:33.180 align:middle line:84% And once we bound sigma 1, then we'll 00:13:33.180 --> 00:13:36.277 align:middle line:84% see that that, in turn, tells us what happens as we keep mixing. 00:13:36.277 --> 00:13:38.360 align:middle line:84% Then we'll be able to say it gets really very well 00:13:38.360 --> 00:13:39.930 align:middle line:90% mixed at some later time. 00:13:39.930 --> 00:13:42.530 align:middle line:90% 00:13:42.530 --> 00:13:45.690 align:middle line:90% OK, but how can we-- 00:13:45.690 --> 00:13:46.950 align:middle line:90% so how can we bound this? 00:13:46.950 --> 00:13:52.700 align:middle line:84% How can we show that mu tensor k is decently well mixed, fairly 00:13:52.700 --> 00:13:54.570 align:middle line:90% well mixed? 00:13:54.570 --> 00:13:56.270 align:middle line:84% So there are two different approaches 00:13:56.270 --> 00:13:58.430 align:middle line:90% that we'll talk about today. 00:13:58.430 --> 00:14:00.890 align:middle line:84% So one approach, which is special for this theorem, 00:14:00.890 --> 00:14:05.060 align:middle line:90% involves either-- 00:14:05.060 --> 00:14:09.800 align:middle line:84% let me say it involves lifting to SL2Z and/or 00:14:09.800 --> 00:14:11.550 align:middle line:90% it involves hyperbolic geometry. 00:14:11.550 --> 00:14:14.095 align:middle line:84% I'll describe it in a couple of slightly different ways. 00:14:14.095 --> 00:14:16.970 align:middle line:90% 00:14:16.970 --> 00:14:23.740 align:middle line:84% So first let's talk about lifting to SL2Z. 00:14:23.740 --> 00:14:27.290 align:middle line:90% 00:14:27.290 --> 00:14:30.650 align:middle line:84% So if you reduce mod p, it gives a group 00:14:30.650 --> 00:14:33.920 align:middle line:90% homomorphism from SL2Z to SL2Fp. 00:14:33.920 --> 00:14:42.186 align:middle line:90% 00:14:42.186 --> 00:14:51.420 align:middle line:84% So let's suppose that M is a probability measure on SL2Z. 00:14:51.420 --> 00:14:55.390 align:middle line:84% And mu is the pushforward pi p of M. 00:14:55.390 --> 00:14:59.160 align:middle line:84% So that's a probability measure here. 00:14:59.160 --> 00:15:03.090 align:middle line:84% And this setup with group homomorphisms commutes, 00:15:03.090 --> 00:15:05.550 align:middle line:90% plays nicely with convolution. 00:15:05.550 --> 00:15:12.990 align:middle line:84% So if I take the pushforward of M convolved with itself k times, 00:15:12.990 --> 00:15:15.935 align:middle line:84% I will get mu convolved with itself k times. 00:15:15.935 --> 00:15:21.870 align:middle line:90% 00:15:21.870 --> 00:15:25.170 align:middle line:84% So one approach to try to understand mu convolved 00:15:25.170 --> 00:15:28.020 align:middle line:84% with itself a bunch of times is to first try 00:15:28.020 --> 00:15:31.030 align:middle line:84% to understand M convolved with itself a bunch of times 00:15:31.030 --> 00:15:33.870 align:middle line:84% and then try to understand what happens when we perform 00:15:33.870 --> 00:15:35.050 align:middle line:90% this push forward. 00:15:35.050 --> 00:15:42.320 align:middle line:90% 00:15:42.320 --> 00:15:51.170 align:middle line:84% So here's our plan, study M convolved with itself k times 00:15:51.170 --> 00:15:55.590 align:middle line:84% to study the push forward operation. 00:15:55.590 --> 00:15:59.840 align:middle line:90% 00:15:59.840 --> 00:16:05.060 align:middle line:84% So this is a reasonable strategy because of the fact, 00:16:05.060 --> 00:16:08.630 align:middle line:84% or to the extent that we feel that SL2Z is easier 00:16:08.630 --> 00:16:10.640 align:middle line:90% to get our hands on then SL2Fp. 00:16:10.640 --> 00:16:14.880 align:middle line:90% 00:16:14.880 --> 00:16:19.220 align:middle line:84% And there are a couple of reasons that SL2Z is easier 00:16:19.220 --> 00:16:22.940 align:middle line:90% to get our hands on then SL2Fp. 00:16:22.940 --> 00:16:31.160 align:middle line:84% Let me call this good features of SL2Z. 00:16:31.160 --> 00:16:33.860 align:middle line:90% 00:16:33.860 --> 00:16:36.800 align:middle line:84% We won't necessarily use all of these, but let me mention them. 00:16:36.800 --> 00:16:42.750 align:middle line:84% So the first feature is that SL2Z is virtually free. 00:16:42.750 --> 00:16:46.090 align:middle line:90% 00:16:46.090 --> 00:16:47.770 align:middle line:84% So that virtually means is that there 00:16:47.770 --> 00:16:58.420 align:middle line:84% is a subgroup H in SL2Z which is a finite index free subgroup. 00:16:58.420 --> 00:17:01.760 align:middle line:84% So in general, if you say that a group is virtually blah, 00:17:01.760 --> 00:17:06.280 align:middle line:84% it means that there's a finite index subgroup, which is blah. 00:17:06.280 --> 00:17:08.750 align:middle line:84% So it's pretty close to being free. 00:17:08.750 --> 00:17:11.560 align:middle line:84% That's not true at all for SL2Fp. 00:17:11.560 --> 00:17:14.210 align:middle line:84% You take SL2Fp, take the simplest generators 00:17:14.210 --> 00:17:18.640 align:middle line:84% you can think of, try to write down all the relations, 00:17:18.640 --> 00:17:19.880 align:middle line:90% it's not super easy. 00:17:19.880 --> 00:17:21.880 align:middle line:84% I don't personally know how to do it, although I 00:17:21.880 --> 00:17:23.172 align:middle line:90% think people know how to do it. 00:17:23.172 --> 00:17:28.150 align:middle line:90% 00:17:28.150 --> 00:17:37.690 align:middle line:84% OK, number two, SL2Z is sitting inside of SL2R. 00:17:37.690 --> 00:17:41.680 align:middle line:84% It's kind of a discrete approximation of SL2R, 00:17:41.680 --> 00:17:46.140 align:middle line:84% so it's pretty closely related to a Lie group. 00:17:46.140 --> 00:17:47.840 align:middle line:90% This is what we'll actually use. 00:17:47.840 --> 00:17:53.520 align:middle line:90% 00:17:53.520 --> 00:18:01.680 align:middle line:84% And number three is that SL2Z acts nicely-- 00:18:01.680 --> 00:18:06.240 align:middle line:84% maybe I should call it 2B, it's quite closely related to two. 00:18:06.240 --> 00:18:08.810 align:middle line:84% It acts nicely on the hyperbolic plane. 00:18:08.810 --> 00:18:13.950 align:middle line:90% 00:18:13.950 --> 00:18:19.365 align:middle line:84% So recall that the isometries of the hyperbolic plane is PSL2R. 00:18:19.365 --> 00:18:22.350 align:middle line:90% 00:18:22.350 --> 00:18:26.250 align:middle line:84% So SL2R, basically SL2R, not quite. 00:18:26.250 --> 00:18:29.440 align:middle line:84% So SL2R acts by isometries on the hyperbolic plane. 00:18:29.440 --> 00:18:31.020 align:middle line:84% And SL2Z is sitting inside there, 00:18:31.020 --> 00:18:35.300 align:middle line:84% so it acts by isometries on the hyperbolic plane. 00:18:35.300 --> 00:18:39.380 align:middle line:84% And that action is captures a lot of SL2Z. 00:18:39.380 --> 00:18:43.250 align:middle line:84% You can also use this to help understand SL2Z. 00:18:43.250 --> 00:18:46.010 align:middle line:84% And so all three of these things are much more complicated 00:18:46.010 --> 00:18:46.670 align:middle line:90% for SL2Fp. 00:18:46.670 --> 00:19:12.080 align:middle line:90% 00:19:12.080 --> 00:19:14.660 align:middle line:84% All right, so I don't-- so one of the things that I 00:19:14.660 --> 00:19:19.460 align:middle line:84% won't completely prove is some thing about doing convolutions 00:19:19.460 --> 00:19:20.430 align:middle line:90% on SL2Z. 00:19:20.430 --> 00:19:22.790 align:middle line:84% But I want to give some intuition 00:19:22.790 --> 00:19:24.710 align:middle line:90% and rough statement about it. 00:19:24.710 --> 00:19:36.150 align:middle line:84% So intuition about convolution on SL2Z. 00:19:36.150 --> 00:19:40.360 align:middle line:90% 00:19:40.360 --> 00:19:50.080 align:middle line:84% So as a warm up, I want to think about convolution on Z. 00:19:50.080 --> 00:19:56.380 align:middle line:84% So let's say nu is a probability measure on Z. 00:19:56.380 --> 00:20:00.470 align:middle line:84% And I take nu convolved with itself k times. 00:20:00.470 --> 00:20:03.830 align:middle line:84% So nu is some fixed simple probability measure. 00:20:03.830 --> 00:20:11.740 align:middle line:84% Maybe let's take nu of x is 1/2 if x is plus minus 1 and 0 00:20:11.740 --> 00:20:14.680 align:middle line:90% otherwise. 00:20:14.680 --> 00:20:19.210 align:middle line:84% So what happens when we convolve it with itself k times? 00:20:19.210 --> 00:20:22.630 align:middle line:84% Well, the central limit theorem tells us 00:20:22.630 --> 00:20:24.495 align:middle line:84% that this thing is almost a Gaussian. 00:20:24.495 --> 00:20:27.300 align:middle line:90% 00:20:27.300 --> 00:20:39.620 align:middle line:84% Gaussian, also closely related to the heat kernel on R. 00:20:39.620 --> 00:20:43.050 align:middle line:84% So in analogy with this, what might we 00:20:43.050 --> 00:20:45.960 align:middle line:84% hope to happen when we take a measure on SL2Z 00:20:45.960 --> 00:20:49.110 align:middle line:84% and we convolve it with itself a lot? 00:20:49.110 --> 00:20:51.690 align:middle line:90% So you might hope-- 00:20:51.690 --> 00:21:00.630 align:middle line:84% so we have M, measure, probability measure on SL2Z. 00:21:00.630 --> 00:21:06.120 align:middle line:84% So you might hope to have some kind of central limit 00:21:06.120 --> 00:21:10.870 align:middle line:90% theorem for matrices. 00:21:10.870 --> 00:21:14.970 align:middle line:84% And it would say that M convolved with itself a lot 00:21:14.970 --> 00:21:22.410 align:middle line:84% is sort of like the heat kernel, which needs to be defined, 00:21:22.410 --> 00:21:23.205 align:middle line:90% on SL2R. 00:21:23.205 --> 00:21:32.990 align:middle line:90% 00:21:32.990 --> 00:21:37.400 align:middle line:84% OK, so this seems like a natural thing 00:21:37.400 --> 00:21:40.070 align:middle line:84% in the intersection of probability theory and matrix 00:21:40.070 --> 00:21:42.770 align:middle line:84% theory and has been studied, although I'm not 00:21:42.770 --> 00:21:51.335 align:middle line:84% an expert on it, but cf work of Furstenberg and. 00:21:51.335 --> 00:21:56.780 align:middle line:90% 00:21:56.780 --> 00:22:00.230 align:middle line:84% So I believe that I know roughly how it works out, although I 00:22:00.230 --> 00:22:01.590 align:middle line:90% don't have a precise statement. 00:22:01.590 --> 00:22:04.500 align:middle line:84% But let me tell you roughly how I think it works out. 00:22:04.500 --> 00:22:05.990 align:middle line:84% And then we'll see how this would 00:22:05.990 --> 00:22:10.200 align:middle line:90% be helpful to understand SL2Fp. 00:22:10.200 --> 00:22:14.180 align:middle line:84% So first of all, we'll have some kind of balls in SL2R. 00:22:14.180 --> 00:22:21.020 align:middle line:84% So BT is the set of matrices A, B, C, D 00:22:21.020 --> 00:22:26.530 align:middle line:84% in SL2R with the property that a squared plus b 00:22:26.530 --> 00:22:31.030 align:middle line:84% squared plus c squared plus d squared is at most R squared-- 00:22:31.030 --> 00:22:31.920 align:middle line:90% is at most T squared. 00:22:31.920 --> 00:22:35.746 align:middle line:90% 00:22:35.746 --> 00:22:38.290 align:middle line:84% So this is some of rough-- some sort of notion 00:22:38.290 --> 00:22:43.360 align:middle line:90% of a ball in the group SL2R. 00:22:43.360 --> 00:22:45.550 align:middle line:84% You could imagine a fancier notion 00:22:45.550 --> 00:22:47.260 align:middle line:84% where you start with the Lie algebra, 00:22:47.260 --> 00:22:48.910 align:middle line:84% you put a ball in the Lie algebra, 00:22:48.910 --> 00:22:50.360 align:middle line:84% and you take the exponential map. 00:22:50.360 --> 00:22:53.530 align:middle line:84% And that's probably a better thing to do than this. 00:22:53.530 --> 00:22:56.900 align:middle line:84% But this is a very low brow definition. 00:22:56.900 --> 00:23:01.610 align:middle line:84% And it is approximately describes balls. 00:23:01.610 --> 00:23:03.320 align:middle line:84% And then if we're interested in SL2Z, 00:23:03.320 --> 00:23:05.260 align:middle line:84% we would take the integer points in this. 00:23:05.260 --> 00:23:10.690 align:middle line:84% BT of Z, that's defined to be BT intersected with SL2Z. 00:23:10.690 --> 00:23:14.890 align:middle line:90% So these are integers. 00:23:14.890 --> 00:23:20.260 align:middle line:84% So for reference, what is the cardinality of BT of Z? 00:23:20.260 --> 00:23:28.530 align:middle line:84% So little lemma, cardinality of BT of Z is roughly T squared. 00:23:28.530 --> 00:23:38.060 align:middle line:84% Sketch, really choose a and d minus T, 00:23:38.060 --> 00:23:42.180 align:middle line:90% T. So that's T squared choices. 00:23:42.180 --> 00:23:48.720 align:middle line:84% And then we have to solve b times c is a times d minus 1. 00:23:48.720 --> 00:23:49.890 align:middle line:90% I think I did that right. 00:23:49.890 --> 00:23:53.900 align:middle line:84% So here, your SL2R, the determinant is 1, so ad minus bc 00:23:53.900 --> 00:23:54.400 align:middle line:90% is 1. 00:23:54.400 --> 00:23:57.340 align:middle line:90% 00:23:57.340 --> 00:24:00.700 align:middle line:84% So we have this integer of size around T squared. 00:24:00.700 --> 00:24:03.060 align:middle line:84% And we want to know how many ways are there 00:24:03.060 --> 00:24:08.290 align:middle line:84% to factor it into two integers of size around T. OK, 00:24:08.290 --> 00:24:14.260 align:middle line:84% so not every integer will factor like this. 00:24:14.260 --> 00:24:15.480 align:middle line:90% It could be prime. 00:24:15.480 --> 00:24:18.150 align:middle line:84% But mostly, they will factor like that. 00:24:18.150 --> 00:24:20.260 align:middle line:84% And how many ways will they factor? 00:24:20.260 --> 00:24:25.430 align:middle line:84% Well, not too many ways, at most, T to the epsilon ways. 00:24:25.430 --> 00:24:32.330 align:middle line:84% So number of choices for b and c is on the order of 1. 00:24:32.330 --> 00:24:35.590 align:middle line:84% So the level of our back of the envelope computation, 00:24:35.590 --> 00:24:40.120 align:middle line:84% there should be about T squared group elements in there. 00:24:40.120 --> 00:24:50.230 align:middle line:84% All right, so there is a vague statement that if you fix M, 00:24:50.230 --> 00:25:00.640 align:middle line:84% then M star to the k is roughly equidistributed on the ball 00:25:00.640 --> 00:25:06.940 align:middle line:84% of radius T for T around exponential of some constant 00:25:06.940 --> 00:25:10.600 align:middle line:90% depending on M times k. 00:25:10.600 --> 00:25:15.670 align:middle line:84% So it's not stated precisely, but this is roughly 00:25:15.670 --> 00:25:18.600 align:middle line:84% what I think the central limit theorem should say in SL2Z. 00:25:18.600 --> 00:25:31.068 align:middle line:90% 00:25:31.068 --> 00:25:33.690 align:middle line:84% In a little bit we'll do the hyperbolic geometry point 00:25:33.690 --> 00:25:35.658 align:middle line:84% of view, where I'll be able to state things 00:25:35.658 --> 00:25:36.950 align:middle line:90% a little bit better if we want. 00:25:36.950 --> 00:25:44.130 align:middle line:90% 00:25:44.130 --> 00:25:48.570 align:middle line:84% OK, so that's step one of our plan, study what the convolution 00:25:48.570 --> 00:25:52.710 align:middle line:90% looks like in upstairs in SL2Z. 00:25:52.710 --> 00:25:55.740 align:middle line:84% Next, we need to think about what happens when we project 00:25:55.740 --> 00:25:57.930 align:middle line:90% from SL2Z down to SL2Fp. 00:25:57.930 --> 00:26:02.730 align:middle line:90% 00:26:02.730 --> 00:26:04.260 align:middle line:84% Why don't we pause there, actually? 00:26:04.260 --> 00:26:07.510 align:middle line:84% I mean, I did some things not completely precisely, 00:26:07.510 --> 00:26:09.930 align:middle line:84% but let me see if people have any questions, or comments, 00:26:09.930 --> 00:26:11.690 align:middle line:90% or are bothered by things. 00:26:11.690 --> 00:26:21.530 align:middle line:90% 00:26:21.530 --> 00:26:22.730 align:middle line:90% Yeah? 00:26:22.730 --> 00:26:25.010 align:middle line:84% AUDIENCE: Question about [INAUDIBLE] to the existence 00:26:25.010 --> 00:26:28.220 align:middle line:90% of the heat kernel that-- 00:26:28.220 --> 00:26:31.400 align:middle line:84% the heat kernel on R, and, in general, 00:26:31.400 --> 00:26:34.790 align:middle line:84% kind of Euclidean looking spaces is kind of closely related 00:26:34.790 --> 00:26:36.210 align:middle line:90% to the Fourier transform. 00:26:36.210 --> 00:26:40.250 align:middle line:84% But in a non-abelian group, is there anything analogous to 00:26:40.250 --> 00:26:45.182 align:middle line:84% that that makes this some sort of heat kernel-ish thing work? 00:26:45.182 --> 00:26:46.640 align:middle line:84% LAWRENCE GUTH: Yeah so the question 00:26:46.640 --> 00:26:52.880 align:middle line:84% is, how does the heat kernel work in a non-abelian group? 00:26:52.880 --> 00:26:56.790 align:middle line:84% Let me punt on that until we get to the hyperbolic space version. 00:26:56.790 --> 00:26:58.460 align:middle line:84% And we'll talk about the heat kernel 00:26:58.460 --> 00:27:01.070 align:middle line:84% there, which is a little bit easier, at least for me, 00:27:01.070 --> 00:27:04.440 align:middle line:90% to make it precise. 00:27:04.440 --> 00:27:08.450 align:middle line:84% So let's do the second step, where we pass from SL2F-- 00:27:08.450 --> 00:27:10.070 align:middle line:90% from SL2Z to SL2Fp. 00:27:10.070 --> 00:27:25.060 align:middle line:90% 00:27:25.060 --> 00:27:32.440 align:middle line:84% So remember our setup that M is a probability measure on SL2Z 00:27:32.440 --> 00:27:37.472 align:middle line:84% And then mu is the pushforward of M, which is 00:27:37.472 --> 00:27:38.680 align:middle line:90% probability measure on SL2Fp. 00:27:38.680 --> 00:27:42.130 align:middle line:90% 00:27:42.130 --> 00:27:47.420 align:middle line:84% Also, let's mention that gamma p could be the kernel of pi p. 00:27:47.420 --> 00:27:54.790 align:middle line:84% So that's a normal subgroup of SL2Z. 00:27:54.790 --> 00:27:57.400 align:middle line:90% 00:27:57.400 --> 00:28:06.520 align:middle line:84% And it's gamma p is the set of a, b, c, d in SL2Z, 00:28:06.520 --> 00:28:15.065 align:middle line:84% so that a, d, b, c is congruent to the identity mod p. 00:28:15.065 --> 00:28:18.420 align:middle line:90% 00:28:18.420 --> 00:28:23.580 align:middle line:84% OK, great, so now our goal is to understand the L2 norm 00:28:23.580 --> 00:28:26.490 align:middle line:90% of some convolution of this. 00:28:26.490 --> 00:28:31.140 align:middle line:84% And here's a cute trick, little lemma. 00:28:31.140 --> 00:28:37.690 align:middle line:84% So if mu is symmetric, so for instance, if M is symmetric, 00:28:37.690 --> 00:28:49.500 align:middle line:84% then mu tensor k L2 squared is mu tensor 2k of the identity. 00:28:49.500 --> 00:28:51.900 align:middle line:90% So here's the proof. 00:28:51.900 --> 00:28:53.700 align:middle line:90% Let's start on the other side. 00:28:53.700 --> 00:28:58.560 align:middle line:84% Mu tensor 2k of the identity is, well, 00:28:58.560 --> 00:29:01.653 align:middle line:90% mu tensor k convolved with-- 00:29:01.653 --> 00:29:03.070 align:middle line:84% so I was saying the word "tensor." 00:29:03.070 --> 00:29:04.237 align:middle line:90% That was not the right word. 00:29:04.237 --> 00:29:05.100 align:middle line:90% I mean convolved. 00:29:05.100 --> 00:29:11.340 align:middle line:84% Mu convolved k convolved with mu convolved k, the identity. 00:29:11.340 --> 00:29:18.020 align:middle line:84% So that's the sum over g in our group, mu convolved k of g mu 00:29:18.020 --> 00:29:20.610 align:middle line:90% convolved k of g inverse. 00:29:20.610 --> 00:29:24.200 align:middle line:84% That's how you convolve two functions. 00:29:24.200 --> 00:29:27.080 align:middle line:84% But what's nice is that mu is symmetric is that these 00:29:27.080 --> 00:29:29.300 align:middle line:90% are the same as each other. 00:29:29.300 --> 00:29:37.070 align:middle line:84% So we just get the sum on g mu convolved k of g squared. 00:29:37.070 --> 00:29:38.260 align:middle line:90% So that's equal to that. 00:29:38.260 --> 00:29:41.540 align:middle line:90% 00:29:41.540 --> 00:29:45.500 align:middle line:84% OK, cool, so that's convenient, because now we just have 00:29:45.500 --> 00:29:49.340 align:middle line:90% to estimate this one thing. 00:29:49.340 --> 00:29:55.640 align:middle line:84% And mu convolved 2k of the identity is-- 00:29:55.640 --> 00:29:57.810 align:middle line:84% that's something that lives here, 00:29:57.810 --> 00:29:59.970 align:middle line:84% but we can relate it to something that lives here. 00:29:59.970 --> 00:30:06.090 align:middle line:84% It's the sum g in gamma p of M convolved 2k of g. 00:30:06.090 --> 00:30:11.251 align:middle line:90% 00:30:11.251 --> 00:30:14.600 align:middle line:84% So in other words, we want to know how much mass-- 00:30:14.600 --> 00:30:18.740 align:middle line:84% what is the measure, using this measure M convolved 2k, 00:30:18.740 --> 00:30:23.320 align:middle line:84% what is the measure of the subgroup gamma p. 00:30:23.320 --> 00:30:27.640 align:middle line:84% All right, now M convolved 2k is approximately evenly distributed 00:30:27.640 --> 00:30:28.870 align:middle line:90% on some ball. 00:30:28.870 --> 00:30:38.440 align:middle line:84% So by the vague statement, this is 00:30:38.440 --> 00:30:42.340 align:middle line:84% equivalent to estimating what fraction of the ball 00:30:42.340 --> 00:30:43.400 align:middle line:90% is in gamma p. 00:30:43.400 --> 00:30:47.920 align:middle line:90% 00:30:47.920 --> 00:30:49.670 align:middle line:84% So we have some big ball in our Lie group. 00:30:49.670 --> 00:30:52.460 align:middle line:84% What fraction of it is in the subgroup gamma p? 00:30:52.460 --> 00:30:55.050 align:middle line:90% 00:30:55.050 --> 00:31:02.170 align:middle line:84% So remark, the index of gamma p is the cardinality 00:31:02.170 --> 00:31:06.260 align:middle line:84% of SL2Fp, which is around p cubed. 00:31:06.260 --> 00:31:11.100 align:middle line:84% So if this were evenly distributed, 00:31:11.100 --> 00:31:13.320 align:middle line:84% then you would expect this quotient to be about 1 00:31:13.320 --> 00:31:15.540 align:middle line:90% over p cubed. 00:31:15.540 --> 00:31:17.970 align:middle line:90% And that's basically true. 00:31:17.970 --> 00:31:19.175 align:middle line:90% So here's another lemma. 00:31:19.175 --> 00:31:22.080 align:middle line:90% 00:31:22.080 --> 00:31:26.010 align:middle line:84% BT intersected with gamma p is bounded 00:31:26.010 --> 00:31:32.010 align:middle line:84% by T squared over p cubed, which is also like the size of BT 00:31:32.010 --> 00:31:35.820 align:middle line:90% of Z over p cubed. 00:31:35.820 --> 00:31:38.140 align:middle line:84% So it is pretty evenly distributed. 00:31:38.140 --> 00:31:49.380 align:middle line:90% 00:31:49.380 --> 00:31:52.080 align:middle line:90% All right, so here's the proof. 00:31:52.080 --> 00:31:58.020 align:middle line:84% So we're looking for a, b, c, d, where 00:31:58.020 --> 00:32:05.040 align:middle line:84% each entry is size smaller than T. And they're integers. 00:32:05.040 --> 00:32:11.940 align:middle line:84% And we want this to be congruent to 1, 0, 0, 1, mod p. 00:32:11.940 --> 00:32:24.821 align:middle line:90% 00:32:24.821 --> 00:32:28.160 align:middle line:84% So first of all, let's try to do a back of the envelope 00:32:28.160 --> 00:32:29.106 align:middle line:90% calculation. 00:32:29.106 --> 00:32:32.480 align:middle line:90% 00:32:32.480 --> 00:32:36.050 align:middle line:84% So if I erase this congruence condition, 00:32:36.050 --> 00:32:38.870 align:middle line:84% then I'm just asking how many integer points are there in BT. 00:32:38.870 --> 00:32:41.960 align:middle line:84% And we said before, that's about T squared. 00:32:41.960 --> 00:32:45.440 align:middle line:84% Now, I'd like a to be congruent to 1 mod p. 00:32:45.440 --> 00:32:49.490 align:middle line:84% So maybe I should divide by p for that. 00:32:49.490 --> 00:32:51.930 align:middle line:84% And I'd like b to be congruent to 0, 00:32:51.930 --> 00:32:54.090 align:middle line:84% so maybe I should divide by p again. 00:32:54.090 --> 00:32:56.130 align:middle line:90% And I have four conditions. 00:32:56.130 --> 00:33:00.230 align:middle line:84% So maybe I should divide by p four times, 00:33:00.230 --> 00:33:02.050 align:middle line:84% T squared divided by P to the fourth. 00:33:02.050 --> 00:33:05.000 align:middle line:90% 00:33:05.000 --> 00:33:08.110 align:middle line:90% Is anybody suspicious? 00:33:08.110 --> 00:33:10.040 align:middle line:90% OK, so that's the wrong answer. 00:33:10.040 --> 00:33:12.740 align:middle line:84% It should be closer to T squared divided by p cubed. 00:33:12.740 --> 00:33:14.878 align:middle line:84% And the reason that was not really OK to do 00:33:14.878 --> 00:33:16.420 align:middle line:84% is that those four conditions are not 00:33:16.420 --> 00:33:17.980 align:middle line:90% independent of each other. 00:33:17.980 --> 00:33:23.412 align:middle line:84% We already know that ad minus bc is 1 because we're in-- 00:33:23.412 --> 00:33:24.620 align:middle line:90% I didn't write this properly. 00:33:24.620 --> 00:33:30.340 align:middle line:84% This is supposed to be in SL2Z, which means that ad minus bc 00:33:30.340 --> 00:33:32.110 align:middle line:90% equals 1. 00:33:32.110 --> 00:33:33.520 align:middle line:84% And I think that probably implies 00:33:33.520 --> 00:33:35.770 align:middle line:84% that once I have three of these congruence conditions, 00:33:35.770 --> 00:33:38.150 align:middle line:90% then I would get the fourth. 00:33:38.150 --> 00:33:40.210 align:middle line:84% But it also brings up the point that I shouldn't 00:33:40.210 --> 00:33:42.444 align:middle line:90% be too fast and loose. 00:33:42.444 --> 00:33:46.600 align:middle line:84% That it's not super obvious that all of these things 00:33:46.600 --> 00:33:49.420 align:middle line:90% are independent of each other. 00:33:49.420 --> 00:33:52.780 align:middle line:84% So there is actually a kind of algebra trick 00:33:52.780 --> 00:33:56.170 align:middle line:84% to make this work that allows us to keep things 00:33:56.170 --> 00:33:57.770 align:middle line:90% more independent of each other. 00:33:57.770 --> 00:33:59.535 align:middle line:90% And here is the algebra trick. 00:33:59.535 --> 00:34:04.760 align:middle line:90% 00:34:04.760 --> 00:34:09.920 align:middle line:90% OK, so b is divisible by p. 00:34:09.920 --> 00:34:14.090 align:middle line:90% And c is divisible by p. 00:34:14.090 --> 00:34:21.170 align:middle line:84% So p divides b, and p divides C. Therefore, p divides bc. 00:34:21.170 --> 00:34:24.560 align:middle line:90% And bc is ad minus 1-- 00:34:24.560 --> 00:34:27.260 align:middle line:90% sorry, p squared divides bc. 00:34:27.260 --> 00:34:30.300 align:middle line:84% And so p squared divides ad minus 1. 00:34:30.300 --> 00:34:32.929 align:middle line:90% These are equal to each other. 00:34:32.929 --> 00:34:38.690 align:middle line:84% All right, similarly, p plus 1 divides a-- 00:34:38.690 --> 00:34:40.430 align:middle line:90% sorry, p divides a minus 1. 00:34:40.430 --> 00:34:43.010 align:middle line:90% 00:34:43.010 --> 00:34:45.139 align:middle line:90% So a is congruent to 1 mod p. 00:34:45.139 --> 00:34:48.540 align:middle line:90% And p divides d minus 1. 00:34:48.540 --> 00:34:54.980 align:middle line:84% So p squared divides a minus 1 times d minus 1. 00:34:54.980 --> 00:34:58.560 align:middle line:84% All right, the leading term here is also a times d. 00:34:58.560 --> 00:35:01.020 align:middle line:84% So we can play these off against each other. 00:35:01.020 --> 00:35:05.835 align:middle line:84% And we conclude that p squared divides a plus d minus 2. 00:35:05.835 --> 00:35:10.330 align:middle line:90% 00:35:10.330 --> 00:35:14.590 align:middle line:84% All right, now let's start choosing things. 00:35:14.590 --> 00:35:17.290 align:middle line:90% So we're going to choose a. 00:35:17.290 --> 00:35:20.960 align:middle line:84% And a has an integer of size at most T. 00:35:20.960 --> 00:35:23.500 align:middle line:90% And it's congruent to 1 mod p. 00:35:23.500 --> 00:35:27.450 align:middle line:84% So I have less than T over p choices. 00:35:27.450 --> 00:35:30.700 align:middle line:90% 00:35:30.700 --> 00:35:32.925 align:middle line:90% Next, I'm going to choose d. 00:35:32.925 --> 00:35:36.130 align:middle line:90% 00:35:36.130 --> 00:35:38.210 align:middle line:90% I've already chosen a. 00:35:38.210 --> 00:35:42.280 align:middle line:84% So I now know the value of d modulo p squared because 00:35:42.280 --> 00:35:44.740 align:middle line:90% of this sneaky algebra trick. 00:35:44.740 --> 00:35:51.520 align:middle line:84% So I get, at most, T over p squared choices. 00:35:51.520 --> 00:35:54.170 align:middle line:84% This was only really OK if T is at least p squared. 00:35:54.170 --> 00:35:56.840 align:middle line:84% So let me say T is at least p squared. 00:35:56.840 --> 00:36:00.040 align:middle line:90% 00:36:00.040 --> 00:36:03.480 align:middle line:84% And now, once I've chosen a and d, 00:36:03.480 --> 00:36:06.030 align:middle line:90% now I have to solve for b and c. 00:36:06.030 --> 00:36:08.920 align:middle line:84% And I forget about all the congruences, 00:36:08.920 --> 00:36:13.530 align:middle line:84% just like above, they're at most around 1 choices. 00:36:13.530 --> 00:36:23.310 align:middle line:84% And so now, they are less than 1 choice, choose b and c. 00:36:23.310 --> 00:36:27.185 align:middle line:84% That's because bc is ad minus 1 as integers. 00:36:27.185 --> 00:36:30.390 align:middle line:90% 00:36:30.390 --> 00:36:31.230 align:middle line:90% Yeah? 00:36:31.230 --> 00:36:32.730 align:middle line:84% AUDIENCE: When you're saying there's 00:36:32.730 --> 00:36:35.350 align:middle line:84% kind of at most one factorization of ad minus 1, 00:36:35.350 --> 00:36:39.953 align:middle line:84% would that actually be a log of, to be precise, 00:36:39.953 --> 00:36:40.870 align:middle line:90% the number of factors? 00:36:40.870 --> 00:36:42.370 align:middle line:84% LAWRENCE GUTH: Yeah, so the question 00:36:42.370 --> 00:36:45.570 align:middle line:84% is, how many factors are there if a number of some size? 00:36:45.570 --> 00:36:50.110 align:middle line:84% And it is a little super constant. 00:36:50.110 --> 00:36:53.353 align:middle line:84% So if this number has size n, It's actually more than log n. 00:36:53.353 --> 00:36:55.270 align:middle line:84% It could potentially could be worse than that. 00:36:55.270 --> 00:36:58.910 align:middle line:84% But it's less than n to the epsilon for any epsilon. 00:36:58.910 --> 00:37:01.890 align:middle line:84% And that's what this symbol was supposed to be. 00:37:01.890 --> 00:37:03.795 align:middle line:90% 00:37:03.795 --> 00:37:06.170 align:middle line:84% AUDIENCE: A whole lot of symbols being slightly different 00:37:06.170 --> 00:37:07.323 align:middle line:90% versions of [INAUDIBLE]. 00:37:07.323 --> 00:37:08.490 align:middle line:90% LAWRENCE GUTH: Yeah, I know. 00:37:08.490 --> 00:37:15.260 align:middle line:90% 00:37:15.260 --> 00:37:17.470 align:middle line:84% So any other questions or comments about the lemma? 00:37:17.470 --> 00:37:24.806 align:middle line:90% 00:37:24.806 --> 00:37:28.910 align:middle line:84% So those are the ingredients of the Sarnak-Xua proof 00:37:28.910 --> 00:37:29.790 align:middle line:90% of theorem one. 00:37:29.790 --> 00:37:32.010 align:middle line:84% All the ingredients are rigorous and precise, 00:37:32.010 --> 00:37:34.010 align:middle line:84% except for the vague statement about how random 00:37:34.010 --> 00:37:36.200 align:middle line:90% walks behave on SL2Z. 00:37:36.200 --> 00:37:39.210 align:middle line:84% And if we're willing to believe that, 00:37:39.210 --> 00:37:41.515 align:middle line:84% then we can put them together and prove theorem one. 00:37:41.515 --> 00:38:03.970 align:middle line:90% 00:38:03.970 --> 00:38:05.850 align:middle line:90% OK, so proof of theorem one. 00:38:05.850 --> 00:38:09.820 align:middle line:90% 00:38:09.820 --> 00:38:17.230 align:middle line:84% So M will be MA for this set A SL2Z. 00:38:17.230 --> 00:38:21.130 align:middle line:90% 00:38:21.130 --> 00:38:28.960 align:middle line:84% Then by the vague statement, M convolves 00:38:28.960 --> 00:38:34.600 align:middle line:84% k times is roughly equidistributed on BT, 00:38:34.600 --> 00:38:38.690 align:middle line:84% where T is around x of some constant times k. 00:38:38.690 --> 00:38:42.290 align:middle line:90% 00:38:42.290 --> 00:38:46.190 align:middle line:90% So now, we will choose k. 00:38:46.190 --> 00:38:49.630 align:middle line:84% So we'll choose T to be a little more than p squared. 00:38:49.630 --> 00:38:52.060 align:middle line:84% So that we can use the lemma at the end, which 00:38:52.060 --> 00:38:55.120 align:middle line:90% means that k is around log p. 00:38:55.120 --> 00:38:56.980 align:middle line:84% So take around round log p steps, 00:38:56.980 --> 00:39:01.620 align:middle line:84% and after that, M convolves k is roughly evenly distributed 00:39:01.620 --> 00:39:06.390 align:middle line:90% on a ball of size T p squared. 00:39:06.390 --> 00:39:12.000 align:middle line:84% And I guess here we can put 2k, I guess. 00:39:12.000 --> 00:39:19.440 align:middle line:84% So then M star 2k of gamma p will be around BT intersect 00:39:19.440 --> 00:39:25.500 align:middle line:84% gamma p over BT, which is smaller than p to the minus 3 00:39:25.500 --> 00:39:26.640 align:middle line:90% by our lemma. 00:39:26.640 --> 00:39:28.420 align:middle line:84% And here, we're losing a little power. 00:39:28.420 --> 00:39:32.130 align:middle line:84% So if I wanted to be more clear about what I was saying, 00:39:32.130 --> 00:39:35.345 align:middle line:84% that would be something like this. 00:39:35.345 --> 00:39:41.100 align:middle line:90% 00:39:41.100 --> 00:39:46.510 align:middle line:84% Therefore, mu tensor k L2 squared, 00:39:46.510 --> 00:39:51.120 align:middle line:84% that's the same thing as m tensor 2k of gamma p. 00:39:51.120 --> 00:39:55.310 align:middle line:84% And so that would be around p to the minus 3 plus epsilon. 00:39:55.310 --> 00:40:00.230 align:middle line:84% In particular, significantly smaller than p to the minus 2. 00:40:00.230 --> 00:40:03.830 align:middle line:90% And then we use the corollary. 00:40:03.830 --> 00:40:07.940 align:middle line:90% That gives us a bound for this. 00:40:07.940 --> 00:40:12.290 align:middle line:84% And we get that sigma 1 of t mu is less than 1 00:40:12.290 --> 00:40:14.575 align:middle line:84% minus c, where the c is independent of p. 00:40:14.575 --> 00:40:32.150 align:middle line:90% 00:40:32.150 --> 00:40:36.920 align:middle line:84% OK, let me tell you just briefly about the hyperbolic geometry 00:40:36.920 --> 00:40:38.090 align:middle line:90% point of view. 00:40:38.090 --> 00:40:40.896 align:middle line:84% And then we'll switch to other choices of generators. 00:40:40.896 --> 00:41:01.060 align:middle line:90% 00:41:01.060 --> 00:41:13.580 align:middle line:84% OK, hyperbolic geometry, all right, 00:41:13.580 --> 00:41:17.620 align:middle line:84% so we mentioned that SL2Z or SL2R acts 00:41:17.620 --> 00:41:21.280 align:middle line:90% on the hyperbolic plane H2. 00:41:21.280 --> 00:41:28.120 align:middle line:84% Let's say that X of p is H2 modulo gamma p. 00:41:28.120 --> 00:41:33.580 align:middle line:84% So X of 1 is the hyperbolic plane modulo SL2Z. 00:41:33.580 --> 00:41:37.900 align:middle line:90% And X of p is a cover of it. 00:41:37.900 --> 00:41:43.570 align:middle line:90% X of p is a cover of X of 1. 00:41:43.570 --> 00:41:51.360 align:middle line:84% And the group of deck transformations is SL2Fp. 00:41:51.360 --> 00:42:01.830 align:middle line:90% 00:42:01.830 --> 00:42:04.900 align:middle line:84% OK, so I would like to try to make a picture of this. 00:42:04.900 --> 00:42:07.930 align:middle line:84% So X of 1 is not quite a manifold, 00:42:07.930 --> 00:42:09.480 align:middle line:84% but it's not super important for this 00:42:09.480 --> 00:42:11.890 align:middle line:84% except for being a technical irritation. 00:42:11.890 --> 00:42:14.280 align:middle line:84% So I'm not going to try to draw that. 00:42:14.280 --> 00:42:20.770 align:middle line:84% So it is almost a non-compact manifold of genus 1. 00:42:20.770 --> 00:42:22.000 align:middle line:90% It looks sort of like this. 00:42:22.000 --> 00:42:26.700 align:middle line:84% The non-compactness is annoying, but it's maybe important. 00:42:26.700 --> 00:42:29.940 align:middle line:84% So then X of p is a cover of this. 00:42:29.940 --> 00:42:37.410 align:middle line:84% So it's going to look sort of like so. 00:42:37.410 --> 00:42:42.245 align:middle line:84% And I guess it has a cusp for each. 00:42:42.245 --> 00:42:45.870 align:middle line:90% 00:42:45.870 --> 00:42:48.970 align:middle line:84% It also has some cusps, not sure exactly how many. 00:42:48.970 --> 00:42:51.020 align:middle line:90% This is X of p. 00:42:51.020 --> 00:42:53.210 align:middle line:90% And this is a cover. 00:42:53.210 --> 00:42:57.380 align:middle line:90% So how to make that look nice? 00:42:57.380 --> 00:43:04.430 align:middle line:84% Well, you could kind of chop this into fundamental domains. 00:43:04.430 --> 00:43:08.390 align:middle line:84% And then each fundamental domain is a bijection onto X of 1. 00:43:08.390 --> 00:43:11.173 align:middle line:84% And if you do this nicely, so then 00:43:11.173 --> 00:43:12.590 align:middle line:84% the different fundamental domains, 00:43:12.590 --> 00:43:15.048 align:middle line:84% they correspond to the group of deck transformations, which 00:43:15.048 --> 00:43:17.120 align:middle line:90% is SL2Fp. 00:43:17.120 --> 00:43:19.940 align:middle line:84% And if you do this nicely, you could put a dot 00:43:19.940 --> 00:43:21.600 align:middle line:90% in each fundamental domain. 00:43:21.600 --> 00:43:24.560 align:middle line:84% And you could put an edge if the fundamental domains 00:43:24.560 --> 00:43:26.210 align:middle line:90% touch each other. 00:43:26.210 --> 00:43:30.620 align:middle line:84% And the graph that I just drew would 00:43:30.620 --> 00:43:37.010 align:middle line:90% be a Cayley graph of SL2Fp. 00:43:37.010 --> 00:43:39.710 align:middle line:90% 00:43:39.710 --> 00:43:44.600 align:middle line:84% So the geometry of these two-dimensional surfaces 00:43:44.600 --> 00:43:47.090 align:middle line:84% is closely related to the geometry of SL2Fp. 00:43:47.090 --> 00:43:53.172 align:middle line:90% 00:43:53.172 --> 00:44:02.130 align:middle line:84% So then, we talk about the spectrum of the Laplacian On 00:44:02.130 --> 00:44:06.210 align:middle line:90% xp, our subject of study. 00:44:06.210 --> 00:44:10.590 align:middle line:84% And 0 is in the spectrum, so the Laplacian of a constant function 00:44:10.590 --> 00:44:11.135 align:middle line:90% is 0. 00:44:11.135 --> 00:44:16.560 align:middle line:90% 00:44:16.560 --> 00:44:18.610 align:middle line:90% But after 0, there's a gap. 00:44:18.610 --> 00:44:21.900 align:middle line:84% So if this is the real line, and here's 0, 00:44:21.900 --> 00:44:23.430 align:middle line:90% that's part of the spectrum. 00:44:23.430 --> 00:44:25.230 align:middle line:90% Then there's a gap. 00:44:25.230 --> 00:44:28.270 align:middle line:84% And this is called lambda 1 of X of p. 00:44:28.270 --> 00:44:34.960 align:middle line:84% And the rest of the spectrum lives here, is contained in. 00:44:34.960 --> 00:44:42.600 align:middle line:90% 00:44:42.600 --> 00:44:46.310 align:middle line:84% So that's what lambda 1 is, the smallest non-zero part 00:44:46.310 --> 00:44:47.900 align:middle line:90% of the spectrum. 00:44:47.900 --> 00:44:51.170 align:middle line:84% All right, so the theorem that Selberg actually proved 00:44:51.170 --> 00:45:00.110 align:middle line:84% in the 1950s is that lambda 1 of X of p is at least 3/16 for all 00:45:00.110 --> 00:45:02.990 align:middle line:90% p. 00:45:02.990 --> 00:45:08.840 align:middle line:84% And the conjecture is that lambda 1 of X of p 00:45:08.840 --> 00:45:12.850 align:middle line:84% is at least a quarter, maybe minus little o of 1, 00:45:12.850 --> 00:45:14.600 align:middle line:84% I'm not sure if you need it, but certainly 00:45:14.600 --> 00:45:17.970 align:middle line:84% if somebody proved that, that would be a big deal, for all p. 00:45:17.970 --> 00:45:22.550 align:middle line:90% 00:45:22.550 --> 00:45:25.075 align:middle line:84% This may not look immediately that it matters very much. 00:45:25.075 --> 00:45:26.450 align:middle line:84% A quarter is a significant number 00:45:26.450 --> 00:45:28.710 align:middle line:84% because it's lambda 1 of hyperbolic plane. 00:45:28.710 --> 00:45:31.820 align:middle line:90% So I would say, it matches that. 00:45:31.820 --> 00:45:34.430 align:middle line:84% Anyway, so this is the theorem that Selberg actually proved. 00:45:34.430 --> 00:45:36.380 align:middle line:84% With modern techniques, it's not that 00:45:36.380 --> 00:45:38.527 align:middle line:84% difficult to get from this theorem to that theorem. 00:45:38.527 --> 00:45:40.735 align:middle line:84% It's much more difficult to prove either one of them. 00:45:40.735 --> 00:45:45.970 align:middle line:90% 00:45:45.970 --> 00:45:49.370 align:middle line:84% OK instead of the random walk, we could use here, 00:45:49.370 --> 00:45:53.470 align:middle line:90% the heat flow on X of p. 00:45:53.470 --> 00:45:57.500 align:middle line:84% And there was a question earlier, what is the heat flow? 00:45:57.500 --> 00:46:00.763 align:middle line:84% So the question earlier was what is the heat flow on SL2R? 00:46:00.763 --> 00:46:02.180 align:middle line:84% But the analysis question here is, 00:46:02.180 --> 00:46:05.050 align:middle line:84% what is the heat flow on this hyperbolic manifold? 00:46:05.050 --> 00:46:08.040 align:middle line:84% Well, you can think of it as solving the heat equation. 00:46:08.040 --> 00:46:12.160 align:middle line:90% 00:46:12.160 --> 00:46:14.740 align:middle line:84% So the heat equation would be del t 00:46:14.740 --> 00:46:18.770 align:middle line:90% of u equals Laplacian of u. 00:46:18.770 --> 00:46:21.470 align:middle line:84% And the Laplacian of k would take a while to write down, 00:46:21.470 --> 00:46:24.310 align:middle line:84% but it's a second order differential operator 00:46:24.310 --> 00:46:26.830 align:middle line:90% on your hyperbolic manifold. 00:46:26.830 --> 00:46:34.120 align:middle line:84% All right, OK, and the solution has a formula 00:46:34.120 --> 00:46:37.570 align:middle line:90% using a heat flow operator. 00:46:37.570 --> 00:46:44.740 align:middle line:84% So let's say u of dot comma t would be Ht of u. 00:46:44.740 --> 00:46:46.830 align:middle line:84% So this is now an operator Ht, which 00:46:46.830 --> 00:46:50.375 align:middle line:84% plays the role of our operator T mu or T mu to the k. 00:46:50.375 --> 00:46:59.540 align:middle line:90% 00:46:59.540 --> 00:47:03.780 align:middle line:84% So everything we talked about with a little bit more analysis 00:47:03.780 --> 00:47:06.480 align:middle line:84% in PDE to set things up, everything we talked about 00:47:06.480 --> 00:47:07.840 align:middle line:90% applies here. 00:47:07.840 --> 00:47:16.830 align:middle line:84% So for instance, SL2 of Fp acts isometrically on X of p. 00:47:16.830 --> 00:47:23.250 align:middle line:84% And therefore, it acts on the eigenspaces of Ht. 00:47:23.250 --> 00:47:26.400 align:middle line:90% 00:47:26.400 --> 00:47:30.150 align:middle line:90% They have multiplicity. 00:47:30.150 --> 00:47:34.830 align:middle line:90% And also, you can do lifting. 00:47:34.830 --> 00:47:43.620 align:middle line:84% So Ht tilde is the heat operator on the universal cover, 00:47:43.620 --> 00:47:45.140 align:middle line:90% the hyperbolic plane. 00:47:45.140 --> 00:47:52.550 align:middle line:84% And then Ht of x1, x2 is the sum over g 00:47:52.550 --> 00:48:00.050 align:middle line:84% in gamma p of Ht tilde x1 tilde gx2 tilde. 00:48:00.050 --> 00:48:02.270 align:middle line:84% So, in other words, the new heat operator 00:48:02.270 --> 00:48:04.700 align:middle line:84% is-- take the heat operator on the hyperbolic plane 00:48:04.700 --> 00:48:07.010 align:middle line:90% and push it forward. 00:48:07.010 --> 00:48:10.070 align:middle line:84% And that follows basically because this 00:48:10.070 --> 00:48:11.220 align:middle line:90% is a local equation. 00:48:11.220 --> 00:48:13.460 align:middle line:84% So solution to the heat equation, you pull back, 00:48:13.460 --> 00:48:15.127 align:middle line:84% you get a solution to the heat equation. 00:48:15.127 --> 00:48:21.290 align:middle line:90% 00:48:21.290 --> 00:48:25.410 align:middle line:84% And Ht tilde, the heat kernel on the hyperbolic plane, 00:48:25.410 --> 00:48:26.815 align:middle line:90% it has an explicit formula. 00:48:26.815 --> 00:48:31.700 align:middle line:90% 00:48:31.700 --> 00:48:34.460 align:middle line:84% And that's the equivalent that plays the role 00:48:34.460 --> 00:48:35.940 align:middle line:90% of the vague statement. 00:48:35.940 --> 00:48:39.338 align:middle line:90% 00:48:39.338 --> 00:48:42.700 align:middle line:84% So this hopefully is a dictionary 00:48:42.700 --> 00:48:44.650 align:middle line:84% of most of the elements of our proof. 00:48:44.650 --> 00:48:49.150 align:middle line:84% They all have natural analogs for this hyperbolic surface. 00:48:49.150 --> 00:48:51.520 align:middle line:84% And the one that was the most troubling 00:48:51.520 --> 00:48:55.420 align:middle line:84% has a less troubling analog, or the one that 00:48:55.420 --> 00:48:59.320 align:middle line:84% was not stated precisely would have a precisely stated analog. 00:48:59.320 --> 00:49:02.140 align:middle line:84% So without doing all the details, if you then 00:49:02.140 --> 00:49:04.690 align:middle line:84% translate the proof that we did into the setting 00:49:04.690 --> 00:49:07.600 align:middle line:84% of hyperbolic surfaces, then that 00:49:07.600 --> 00:49:10.390 align:middle line:84% would give a proof of a theorem like this one 00:49:10.390 --> 00:49:12.200 align:middle line:84% with a constant that's not quite as good. 00:49:12.200 --> 00:49:14.020 align:middle line:90% But usually, it doesn't matter. 00:49:14.020 --> 00:49:14.920 align:middle line:90% Yeah? 00:49:14.920 --> 00:49:16.960 align:middle line:84% AUDIENCE: I have a question about the heat flow. 00:49:16.960 --> 00:49:20.150 align:middle line:84% Since the spaces X of p are not compact, 00:49:20.150 --> 00:49:22.390 align:middle line:84% does there have to be some of decay condition 00:49:22.390 --> 00:49:25.937 align:middle line:84% to make the heat flow well-defined in these spaces? 00:49:25.937 --> 00:49:26.770 align:middle line:90% LAWRENCE GUTH: Yeah. 00:49:26.770 --> 00:49:30.170 align:middle line:84% So the question is, since X of p is not compact, 00:49:30.170 --> 00:49:32.290 align:middle line:84% should we have some kind of decay condition 00:49:32.290 --> 00:49:34.220 align:middle line:90% to make the heat flow work? 00:49:34.220 --> 00:49:36.230 align:middle line:90% Yeah, I think the answer is yes. 00:49:36.230 --> 00:49:38.410 align:middle line:84% So these spaces have finite area. 00:49:38.410 --> 00:49:41.150 align:middle line:84% So it's not as bad as it would be in some other settings. 00:49:41.150 --> 00:49:43.930 align:middle line:84% But I think we should say something about the solution 00:49:43.930 --> 00:49:46.310 align:middle line:90% not going crazy near infinity. 00:49:46.310 --> 00:49:52.210 align:middle line:90% 00:49:52.210 --> 00:49:57.370 align:middle line:84% Yeah, so this is the end of our discussion 00:49:57.370 --> 00:50:01.310 align:middle line:84% of the Selberg theorem for this particular set of generators. 00:50:01.310 --> 00:50:02.920 align:middle line:84% So take a moment to see if you have 00:50:02.920 --> 00:50:04.720 align:middle line:90% questions or comments about it. 00:50:04.720 --> 00:50:06.400 align:middle line:84% And then we'll explore what would 00:50:06.400 --> 00:50:09.276 align:middle line:84% happen if we didn't have that particular set of generators. 00:50:09.276 --> 00:50:25.540 align:middle line:90% 00:50:25.540 --> 00:50:27.970 align:middle line:84% Yeah, anything on people's minds? 00:50:27.970 --> 00:50:28.570 align:middle line:90% Yeah? 00:50:28.570 --> 00:50:30.112 align:middle line:84% AUDIENCE: Can you give some intuition 00:50:30.112 --> 00:50:34.150 align:middle line:84% why the support grows exponentially? 00:50:34.150 --> 00:50:37.510 align:middle line:84% LAWRENCE GUTH: Yeah, so the question was, if we go back to-- 00:50:37.510 --> 00:50:41.120 align:middle line:90% 00:50:41.120 --> 00:50:45.300 align:middle line:84% we were thinking about, we had a probability measure on SL2Z, 00:50:45.300 --> 00:50:47.910 align:middle line:84% we convolve it with itself a lot of times, 00:50:47.910 --> 00:50:52.040 align:middle line:84% and then we discovered that the support is roughly 00:50:52.040 --> 00:50:54.180 align:middle line:84% evenly distributed on a ball of size T, 00:50:54.180 --> 00:50:58.520 align:middle line:84% But T grows exponentially with k. 00:50:58.520 --> 00:51:05.150 align:middle line:84% OK, yeah, so this is a kind of different behavior from what 00:51:05.150 --> 00:51:07.130 align:middle line:90% happens on the real line. 00:51:07.130 --> 00:51:09.000 align:middle line:84% And let's think about what's going on. 00:51:09.000 --> 00:51:11.540 align:middle line:84% So M, let's say, is just evenly distributed 00:51:11.540 --> 00:51:14.000 align:middle line:84% between a few matrices, maybe the matrices 00:51:14.000 --> 00:51:15.410 align:middle line:90% in the Selberg thing. 00:51:15.410 --> 00:51:18.000 align:middle line:84% So now, I'm going to take k of those matrices, 00:51:18.000 --> 00:51:20.090 align:middle line:84% and I'm going to multiply them together. 00:51:20.090 --> 00:51:23.060 align:middle line:84% Now, if I multiply k matrices together, 00:51:23.060 --> 00:51:25.550 align:middle line:84% the biggest that the entries could be at the end 00:51:25.550 --> 00:51:28.400 align:middle line:90% is exponential in k. 00:51:28.400 --> 00:51:32.240 align:middle line:84% On the other hand, you might wonder, is it really that big? 00:51:32.240 --> 00:51:35.480 align:middle line:90% Or is there some cancellation? 00:51:35.480 --> 00:51:39.700 align:middle line:84% Suppose I had 1 by 1 matrices, and I had a measure, 00:51:39.700 --> 00:51:43.630 align:middle line:84% at 50% I was 1/2, and 50%, it was 2. 00:51:43.630 --> 00:51:46.100 align:middle line:84% Take k of those, and I multiply them together. 00:51:46.100 --> 00:51:47.500 align:middle line:90% What would happen? 00:51:47.500 --> 00:51:49.310 align:middle line:84% Well, there would a lot of cancellation. 00:51:49.310 --> 00:51:52.660 align:middle line:84% There are a lot of cancellations between the 1/2s and the 2s 00:51:52.660 --> 00:51:54.970 align:middle line:84% It would get clearer to us if we took 00:51:54.970 --> 00:51:57.220 align:middle line:90% the logarithm of everything. 00:51:57.220 --> 00:51:59.470 align:middle line:84% And the size of that would actually 00:51:59.470 --> 00:52:03.610 align:middle line:84% end up being exponential of the square root of k. 00:52:03.610 --> 00:52:07.540 align:middle line:84% So the 2 by 2 case is different from that. 00:52:07.540 --> 00:52:11.230 align:middle line:84% The actual size of this is typically exponential in k. 00:52:11.230 --> 00:52:14.200 align:middle line:84% There's not as much cancellation of that kind. 00:52:14.200 --> 00:52:20.582 align:middle line:84% And let's see, how could we get some intuition about that? 00:52:20.582 --> 00:52:25.141 align:middle line:84% AUDIENCE: Is it in some sense because of the [INAUDIBLE] 00:52:25.141 --> 00:52:31.368 align:middle line:90% 00:52:31.368 --> 00:52:33.160 align:middle line:84% LAWRENCE GUTH: Yeah, that's a good comment. 00:52:33.160 --> 00:52:37.210 align:middle line:84% So the comment was SL2Z is virtually free. 00:52:37.210 --> 00:52:42.790 align:middle line:84% So it's not very likely to go out in SL2Z and then come back. 00:52:42.790 --> 00:52:46.320 align:middle line:90% And that's a related fact. 00:52:46.320 --> 00:52:50.230 align:middle line:84% Yeah, so it's also related to hyperbolic geometry. 00:52:50.230 --> 00:52:54.630 align:middle line:84% So you do a random walk in the hyperbolic plane, if you take 00:52:54.630 --> 00:52:57.420 align:middle line:84% s steps, you'll typically be around a distance 00:52:57.420 --> 00:52:59.100 align:middle line:90% s from the origin. 00:52:59.100 --> 00:53:01.230 align:middle line:84% And that's because the hyperbolic plane 00:53:01.230 --> 00:53:03.370 align:middle line:90% is branching out. 00:53:03.370 --> 00:53:06.780 align:middle line:84% And so at every moment, you're not equally likely to be going 00:53:06.780 --> 00:53:10.470 align:middle line:84% back towards the origin or away, you're kind of 3/4 00:53:10.470 --> 00:53:11.920 align:middle line:90% of the directions are away. 00:53:11.920 --> 00:53:15.420 align:middle line:84% And that's what life is like inside of the matrices. 00:53:15.420 --> 00:53:17.730 align:middle line:90% Yeah, great question. 00:53:17.730 --> 00:53:19.370 align:middle line:90% Anything else on people's minds? 00:53:19.370 --> 00:53:23.640 align:middle line:90% 00:53:23.640 --> 00:53:36.790 align:middle line:84% OK, all right, so let's look back at this Selberg theorem, 00:53:36.790 --> 00:53:39.040 align:middle line:84% now that we understand it better. 00:53:39.040 --> 00:53:43.900 align:middle line:84% And we had this particular A listed here. 00:53:43.900 --> 00:53:45.920 align:middle line:90% What did we really use about A? 00:53:45.920 --> 00:53:47.295 align:middle line:90% What was special about A? 00:53:47.295 --> 00:53:50.860 align:middle line:90% 00:53:50.860 --> 00:53:51.970 align:middle line:90% Yeah? 00:53:51.970 --> 00:53:56.170 align:middle line:84% AUDIENCE: We used the fact that it was a generator of SL2Fp. 00:53:56.170 --> 00:53:57.700 align:middle line:84% LAWRENCE GUTH: Yeah, so we used that 00:53:57.700 --> 00:54:00.580 align:middle line:90% as a generator of SL2 of Fp. 00:54:00.580 --> 00:54:02.050 align:middle line:84% And what I'd like us all to think 00:54:02.050 --> 00:54:06.760 align:middle line:84% about is if we had any set of generators of SL2Fp, 00:54:06.760 --> 00:54:08.780 align:middle line:84% do we think that this would still be true? 00:54:08.780 --> 00:54:11.025 align:middle line:84% And do we think that this proof would still work? 00:54:11.025 --> 00:54:13.728 align:middle line:90% 00:54:13.728 --> 00:54:15.270 align:middle line:84% I'll write that down while you think. 00:54:15.270 --> 00:54:22.320 align:middle line:90% 00:54:22.320 --> 00:54:22.820 align:middle line:90% it. 00:54:22.820 --> 00:54:26.710 align:middle line:90% 00:54:26.710 --> 00:54:41.970 align:middle line:84% So question, can we replace A by any set of generators of SL2Fp? 00:54:41.970 --> 00:54:46.540 align:middle line:90% 00:54:46.540 --> 00:54:47.040 align:middle line:90% Yeah? 00:54:47.040 --> 00:54:50.940 align:middle line:84% AUDIENCE: So at minimum, they need to be symmetric, right? 00:54:50.940 --> 00:54:54.450 align:middle line:84% LAWRENCE GUTH: Yeah, let's say they're symmetric. 00:54:54.450 --> 00:54:56.038 align:middle line:84% Let's focus on the symmetric case. 00:54:56.038 --> 00:54:57.580 align:middle line:84% That makes things technically easier. 00:54:57.580 --> 00:55:01.390 align:middle line:84% And I would be happy with any symmetric set of generators. 00:55:01.390 --> 00:55:01.980 align:middle line:90% Yeah? 00:55:01.980 --> 00:55:03.730 align:middle line:84% AUDIENCE: Well, does the set of generators 00:55:03.730 --> 00:55:07.505 align:middle line:84% still need to be kind of at least close to virtually free? 00:55:07.505 --> 00:55:08.880 align:middle line:84% LAWRENCE GUTH: Does it still need 00:55:08.880 --> 00:55:12.450 align:middle line:90% to be close to virtually free? 00:55:12.450 --> 00:55:14.670 align:middle line:90% Yeah, that's a good question. 00:55:14.670 --> 00:55:17.170 align:middle line:84% So although I mentioned being virtually free, 00:55:17.170 --> 00:55:21.900 align:middle line:84% we didn't actually use virtually free in the proof. 00:55:21.900 --> 00:55:23.446 align:middle line:90% That was for intuition. 00:55:23.446 --> 00:55:26.580 align:middle line:90% 00:55:26.580 --> 00:55:29.690 align:middle line:84% But we used something in the proof. 00:55:29.690 --> 00:55:31.060 align:middle line:90% Yeah, well, it's-- 00:55:31.060 --> 00:55:33.560 align:middle line:84% AUDIENCE: They're conjugate to their own powers or something 00:55:33.560 --> 00:55:34.925 align:middle line:90% like that, right? 00:55:34.925 --> 00:55:36.300 align:middle line:84% LAWRENCE GUTH: They're conjugate. 00:55:36.300 --> 00:55:38.275 align:middle line:84% So a nifty feature of these guys, 00:55:38.275 --> 00:55:39.650 align:middle line:84% so the comment was these guys are 00:55:39.650 --> 00:55:41.780 align:middle line:90% conjugate to their own powers. 00:55:41.780 --> 00:55:43.530 align:middle line:90% Yeah, so that's important. 00:55:43.530 --> 00:55:45.540 align:middle line:90% But we didn't use that either. 00:55:45.540 --> 00:55:52.110 align:middle line:84% So when we proved that every representation of SL2Fp is big, 00:55:52.110 --> 00:55:54.150 align:middle line:90% we considered these elements. 00:55:54.150 --> 00:55:55.880 align:middle line:84% And we used the fact that they're 00:55:55.880 --> 00:55:57.360 align:middle line:90% conjugate to their own powers. 00:55:57.360 --> 00:56:00.800 align:middle line:84% But once we knew abstractly that every representation of SL2Fp 00:56:00.800 --> 00:56:04.010 align:middle line:84% is big, we just used it as a black box. 00:56:04.010 --> 00:56:06.548 align:middle line:84% So it's not important that these elements 00:56:06.548 --> 00:56:08.090 align:middle line:84% in the statement of Selberg's theorem 00:56:08.090 --> 00:56:09.923 align:middle line:84% are the same elements we used in that proof. 00:56:09.923 --> 00:56:13.490 align:middle line:90% 00:56:13.490 --> 00:56:15.340 align:middle line:90% Yeah, good. 00:56:15.340 --> 00:56:18.890 align:middle line:84% But well, so one part of the answer 00:56:18.890 --> 00:56:29.020 align:middle line:84% is that we used that A is a set of generators of SL2Z. 00:56:29.020 --> 00:56:36.890 align:middle line:90% 00:56:36.890 --> 00:56:40.310 align:middle line:84% So in the proof where we lifted things to SL2Z, 00:56:40.310 --> 00:56:44.260 align:middle line:84% we took this set A, we lifted it to SL2Z, 00:56:44.260 --> 00:56:47.000 align:middle line:84% and then we did a random walk on SL2Z. 00:56:47.000 --> 00:56:50.120 align:middle line:84% And we needed to know that thing was kind of evenly distributed. 00:56:50.120 --> 00:56:52.660 align:middle line:84% And the only way we could hope for that to be true 00:56:52.660 --> 00:56:56.980 align:middle line:84% is if this was a set of generators of SL2Z. 00:56:56.980 --> 00:56:58.568 align:middle line:90% Yeah? 00:56:58.568 --> 00:57:00.610 align:middle line:84% AUDIENCE: Are there any even straightforward sets 00:57:00.610 --> 00:57:05.230 align:middle line:84% of generators of SL2Fp that would not be generators of SL2Z? 00:57:05.230 --> 00:57:06.670 align:middle line:84% LAWRENCE GUTH: OK, so the question 00:57:06.670 --> 00:57:11.650 align:middle line:84% is, could it happen that we have generators of SL2Fp, 00:57:11.650 --> 00:57:13.750 align:middle line:90% but they don't generate SL2Z? 00:57:13.750 --> 00:57:15.620 align:middle line:90% And the answer is yes. 00:57:15.620 --> 00:57:17.860 align:middle line:90% That happens fairly commonly. 00:57:17.860 --> 00:57:21.435 align:middle line:90% And let me show you an example. 00:57:21.435 --> 00:57:33.940 align:middle line:90% 00:57:33.940 --> 00:57:36.420 align:middle line:90% So here's an example. 00:57:36.420 --> 00:57:40.650 align:middle line:84% Maybe we won't prove everything, but let's say that Ak-- 00:57:40.650 --> 00:57:42.250 align:middle line:84% no, I should use a different letter. 00:57:42.250 --> 00:57:52.132 align:middle line:84% Am is the set of 1 plus or minus m, 1, 0, and 1 plus or minus m, 00:57:52.132 --> 00:57:54.900 align:middle line:90% 1, 0. 00:57:54.900 --> 00:57:57.690 align:middle line:84% So this is still a set of four elements. 00:57:57.690 --> 00:58:00.180 align:middle line:90% Size of A equals 4. 00:58:00.180 --> 00:58:06.420 align:middle line:84% And if m is 1 or 2, it generates SL2Z. 00:58:06.420 --> 00:58:09.810 align:middle line:84% But if m is at least 3, it does not 00:58:09.810 --> 00:58:16.500 align:middle line:84% generate-- does not generate SL2Z. 00:58:16.500 --> 00:58:22.210 align:middle line:84% On the other hand, for every m, for all almost every p, 00:58:22.210 --> 00:58:37.100 align:middle line:84% so for all but finitely many p, this Am generates SL2Fp. 00:58:37.100 --> 00:58:43.880 align:middle line:90% 00:58:43.880 --> 00:58:45.700 align:middle line:84% All right, so why does this happen? 00:58:45.700 --> 00:58:54.260 align:middle line:90% 00:58:54.260 --> 00:58:59.426 align:middle line:84% All right, so notice that this guy is 1, 0, 1, 00:58:59.426 --> 00:59:02.800 align:middle line:90% 1 to the plus or minus m right. 00:59:02.800 --> 00:59:07.280 align:middle line:90% 00:59:07.280 --> 00:59:10.080 align:middle line:84% So we assume that these guys generate SL2Z. 00:59:10.080 --> 00:59:12.050 align:middle line:90% It's not that hard to show. 00:59:12.050 --> 00:59:15.860 align:middle line:84% Take a word in these guys, it's only 00:59:15.860 --> 00:59:19.910 align:middle line:84% included in the group generated by these guys 00:59:19.910 --> 00:59:26.860 align:middle line:84% if it has a rather special structure that each letter is 00:59:26.860 --> 00:59:31.000 align:middle line:84% raised to the plus or minus n-th power, so maybe m is 10. 00:59:31.000 --> 00:59:36.790 align:middle line:84% So now it's clear that most words do not look like that. 00:59:36.790 --> 00:59:38.590 align:middle line:84% Now, it's not immediately clear that we 00:59:38.590 --> 00:59:41.200 align:middle line:84% don't have the whole group, because an element in the group 00:59:41.200 --> 00:59:44.290 align:middle line:84% might be represented by many words, 00:59:44.290 --> 00:59:47.800 align:middle line:84% but we mentioned before that there has a finite index 00:59:47.800 --> 00:59:49.250 align:middle line:90% subgroup which is free. 00:59:49.250 --> 00:59:53.200 align:middle line:84% So there are not that many ways to represent most words. 00:59:53.200 --> 00:59:57.830 align:middle line:84% So this tiny subset of the words, 00:59:57.830 --> 01:00:00.060 align:middle line:84% we don't expect it to cover very much of the group. 01:00:00.060 --> 01:00:04.684 align:middle line:90% 01:00:04.684 --> 01:00:07.480 align:middle line:90% And life is different in SL2Fp. 01:00:07.480 --> 01:00:09.250 align:middle line:84% And a little later, we'll talk about who 01:00:09.250 --> 01:00:11.230 align:middle line:90% are the subgroups of SL2Fp. 01:00:11.230 --> 01:00:13.430 align:middle line:84% But there aren't a whole lot of them. 01:00:13.430 --> 01:00:15.640 align:middle line:84% And once you have enough elements that they're not 01:00:15.640 --> 01:00:18.270 align:middle line:84% trapped in a subgroup, then they must generate the whole group. 01:00:18.270 --> 01:00:23.130 align:middle line:90% 01:00:23.130 --> 01:00:29.400 align:middle line:84% OK, all right, so there are other choices 01:00:29.400 --> 01:00:30.862 align:middle line:90% of generators for SL2Z. 01:00:30.862 --> 01:00:32.820 align:middle line:84% Any choice of generators would have worked fine 01:00:32.820 --> 01:00:36.120 align:middle line:84% in our argument, but it's actually quite common 01:00:36.120 --> 01:00:39.480 align:middle line:84% that you have some generators for SL2Fp, 01:00:39.480 --> 01:00:40.815 align:middle line:90% and they don't generate SL2Z. 01:00:40.815 --> 01:00:45.270 align:middle line:90% 01:00:45.270 --> 01:00:51.000 align:middle line:84% In this case, our argument doesn't work very well. 01:00:51.000 --> 01:00:58.500 align:middle line:84% Let's say that Gm is the group generated by Am. 01:00:58.500 --> 01:01:04.830 align:middle line:84% This group is a lot harder to understand than SL2Z. 01:01:04.830 --> 01:01:08.610 align:middle line:84% So for instance, if you were to write down a matrix 01:01:08.610 --> 01:01:10.800 align:middle line:84% and then you'd like to know is it 01:01:10.800 --> 01:01:15.900 align:middle line:84% in Gm, that's not super easy to figure out. 01:01:15.900 --> 01:01:18.980 align:middle line:84% SL2Z are all the integer matrices 01:01:18.980 --> 01:01:24.020 align:middle line:84% where the entries satisfy the equation ad minus bc equals 1. 01:01:24.020 --> 01:01:26.030 align:middle line:84% Whether or not a matrix is in this group 01:01:26.030 --> 01:01:28.580 align:middle line:84% does not have a simple criterion like that. 01:01:28.580 --> 01:01:30.780 align:middle line:84% As far as I know, the best thing-- well, 01:01:30.780 --> 01:01:33.560 align:middle line:84% I don't know anything faster to do than to take the generators 01:01:33.560 --> 01:01:35.393 align:middle line:84% and multiply them together in different ways 01:01:35.393 --> 01:01:37.080 align:middle line:90% and see if you get this matrix. 01:01:37.080 --> 01:01:37.640 align:middle line:90% Yeah? 01:01:37.640 --> 01:01:41.090 align:middle line:84% AUDIENCE: How close is Gm to gamma m? 01:01:41.090 --> 01:01:43.650 align:middle line:90% Do they have anything in common? 01:01:43.650 --> 01:01:45.590 align:middle line:84% Because at first, it sort of look 01:01:45.590 --> 01:01:50.480 align:middle line:84% like things congruent to the identity mod m. 01:01:50.480 --> 01:01:53.400 align:middle line:84% LAWRENCE GUTH: Right, so Gm is a subgroup. 01:01:53.400 --> 01:01:56.610 align:middle line:84% So OK, so the question was how close is Gm to gamma m? 01:01:56.610 --> 01:02:01.970 align:middle line:84% Remember that gamma m was the subgroup of matrices in SL2Z 01:02:01.970 --> 01:02:05.030 align:middle line:84% that are congruent to the identity mod m. 01:02:05.030 --> 01:02:07.980 align:middle line:84% So these guys are congruent to the identity mod m. 01:02:07.980 --> 01:02:11.390 align:middle line:84% So therefore, Gm is a subgroup of gamma m. 01:02:11.390 --> 01:02:25.780 align:middle line:84% However, this group has infinite index In SL2Z for m at least 3. 01:02:25.780 --> 01:02:28.090 align:middle line:84% Whereas, gamma m had finite index 01:02:28.090 --> 01:02:33.070 align:middle line:84% because it was the kernel of a map to a finite group. 01:02:33.070 --> 01:02:37.570 align:middle line:84% So gamma m, which is already not a super easy thing 01:02:37.570 --> 01:02:44.450 align:middle line:84% to understand if m is large, this is well-- 01:02:44.450 --> 01:02:47.030 align:middle line:84% objectively, this is a smaller group than gamma m, 01:02:47.030 --> 01:02:49.450 align:middle line:84% and subjectively, I think it is a harder group 01:02:49.450 --> 01:02:50.775 align:middle line:90% to understand than gamma m. 01:02:50.775 --> 01:02:55.190 align:middle line:90% 01:02:55.190 --> 01:02:58.400 align:middle line:84% So this whole method has not really worked. 01:02:58.400 --> 01:03:01.510 align:middle line:84% And in fact, it goes in the opposite direction. 01:03:01.510 --> 01:03:06.550 align:middle line:84% So namely, for generators of SL2Z, 01:03:06.550 --> 01:03:08.810 align:middle line:84% we first understand SL2Z really well, 01:03:08.810 --> 01:03:10.960 align:middle line:84% and then we can use that to understand SL2Fp 01:03:10.960 --> 01:03:13.030 align:middle line:90% with those generators. 01:03:13.030 --> 01:03:17.320 align:middle line:84% And with a situation like this, there's some progress. 01:03:17.320 --> 01:03:20.160 align:middle line:84% And it goes, first, you try to understand SL2Fp 01:03:20.160 --> 01:03:21.900 align:middle line:84% with those generators, and you use 01:03:21.900 --> 01:03:24.065 align:middle line:84% that to try to understand G of m. 01:03:24.065 --> 01:03:25.440 align:middle line:84% There's actually a big literature 01:03:25.440 --> 01:03:26.830 align:middle line:90% now about these groups G of m. 01:03:26.830 --> 01:03:29.130 align:middle line:84% And first, you try to figure out things about SL2Fp 01:03:29.130 --> 01:03:30.330 align:middle line:90% with those generators. 01:03:30.330 --> 01:03:35.710 align:middle line:90% 01:03:35.710 --> 01:03:38.200 align:middle line:84% So that's one part of the answer to this question. 01:03:38.200 --> 01:03:41.130 align:middle line:84% And what we said so far is that our proof doesn't usually 01:03:41.130 --> 01:03:43.330 align:middle line:84% work for other sets of generators, 01:03:43.330 --> 01:03:45.540 align:middle line:84% but that's not an answer to the question 01:03:45.540 --> 01:03:47.730 align:middle line:90% whether the theorem is true. 01:03:47.730 --> 01:03:50.465 align:middle line:84% And as far as we know, the theorem may well be true. 01:03:50.465 --> 01:03:51.840 align:middle line:84% So there's a conjecture you could 01:03:51.840 --> 01:03:54.520 align:middle line:84% take any set of generators of SL2Fp, 01:03:54.520 --> 01:03:56.105 align:middle line:84% you just always have a spectral gamma. 01:03:56.105 --> 01:03:58.800 align:middle line:90% 01:03:58.800 --> 01:04:00.240 align:middle line:84% That's not proven yet, but there's 01:04:00.240 --> 01:04:01.510 align:middle line:90% some significant progress. 01:04:01.510 --> 01:04:04.050 align:middle line:84% And I will tell you something about it. 01:04:04.050 --> 01:04:04.830 align:middle line:90% Yeah? 01:04:04.830 --> 01:04:08.340 align:middle line:84% AUDIENCE: What's the equivalent spectral statement? 01:04:08.340 --> 01:04:11.170 align:middle line:84% LAWRENCE GUTH: What is the equivalent spectral statement? 01:04:11.170 --> 01:04:19.480 align:middle line:90% 01:04:19.480 --> 01:04:20.962 align:middle line:90% So if you take-- 01:04:20.962 --> 01:04:22.670 align:middle line:84% so I'm not sure if this is your question, 01:04:22.670 --> 01:04:24.720 align:middle line:84% but let me say something, and then-- 01:04:24.720 --> 01:04:28.010 align:middle line:84% so if we take another set of generators here, 01:04:28.010 --> 01:04:30.340 align:middle line:84% we have a different operator, and we can 01:04:30.340 --> 01:04:32.510 align:middle line:90% ask what is sigma 1 and so on. 01:04:32.510 --> 01:04:34.930 align:middle line:84% But so I think what you were wondering 01:04:34.930 --> 01:04:37.820 align:middle line:84% is if we took another set of generators there, 01:04:37.820 --> 01:04:39.940 align:middle line:84% would that be related to the spectrum 01:04:39.940 --> 01:04:41.800 align:middle line:90% of the Laplacian of something? 01:04:41.800 --> 01:04:42.760 align:middle line:90% Is that your question? 01:04:42.760 --> 01:04:43.260 align:middle line:90% OK. 01:04:43.260 --> 01:04:49.480 align:middle line:90% 01:04:49.480 --> 01:04:53.110 align:middle line:90% OK, so not as far as I know. 01:04:53.110 --> 01:04:55.990 align:middle line:84% And so you might say this theorem is great 01:04:55.990 --> 01:04:58.490 align:middle line:84% because it connects to this hyperbolic geometry, 01:04:58.490 --> 01:05:00.730 align:middle line:84% and it tells us really interesting estimate 01:05:00.730 --> 01:05:02.500 align:middle line:84% for the spectrum of some surface. 01:05:02.500 --> 01:05:06.730 align:middle line:84% With other generators, maybe it doesn't exactly do that. 01:05:06.730 --> 01:05:10.270 align:middle line:84% But on the other hand, it says some interesting stuff 01:05:10.270 --> 01:05:12.540 align:middle line:84% about this group, which is interesting 01:05:12.540 --> 01:05:13.505 align:middle line:90% in some other problems. 01:05:13.505 --> 01:05:20.932 align:middle line:90% 01:05:20.932 --> 01:05:21.890 align:middle line:90% Yeah, let's leave that. 01:05:21.890 --> 01:05:53.760 align:middle line:90% 01:05:53.760 --> 01:05:57.490 align:middle line:84% All right, so suppose we're just given any set A inside of SL2Fp, 01:05:57.490 --> 01:06:00.690 align:middle line:84% and we'd like to figure out if there's a spectral gap. 01:06:00.690 --> 01:06:02.550 align:middle line:84% One thing that we know could go wrong 01:06:02.550 --> 01:06:05.730 align:middle line:84% is that A could be contained in a subgroup. 01:06:05.730 --> 01:06:13.100 align:middle line:84% So if A is contained in H, which is a proper subgroup, 01:06:13.100 --> 01:06:17.290 align:middle line:90% then sigma 1 of TA is 1. 01:06:17.290 --> 01:06:21.380 align:middle line:90% 01:06:21.380 --> 01:06:25.410 align:middle line:84% All right, so that's elementary to see. 01:06:25.410 --> 01:06:27.290 align:middle line:84% But in order to really use this, we 01:06:27.290 --> 01:06:29.750 align:middle line:84% would actually-- it helps to know what 01:06:29.750 --> 01:06:32.150 align:middle line:90% are all the proper subgroups. 01:06:32.150 --> 01:06:43.580 align:middle line:84% So let's talk a little bit about the classification of subgroups 01:06:43.580 --> 01:06:44.780 align:middle line:90% of SL2Fp. 01:06:44.780 --> 01:06:53.100 align:middle line:90% 01:06:53.100 --> 01:06:56.960 align:middle line:84% So the classification was done by someone 01:06:56.960 --> 01:06:59.190 align:middle line:90% named Dickson in 1901. 01:06:59.190 --> 01:07:04.790 align:middle line:84% And it's not that hard, but it's not that easy either. 01:07:04.790 --> 01:07:07.850 align:middle line:84% And I was a little surprised to learn 01:07:07.850 --> 01:07:12.490 align:middle line:84% that if you'd like to classify the subgroups of SLDFp, 01:07:12.490 --> 01:07:14.830 align:middle line:90% that was only done-- 01:07:14.830 --> 01:07:16.670 align:middle line:84% first of all, it's only done approximately. 01:07:16.670 --> 01:07:20.020 align:middle line:84% And it was only done around the year 2000. 01:07:20.020 --> 01:07:24.160 align:middle line:84% So some of these problems actually are really pretty hard. 01:07:24.160 --> 01:07:31.810 align:middle line:90% So it's done by Dixon in 1901. 01:07:31.810 --> 01:07:34.390 align:middle line:84% OK, well, let's think of some subgroups. 01:07:34.390 --> 01:07:37.390 align:middle line:84% We could have the diagonal subgroup, which is typically 01:07:37.390 --> 01:07:40.060 align:middle line:90% called T for torus. 01:07:40.060 --> 01:07:41.080 align:middle line:90% There's this guy. 01:07:41.080 --> 01:07:48.460 align:middle line:84% We have the upper triangular, unipotent subgroup, like that. 01:07:48.460 --> 01:07:50.950 align:middle line:90% Let me put a little 0 here. 01:07:50.950 --> 01:07:54.610 align:middle line:84% And we have the more general upper triangular 01:07:54.610 --> 01:08:00.070 align:middle line:90% guys, which have that. 01:08:00.070 --> 01:08:03.370 align:middle line:84% Now, if you take a subgroup, you can conjugate it, 01:08:03.370 --> 01:08:04.988 align:middle line:84% and you can get another subgroup. 01:08:04.988 --> 01:08:06.405 align:middle line:84% So you can conjugate all of these. 01:08:06.405 --> 01:08:13.800 align:middle line:90% 01:08:13.800 --> 01:08:18.215 align:middle line:84% And so let me write B, that means a conjugate of B0. 01:08:18.215 --> 01:08:23.170 align:middle line:90% 01:08:23.170 --> 01:08:24.743 align:middle line:84% So those are a bunch of subgroups. 01:08:24.743 --> 01:08:26.160 align:middle line:84% Those are the only ones that I was 01:08:26.160 --> 01:08:30.090 align:middle line:90% able to think of without effort. 01:08:30.090 --> 01:08:35.830 align:middle line:84% There are a few other small subgroups, like for instance, 01:08:35.830 --> 01:08:40.979 align:middle line:84% apparently the icosahedral group appears in SL2Fp for some p's. 01:08:40.979 --> 01:08:43.050 align:middle line:84% And the statement of the classification 01:08:43.050 --> 01:08:46.569 align:middle line:84% is a lot easier if we don't say exactly what groups appear, 01:08:46.569 --> 01:08:50.670 align:middle line:84% but just kind of roughly and ignore small groups. 01:08:50.670 --> 01:08:53.800 align:middle line:90% So there's a theorem about this. 01:08:53.800 --> 01:09:05.240 align:middle line:84% The rough theorem would say if H in SL2Fp is a proper subgroup, 01:09:05.240 --> 01:09:13.580 align:middle line:84% then there exists a Borel subgroup B, one of these guys, 01:09:13.580 --> 01:09:22.189 align:middle line:84% and B intersect H is a large fraction of H. So this c 01:09:22.189 --> 01:09:24.295 align:middle line:84% bigger than 0 is a universal constant. 01:09:24.295 --> 01:09:27.470 align:middle line:90% 01:09:27.470 --> 01:09:31.160 align:middle line:84% So more or less, H is not quite in B, but almost all. 01:09:31.160 --> 01:09:35.359 align:middle line:84% H has a very large part that's in B. 01:09:35.359 --> 01:09:37.529 align:middle line:90% And B is a conjugate of B0. 01:09:37.529 --> 01:09:42.800 align:middle line:84% And the fine print is conjugate of B0 conjugated 01:09:42.800 --> 01:09:46.920 align:middle line:84% by not necessarily someone in SL2Fp, 01:09:46.920 --> 01:09:50.420 align:middle line:84% but maybe the matrix we conjugate by 01:09:50.420 --> 01:09:52.680 align:middle line:84% might involve the algebraic closure. 01:09:52.680 --> 01:09:53.430 align:middle line:90% Anyway, it's fine. 01:09:53.430 --> 01:10:01.160 align:middle line:90% 01:10:01.160 --> 01:10:03.710 align:middle line:90% Now, let's make a remark. 01:10:03.710 --> 01:10:06.363 align:middle line:90% The analog of this is not true. 01:10:06.363 --> 01:10:09.670 align:middle line:90% 01:10:09.670 --> 01:10:18.430 align:middle line:84% For SL2Fq, q is a prime power because SL2Fp 01:10:18.430 --> 01:10:21.820 align:middle line:90% is a subgroup of SL2Fq. 01:10:21.820 --> 01:10:25.675 align:middle line:84% but SL2Fp is not contained in any upper triangular 01:10:25.675 --> 01:10:26.570 align:middle line:90% or something. 01:10:26.570 --> 01:10:30.760 align:middle line:90% 01:10:30.760 --> 01:10:34.100 align:middle line:84% So there's an analogous theorem for SL2Fq, 01:10:34.100 --> 01:10:36.610 align:middle line:84% but it needs to be stated a little bit different way to take 01:10:36.610 --> 01:10:38.770 align:middle line:90% account of the subfields. 01:10:38.770 --> 01:10:41.430 align:middle line:84% So this may already start to remind us of something 01:10:41.430 --> 01:10:43.180 align:middle line:84% that we've seen, that projection theory is 01:10:43.180 --> 01:10:44.410 align:middle line:90% different over Fp and Fq. 01:10:44.410 --> 01:10:58.180 align:middle line:90% 01:10:58.180 --> 01:11:02.100 align:middle line:84% OK, so now, I can state the second main theorem 01:11:02.100 --> 01:11:03.840 align:middle line:84% that I wanted to say about random walks 01:11:03.840 --> 01:11:08.160 align:middle line:84% on groups, which is a generalization of Selberg's 01:11:08.160 --> 01:11:08.890 align:middle line:90% theorem. 01:11:08.890 --> 01:11:12.270 align:middle line:84% And it allows to take any set of generators 01:11:12.270 --> 01:11:16.950 align:middle line:84% where it's not trapped in a Borel subgroup, 01:11:16.950 --> 01:11:19.210 align:middle line:84% but it requires a little quantification, 01:11:19.210 --> 01:11:21.510 align:middle line:84% which is perhaps not quite as strong 01:11:21.510 --> 01:11:23.360 align:middle line:90% as one might fantasize about. 01:11:23.360 --> 01:11:30.510 align:middle line:90% 01:11:30.510 --> 01:11:34.920 align:middle line:84% All right, so for every epsilon bigger than 0, there 01:11:34.920 --> 01:11:46.620 align:middle line:84% are a couple of positive constants every p. 01:11:46.620 --> 01:11:53.850 align:middle line:84% So if I have a subset of SL2Fp, and the subset is not 01:11:53.850 --> 01:11:55.920 align:middle line:84% contained in a Borel subgroup, and also it 01:11:55.920 --> 01:11:59.010 align:middle line:84% doesn't concentrate too much in a Borel subgroup, 01:11:59.010 --> 01:12:06.980 align:middle line:84% so for every B Borel, if I take A intersected with B, 01:12:06.980 --> 01:12:14.990 align:middle line:84% this is, at most, p to the minus epsilon size of B 01:12:14.990 --> 01:12:17.330 align:middle line:90% and with this small constant. 01:12:17.330 --> 01:12:23.390 align:middle line:84% So a smallish-- sorry, size of A. So only a small fraction of A 01:12:23.390 --> 01:12:26.210 align:middle line:90% is in this Borel subgroup. 01:12:26.210 --> 01:12:29.220 align:middle line:84% And it's a p to the epsilon kind of fraction where you-- 01:12:29.220 --> 01:12:30.540 align:middle line:90% that's a pretty small fraction. 01:12:30.540 --> 01:12:32.082 align:middle line:84% You might hope to do a little better, 01:12:32.082 --> 01:12:35.030 align:middle line:90% but anyway, that's what it says. 01:12:35.030 --> 01:12:39.170 align:middle line:90% And here, I wasn't positive-- 01:12:39.170 --> 01:12:42.090 align:middle line:84% so when I proved it, I needed to say coset, 01:12:42.090 --> 01:12:45.030 align:middle line:84% but I'm not sure if it's coset or subgroup, anyway. 01:12:45.030 --> 01:12:47.420 align:middle line:84% So if it doesn't concentrate in something closely related 01:12:47.420 --> 01:12:50.430 align:middle line:90% to a subgroup, then it expands. 01:12:50.430 --> 01:12:54.050 align:middle line:84% So then it has a mixed signal 1 of TA is less than 1 01:12:54.050 --> 01:12:55.080 align:middle line:90% minus c of epsilon. 01:12:55.080 --> 01:13:07.170 align:middle line:90% 01:13:07.170 --> 01:13:12.870 align:middle line:84% So up to a little bit of quantification, 01:13:12.870 --> 01:13:15.340 align:middle line:84% if A is contained in a Borel subgroup, 01:13:15.340 --> 01:13:17.620 align:middle line:84% definitely we won't have a spectral gap. 01:13:17.620 --> 01:13:22.080 align:middle line:84% And if A doesn't intersect a Borel subgroup too much, 01:13:22.080 --> 01:13:24.010 align:middle line:84% then we will have a spectral gap. 01:13:24.010 --> 01:13:26.550 align:middle line:84% It's almost if and only if, although you might hope 01:13:26.550 --> 01:13:27.785 align:middle line:90% to improve this a little bit. 01:13:27.785 --> 01:13:32.094 align:middle line:90% 01:13:32.094 --> 01:13:35.710 align:middle line:84% So this is kind of its own story to talk about. 01:13:35.710 --> 01:13:38.520 align:middle line:84% So since we're almost done, let me just 01:13:38.520 --> 01:13:41.175 align:middle line:84% make a couple of high level comments about it. 01:13:41.175 --> 01:13:41.675 align:middle line:90% Yeah? 01:13:41.675 --> 01:13:43.800 align:middle line:84% AUDIENCE: What's a Borel subgroup? 01:13:43.800 --> 01:13:44.880 align:middle line:90% LAWRENCE GUTH: A Borel-- 01:13:44.880 --> 01:13:47.640 align:middle line:84% so I don't know why-- so Borel is involved in this. 01:13:47.640 --> 01:13:50.670 align:middle line:84% A Borel subgroup is these guys B. 01:13:50.670 --> 01:13:54.270 align:middle line:84% Take an upper triangular matrix, that's a Borel subgroup, 01:13:54.270 --> 01:13:55.370 align:middle line:90% or conjugate that. 01:13:55.370 --> 01:13:58.130 align:middle line:90% 01:13:58.130 --> 01:14:01.460 align:middle line:84% So one comment about this theorem 01:14:01.460 --> 01:14:05.455 align:middle line:84% is that we had better know this theorem. 01:14:05.455 --> 01:14:06.955 align:middle line:84% You better know something like this. 01:14:06.955 --> 01:14:09.740 align:middle line:90% 01:14:09.740 --> 01:14:13.490 align:middle line:84% And so the first part of this theorem 01:14:13.490 --> 01:14:17.870 align:middle line:84% is to prove that if you don't have a spectral gap, 01:14:17.870 --> 01:14:20.780 align:middle line:84% then there must be a subset that's 01:14:20.780 --> 01:14:26.705 align:middle line:84% kind of approximately a subgroup that intersects A a lot. 01:14:26.705 --> 01:14:28.080 align:middle line:84% If you don't have a spectral gap, 01:14:28.080 --> 01:14:31.610 align:middle line:84% there must be a subset that's kind of approximately a subgroup 01:14:31.610 --> 01:14:34.610 align:middle line:84% where maybe A convolved with itself many times 01:14:34.610 --> 01:14:39.380 align:middle line:84% is heavily concentrated in this approximate subgroup. 01:14:39.380 --> 01:14:42.480 align:middle line:90% And that's a very nice argument. 01:14:42.480 --> 01:14:44.540 align:middle line:84% And it turns out to use the tools 01:14:44.540 --> 01:14:45.960 align:middle line:90% from additive combinatorics. 01:14:45.960 --> 01:14:47.720 align:middle line:84% It uses like Balog-Szemeredi-Gowers 01:14:47.720 --> 01:14:52.790 align:middle line:84% and things like that in a non-commutative version. 01:14:52.790 --> 01:14:55.880 align:middle line:84% And then the second part of the proof is to say, 01:14:55.880 --> 01:14:58.550 align:middle line:84% is kind of a robust version of this theorem. 01:14:58.550 --> 01:15:01.043 align:middle line:84% This is a classification of actual subgroups. 01:15:01.043 --> 01:15:02.710 align:middle line:84% We want to make it a little bit stronger 01:15:02.710 --> 01:15:06.550 align:middle line:84% and have a classification of approximate subgroups. 01:15:06.550 --> 01:15:09.580 align:middle line:84% So not only any actual proper subgroup 01:15:09.580 --> 01:15:12.410 align:middle line:84% would have to intersect a Borel subgroup a lot, 01:15:12.410 --> 01:15:15.590 align:middle line:84% but actually, any kind of approximate subgroup, 01:15:15.590 --> 01:15:18.340 align:middle line:84% so any subset where when you multiply it by itself two 01:15:18.340 --> 01:15:21.610 align:middle line:84% or three times, it doesn't get very much bigger, 01:15:21.610 --> 01:15:26.410 align:middle line:84% the only way that can happen is if your subset heavily 01:15:26.410 --> 01:15:29.590 align:middle line:90% intersects a Borel subgroup. 01:15:29.590 --> 01:15:36.130 align:middle line:84% Now, that second part, you'll notice, is not true over Fp-- 01:15:36.130 --> 01:15:39.670 align:middle line:90% sorry, it's not true over Fq. 01:15:39.670 --> 01:15:43.600 align:middle line:84% And so the second part will draw on some product theory 01:15:43.600 --> 01:15:45.970 align:middle line:84% and our theorems that we worked very 01:15:45.970 --> 01:15:50.770 align:middle line:84% hard on that give a projection estimate over Fp that's 01:15:50.770 --> 01:15:52.780 align:middle line:90% not true over Fq. 01:15:52.780 --> 01:15:53.645 align:middle line:90% We will need that. 01:15:53.645 --> 01:15:56.715 align:middle line:90% 01:15:56.715 --> 01:16:00.540 align:middle line:84% And so I think this is a nice example of how 01:16:00.540 --> 01:16:03.150 align:middle line:84% those theorems are only an epsilon 01:16:03.150 --> 01:16:04.890 align:middle line:84% improvement on a trivial estimate where 01:16:04.890 --> 01:16:06.630 align:middle line:90% epsilon is very small. 01:16:06.630 --> 01:16:08.752 align:middle line:90% But it still is meaningful. 01:16:08.752 --> 01:16:10.710 align:middle line:84% There's an important difference between epsilon 01:16:10.710 --> 01:16:12.450 align:middle line:90% being zero or not zero. 01:16:12.450 --> 01:16:17.400 align:middle line:84% And they're used to prove kind of this c of epsilon 01:16:17.400 --> 01:16:18.790 align:middle line:90% is pretty small. 01:16:18.790 --> 01:16:24.030 align:middle line:84% But there's just a really important difference 01:16:24.030 --> 01:16:25.570 align:middle line:84% between whether you had zero here, 01:16:25.570 --> 01:16:27.640 align:middle line:84% which is like a completely vacuous statement, 01:16:27.640 --> 01:16:30.070 align:middle line:84% and whether you had 1 minus epsilon, 01:16:30.070 --> 01:16:32.470 align:middle line:84% which even if epsilon is a small constant, 01:16:32.470 --> 01:16:37.230 align:middle line:84% is a qualitatively sharp statement about how 01:16:37.230 --> 01:16:38.435 align:middle line:90% these random walk mixes. 01:16:38.435 --> 01:16:41.580 align:middle line:90% 01:16:41.580 --> 01:16:42.550 align:middle line:90% OK, cool. 01:16:42.550 --> 01:16:46.310 align:middle line:84% So we'll talk more about those things on Thursday. 01:16:46.310 --> 01:16:53.000 align:middle line:90%