 We piloted this year was this Park City Experimental Math Lab, which is a long running thing that our steering committee member, Jai Devathriya, has carried out in a semester long format in Champaign Urbana and the University of Washington. And we thought we'd give it a try here, focused on people from the undergraduate program, but really ultimately to try to perhaps also get people involved together from various groups. So this is our first year, our trial run. And I have to say that having been involved in this through the three weeks, it came out even better than any of us could have expected. I was extremely pleased, and I think it's been a big success. But you can judge for yourself after the next hour. So the format of what we're going to do is we have seven groups. Seven groups? Seven groups. And so they each have how many minutes? Seven minutes. So we're going to be out of here in 49 minutes. So I think that what we're going to do is I'm going to introduce each group. And I will tell you the list of people in the group and the mentor for the group. And I think that we'll maybe have time for just a short question or two, but we're not going to have time for extensive questions at the end of each thing. But everybody's going to be around, and I'm sure that they'd love to answer questions about what they've done. These are little brief snippets into what they've done. You can only say so much in seven minutes, though some people I know are packing in a lot of words. Anyway, so we'll see how it goes. And so the format is going to be the each group is going to sit here. So everybody from each group comes up and sits. And then most of these presentations are going to be each person doing a couple of minutes and then rotating a microphone around. So the first group is about Benford's law, as the title says here. So the advisor mentor on this was Jay Devathraya himself. And the presenters are, so Michael Ouijaoua. Did I say that right? Ouijaoua, excuse me, I'm so sorry. Ann Palletti, Aziz Jumash, Liz McGrath, Ann Carlstein, Rebecca Czarnitowski. How'd I do? And Jaime Hernandez. Good. OK, take it away. Hi, everybody. Can you hear me? So my group examined various sequences to try to decide if they satisfy Benford's law. So I figured I'll start with what Benford's law is, and then I'm going to pass to my team members to talk about what we did. So closer. OK, so let's start with a sequence of powers of two. The catch is that I'm only interested in the first digits. So after 1, 2, 4, 8, and 16, instead of 16, I'm going to keep just the 1. Instead of 32, I'm just going to keep the first digit of 3. And so on. So that's my sequence. So the question would be, what is the frequency of the digits? If I start computing the first 10,000 digits, how many of those, what proportion of those would be digit 2, which one would be digit 7, and so on? So the first time I saw this problem many years ago. So my first thought was, shouldn't it be equally distributed? Shouldn't the digit 1 show up about 1 ninth of the time? Well, I was wrong. So if you look at the first 10,000 powers of 2, roughly 30% of those would be digit 1, starts with digit 1, and about 70% will start with digit 2. And it keeps going on until you get to digit 9, which is pretty small. And here I drew a curve to, I superimposed the curve to suggest that there's some law governing the distribution. And oh, by the way, this is the example of a sequence that satisfies Benford's law. So if you look at the first digit of the sequence satisfy Benford's law, if when you look at the first digits, the first digit 1 shows up about 30% of the time, digit 2, 70% of the time, and so on. So how can we prove that a sequence satisfies Benford's law? So of course, we need a mathematical way to extract the first digit. So there's a lot of stuff here, but just focus on the one in blue. We are gonna take the log base 10 of this number, and then we take the fractional part. I'm gonna show you why this works. So look at this table. I wrote down the numbers that start with digit 1, and they are of varying magnitude. So I have one digit up to millions. If you look at the log base 10, the integer part is just going to measure the number of digits. So I don't really need that. So I'm gonna take the fractional part. The key thing is when I plot it on the interval between 0 and 1, all of them are concentrated on the interval between 0 and 0.3. So if I do the same thing for digits, the ones that start with digit 2, they are all concentrated between 0.3 and about 0.47. I can keep going, right? If I keep going, the ones that start with digit 9 is near the end, and the interval is pretty small, okay? And you can start to see that why I have 30% that start with digit 1 in the powers of 2, right? And 17% for digit 2, okay? Except that I haven't really used the fact that I'm studying powers of 2, okay? So why does the powers of 2 come in? So I need to ensure that the fractional part of the log is going to be equally distributed, right? Just think about it. If I take the fractional part of the log and they're all bunched together near the end, then I'm gonna get lots of digits 8s and 9s, right? But that clearly doesn't happen. So what's going on is there's some equidistribution of the fractional part of the log. And what this means is if you look at interval of length 0.4, that's about 40% of the unit interval. About 40% of the points should drop in that interval, okay? And so thankfully there's a while's equidistribution theorem which says that if you have an irrational number, you take the fractional part, sorry, if you take the irrational number, take the multiples, integer multiples, and you take the fractional part, then you'll be equally distributed on the unit interval. And thankfully log base 10 of 2 is such a number. It's an irrational number. So that's why on an interval of length 0.3, you expect about 30% of those to fall in there. That's why Benford's law hold in that case. So I have a more exact formula for the length of the interval, but now I'm gonna pass to Ann. Okay, so one thing that's interesting about Benford's law is we saw that it, oh, wait, is this better? Okay, so one of the interesting things that we saw with Benford's law is that, so in addition to having these leading digit frequencies for something like two to the n, or other numbers to the nth power, it works for a lot of other leading digit frequencies for different sequences of numbers. This is including Fibonacci numbers or some power sequences, et cetera. But in addition, it also holds for a lot of real number, I mean real world data sets. So for example, if you look at the leading digit frequencies of the populations of various countries around the world, if you look at GDPs of different countries, if you look at the number of print materials in US libraries, if you look at financial records, it also holds. So one thing that we decided to check is a real world data set for ourselves. So looking at some of the most recent Sotheby's auctions, we looked at the results of how much everything the auction went for, and we put it into this graph. And as you can see, the blue bars show what actually happened in the orange bars show the predicted leading digit frequencies based on Benford's law. And as you can see, it's pretty similar. So that was interesting to see. Another thing that was interesting to consider is, okay, so we know the frequencies of the leading first digits. So what about if we look at the first two digits? So what we did is we looked at, we looked at sequences that we know satisfied Benford's law, like two to the N, Fibonacci numbers, Fibonacci numbers squared, X to the fourth, et cetera, and looked to see, okay, so how often does 10 appear as the leading two digits, how often does 11, how often does 12, et cetera, all the way up to 99. And as you can see here in this graph, the results are very similar to each other. And it's also pretty reminiscent of the logarithmic curve that we saw before. So in fact, the blue line shows what we know is true from Benford's law on single digits and the orange line shows what we found for two digits. So in fact, this pattern continues and Benford's law, the same equation that we saw earlier on, holds for two digits. So next we looked at polynomial sequences and one of the tools we used was Taylor expansions. And fun thing about Taylor expansions is we can take sequences that are form A to the N, which we have already shown follow Benford's law and make polynomial for approximations. And so if we look at sequences of this form, we should expect them to follow Benford's, which if you look at this graph, as we take the degree of the polynomial up of this form, we start to approximate Benford's law more. So overall, what we learned was basically that some sets do follow Benford's law, some don't. For example, powers of different bases seem to follow them, but some examples that don't follow are really low degree polynomials like degree one, or the appearance of letters in the English language where we represent letters A equals one, B equals two and so forth.